title: Indicators on how even the heat is distributed?

text: I'm wondering if there are any good indicators on how even the heat is distributed on an object (for simplicity, a flat object maybe)? What are the possibly reasonable ways to maximize the evenness if varying the shape of the object is allowed.


title: How does a spinning electron produce a magnetic field?

text: I learned in my undergraduate physics class that atoms have magnetic fields produced by the orbit of electrons and the spin of electrons. I understand how an orbit can induce a magnetic field because a charge moving in a circle is the same as a loop of current. What I do not understand is how a spinning ball of charge can produce a magnetic field. Can someone explain how spin works, preferably in a way I can understand?


title: Difference in timbre between 'quiet' and 'far away'

text: I'd like to know what are the differences in timbre - or the acoustic properties of a sound - that allow us to differentiate between a sound which is quiet (but close-by) and one which is far away. For example, you can tell when someone near to you is playing an instrument quietly even without looking to see where they are - they don't sound 'far away'. Hearing a loud gig or a car stereo playing from the next street doesn't sound like it's quiet - it sounds loud, but far away. But other times we can't differentiate - I sometimes hear a siren on TV and think it's on the street! I thought only the amplitude (i.e. volume) of a sound wave diminished with distance - does the shape/frequency alter too? Is this ability just to do with having two ears to locate the source - surely someone who is deaf in one ear can still tell an orchestra is playing a _diminuendo_ and not gradually getting further away?!


title: What are the physycal meaning of universal constants such as the magnetic, electric and gravitational constants?

text: I know that: $G=6.67300 × 10^{-11}\dfrac{Nm^2}{kg^2}$(Gravitational constant) $K_e=9×10^9\dfrac{Nm^2}{C^2}$(Electric constant or Coloumb's constant) $k_m=1×10^{-7}\dfrac{Ns^2}{C^2}$(Magnetic constant) * * * But what are the meaning of this values? Would the universe be much different with a little change on this values? What if we hadn't known them?


title: Question about entangled states

text: I have a question about entangled state. Suppose I consider the entangled state $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. I saw an argument for how measurement of the first bit is affected by whether or not the second bit is measured because this is an entangled state in an article I am reading. The argument is clear to me and I describe it below, so please keep reading: If I measure the first bit without first measuring the 2nd bit, then the probability that I get $|0\rangle$ is 1/2. This is clear to me. If I measure the 1st bit after measuring the 2nd bit, the probability that it is $|0\rangle$ is 0 or 1 depending on whether the measured value of the 2nd bit was $|0\rangle$ or $|1\rangle$. This is clear to me too. But what I am confused about is this: If I measure the first bit after the second bit, this is what I would calculate the probability of measuring the first bit being $|0\rangle$ to be: $\frac{1}{2}\times 1 + \frac{1}{2}\times 0 = \frac{1}{2}$. The first term is the prob. of measuring the second bit as $|0\rangle$ and then the 1st bit as 0 and the 2nd term is the prob of measuring the 2nd bit as 1 and then measuring the 1st bit as zero. If the probability of measuring the first bit as 0 is to same regardless of whether the 2nd bit was measured or not is unchanged, what is the effect of the entangled state? Or am I missing something?


title: Is spacetime an illusion?

text: In consistent histories, for gauge theories, can the projection operators used in the chains be not gauge invariant? In quantum gravity, for a projection operator to be gauge invariant means it has to be diffeomorphism invariant. There are no localized diffeomorphism invariant operators, not even localized in time. No localized observables. So, what are the permissible chains in quantum gravity? Do they have to be delocalized, or localized on the boundaries of spacetime? Are space and time illusions, as Andrew Strominger claimed? A Cosmic Hologram projecting out the illusion that is spacetime from the future boundary of time?


title: Thermal physics question, final temperature of ice and water

text: Here is the question from my textbook: "In an insulated vessel, 250g of ice at $0^{\circ}$C is added to 600g of water at $18^{\circ}$C. What is the final temperature of the system?" The solution says that it takes more heat to melt ice than to cool water, so some ice will remain. It says the final temperature should be $0^{\circ}$C, but no equation is given to explain that conclusion. Can the final temperature of this system be determined by reasoning alone?


title: Can Jupiter be ignited?

text: Our solar system itself contains two candidate "Earths" One is Jupiter's moon Europa and another is Saturn's moon Titan. Both of them have the problem of having at low temperature as Sun's heat cannot reach them. Jupiter is made up mostly of Hydrogen so is our Sun. Can we ignite Jupiter so that it will produce enough heat to warm these two earth like planets/satellites? What can be consequences of any of such attempt?


title: How is it possible the speed of light is not constant?

text: I was reading this article recently, which summarizes a couple of new studies into the speed of light. > In one paper, Marcel Urban from the University of Paris-Sud, located in > Orsay, France and his colleagues identified a quantum level mechanism for > interpreting vacuum as being filled with pairs of virtual particles with > fluctuating energy values. > > As a result, the inherent characteristics of vacuum, like the speed of > light, may not be a constant after all, but fluctuate. > > Meanwhile, in another study, Gerd Leuchs and Luis L. Sánchez-Soto, from the > Max Planck Institute for the Physics of Light in Erlangen, Germany, suggest > that physical constants, such as the speed of light and the so-called > impedance of free space, are indications of the total number of elementary > particles in nature. Now, my knowledge and understanding of electromagnetic waves is a bit rusty, but I guess I don't follow: **how does fluctuating energy values affect the speed of light?**


title: Space time curvature real or theoretical (mathematical)?

text: Assuming one were in a capsule of some kind, with no window or instruments, and you swung into the gravitational field of a massive object (planet). Assuming no atmosphere to provide friction, could you tell you were curving...there wouldn't be like centrifugal force, would there? Would you even feel gravity until you slammed into the planet's surface? Addendum: Is gravitational lensing proof of space time curvature?


title: Comparing interaction potential in standard $ϕ^4 $theory

text: I am posting this question again because, Willie Wong asked me to do it. So it is a continuing post of the Interaction potential in standard ϕ4 theory. I have been studying about solitions so I had to deal with scalar field theory. The problem I faced in Lagrangian of Scalar field is interacting potential. According to Scalar field theory we can write: $$\mathcal{L}=\frac{1}{2}\partial^\mu \phi \partial_\mu \phi -\frac{m^2}{2}\phi^2 -\frac{\lambda}{4!}\phi^4 \tag {1}$$ The potential can be written separately $$ V(\phi)= \frac{m^2}{2}\phi^2 +\frac{\lambda}{24}\phi^4 \tag {2}$$ I found on Srednicki (Quantum Field Theory, page 188) that, the author wrote the potential as, $$ V(\phi)= \frac{1}{24} \lambda (\phi^2- v^2 )^2-\frac{\lambda}{24} v^4 \tag {3}$$ After that the author excluded the term $-\frac{\lambda}{24} v^4 $. **why is that?** In another paper an author wrote the potential as $$V(\phi)= \frac{1}{8} \phi^2 (\phi -2)^2 \tag{4}$$   **I don't see coupling constant $\lambda$ in the equation (4).** **What I'm trying to find is to get the potential in equation (4) from the equation (2)**


title: Near-Earth supernova

text: There are 51 stars within 17 light years of the Earth (source). If one of these stars was to become a supernova, how would they effect the Earth? I have read the Wikipedia article _Near-Earth supernova_ , but what would the explanation be in more layman's terms? What would happen if one of the 51 stars above became a supernova?


title: Correlated three-particle Green Function

text: I know the relationship between normal and correlated two-particle Green Functions for fermions: $$G_c(1,2,3,4)=\Gamma(1,2,3,4)=G(1,2,3,4)+G(1,3)G(2,4)-G(1,4)G(2,3)$$ Also known as irreducible n-particle Green function or n-particle vertex (there are many definitions so I'm confused). Definitions for G also vary, I use this one: $G(1..2n) = (-1)^{n}\langle T\ c_1\dots c_n c_{n+1}^\dagger\dots c_{2n}^\dagger \rangle$ I need a similar relationship for three-particle functions. Now I have this formula: $$ \Gamma(1,2,3,4,5,6) = G(1, 2, 3, 4, 5, 6) -2 G(1, 4) G(2, 5) G(3, 6) +2 G(1, 4) G(2, 6) G(3, 5) -2 G(1, 5) G(2, 6) G(3, 4) +2 G(1, 5) G(2, 4) G(3, 6) -2 G(1, 6) G(2, 4) G(3, 5) +2 G(1, 6) G(2, 5) G(3, 4) -G(1, 4) G(2, 3, 5, 6) +G(1, 5) G(2, 3, 4, 6) -G(1, 6) G(2, 3, 4, 5) +G(2, 4) G(1, 3, 5, 6) -G(2, 5) G(1, 3, 4, 6) +G(2, 6) G(1, 3, 4, 5) -G(3, 4) G(1, 2, 5, 6) +G(3, 5) G(1, 2, 4, 6) -G(3, 6) G(1, 2, 4, 5)$$ But it doesn't always satisfy Wick's theorem for the Hamiltonian I'm working with. Is this formula correct? Also it would be great to get an explanation of how these formulas appear in many-body theory, from logarithm of time-ordered exponential.


title: Are we capable of discovering planets in the Andromeda galaxy?

text: I just watched this SpaceRip video on YouTube which shows pictures taken by Hubble while looking into the disk of the Andromeda galaxy to study a certain type of variable star. It occurred to me that if we can study individual stars in Andromeda, we must be able to use most of our methods of exoplanet detection on them. Is this true? If so, why not look for exoplanets _in another galaxy_? My logic tells me that since it's difficult to look through the disk of our own galaxy (since we're inside it), we're limited to a relatively local region for detecting exoplanets. But if we turned our sights to Andromeda, couldn't we study the differences in planet formation between different regions of a galaxy? I also suspect that the reason we're not doing this is because we would need _much_ more sensitive instruments for this, and because we already have one planet finding telescope in space (so it'd be tough to justify sending up another one). Am I mostly right in this reasoning? Are we capable of searching for exoplanets in Andromeda? Would there be a benefit to doing it (besides the obvious cool factor)? Is cost pretty much the only limiting factor?


title: The speed of light, objects moving against each other and the effects of a clash

text: Physics is a hobby interest of mine, so I may be asking a senseless question, but there is something I cannot get my head around when it comes to relative frames, the **constant** speed of light (as a top-limit for any speed) and the following situation: Imagine two spaceships, capable of reaching speed close to $c$, flying in opposite directions and passing each other by. According to the special relativity, we will calculate the speeds this way: $$ V_{\rm total} = \frac{u+v}{1+\frac{uv}{c^2}} $$ where ${u}$ and $v$ would be the speeds of each spaceship, both close to $c$. According to this, I perfectly understand that each ship will "see" the other moving with near the speed of light, not faster. An observer in a separate reference frame will see each ship moving with a speed near that of the light too, and it is OK. Now, let's assume the ships crash. Would the effect of the crash be the same as if only one of them was moving, as opposed to both speeding against each other? I am thinking there would not be much difference, as if the first ship is stationary, it will still experience the other crashing at him with the speed near that of the light. The external observer would also confirm this. If this is so, would there be any reason to accelerate particles near the speed of light in opposite directions as it is being done in CERN? I assume (may be wrongly) that the above observation for the spaceships can be applied to particles also. So, would it be enough if the particles are just accelerated in one direction to "bomb" a stationary object, in order for them to get scattered? Am I missing something that would apply to particles but not to larger objects (like ships) or even something more-fundamental?


title: Why aren't there more natural satellites orbiting the Earth than just the Moon?

text: Considering that there's a lot of debris in space and that impacts fling out rocks into space all the time, why do we only have one large natural satellite - the Moon? Shouldn't there be all kinds of rocks in all shapes and sizes orbiting the Earth?


title: Is spectrum of Discrete-Time Fourier Transform (DTFT) periodic or not

text: I can't think of any better title. Here is the content that I got question http://cnx.org/content/m10247/2.31/ As it state the nature of DTFT's spectrum is periodic as it show in figure 1 ![enter image description here](http://i.stack.imgur.com/RHr9S.png) ![enter image description here](http://i.stack.imgur.com/892ct.png) However, in figure 3, spectrum is non-periodic ![enter image description here](http://i.stack.imgur.com/lO3dP.png) So, the spectrum of DTFT is periodic or not or it is dependent on each case of $s(t)$


title: Shadow of a Jovian moon over the Great Red Spot

text: Where can I find pictures of the shadow of any of the Jovian moons partially covering the Great Red Spot? A series of such pictures over time would even be better. The idea is to learn more about the structure of the storm by observing how the shadow behaves as it is passing over the boundaries and body of the Great Red Spot storm.


title: Are there Trojan family or Hilda family satellites locked in Earth's orbit?

text: Jupiter has many Trojan asteroids located at Lagrangian points L4 and L5 and Hilda asteroids dispersed between points L3, L4, and L5. Does the Earth have similar satellites? If so, how many?


title: "Consciousness causes collapse"-interpretation or free will excluded by quantum zeno effect?

text: This question is if the von Neumann–Wigner interpretation, aka "consciousness cause collapse" (see e.g. John von Neumann's 1932 book The Mathematical Foundations of Quantum Mechanics) is compatible with the quantum Zeno effect (see Sudarshan, E. C. G.; Misra, B. (1977) "The Zeno's paradox in quantum theory", Journal of Mathematical Physics 18 (4): 756–763) respectively its empirical observation (such as by Wineland 1989 or by Raizen 2002). Or if the Quantum Zeno effect falsies or created the possibility to empirically verify the von Neumann-Wigner interpretation. Assume measuring the state of a system every second causes a zeno effect and measuring every three seconds causes an anti-zeno effect. Now let's assume the state is measured every second and recorded. A proponent of the "consciousness causes collapse" interpretation would argue that the wave function only collapses when a conscious observer actually looks at the measurement results. But what if the conscious observer first looks at only every third result and decides only afterwards on whether to look at the other results too. Will she observe the zeno effect depending on her future decision? Wouldn't such an experiment show that either (1) the consciousness of the observer has no impact and there is a zeno effect even if two out of three measurement results are destroyed before a conscious observer looks at them or that (2) the observer has no free will and whenever she observes a zeno effect looking at every third measurement she can't help but looking at the other results too, and conversely if the first set of results she looks at shows an anti-zeno effect she has an irresistible urge to destroy the other results? Well the first option looks more plausible to me, but it seems like something that could be experimentally tested.


title: Capillary in a slanted tube with a bend

text: So I've read a question on here about Capillary action in a slanted tube, but my question goes a little further. Firstly though, could someone explain to me what is meant by the only answer to that question. The answer states that the lift force isn't dependant on the weight of the fluid whereas the capillary lift hight equation is a combination of the forces from gravity and capillary action. When I have tried to balance these equation for both a upright tube and a slanted tube I get different lift heights and masses of fluid lifted. I'll explain some more at the bottom of the question. If we have a slanted tube that slants 15 degree to the right for 10mm and then 15 degree to the left (from normal - not going back to upright)for 10mm would this tube have the same lift height as a tube that slanted 15 degrees for 20mm? (Assuming the tube radius is constant in and equal). Essentially is there any issue with the fluid going around the bend? As promised more explanation of what I've done thus far. A tube filled to a given height has the same volume irrespective of the angle the tub is at (see Volume of a slanted tube). So with this in mind and knowing that the side length of the cylinder up to the water level is longer than the height of the water (if the cylinder is slanted), we can comfortably say that to lift the same mass of water the height will be the same, but the length up the pipe will be lower. However the lifting force from capillary action is based on the circumference of the pipe (2*pi*r), but **this** changes dependant on angle since a slanted pipe will have a cross section that is a ellipse. So I expect the lifting force to be greater, allowing for a greater amount of fluid to be lifted, thereby increasing the height that the fluid is lifted to.


title: Calculating the ideal mixing entropy using Gibbs' entropy formula

text: Two distinguishable gases are in separate volumes $xV$ and $(1-x)V$ $(x\in [0,1])$ respectively, and the number of particles on each side is $xN$ and $(1-x)N$ respectively. The volumes are separated by some tap or something at first. If we then open the tap, the gases will mix, the ideal mixing entropy is given by $\Delta S=-Nk_B(x\ln x + (1-x)\ln(1-x))$. I'm trying to get this result using Gibbs' entropy formula $S=-k_B\sum_i P_i\ln P_i$. My first way is considering the case of 1 particle, such that we only have 2 states with probabilities $x$ and $1-x$, so then $S=-k_B(x\ln x + (1-x)\ln(1-x))$. Then using that entropy is extensive we get the answer. Correct? Now I'm trying to derive the entropy by using the binomial distribution: the chance of $n$ particles being in the volume $Vx$ is $P(n)=\binom{N}{n}x^n(1-x)^{N-n}$. Inserting this in Gibbs' formula yields: $S=-k_B\sum_nP(n)\ln\left(\binom{N}{n}x^n(1-x)^{N-n}\right)$, then expanding the log gives $S=-k_B\sum_nP(n)\left(\ln\left(\binom{N}{n}\right)+n\ln x +(N-n)\ln(1-x)\right)$, then using that $<n>=xN$ we get $S=-Nk_B(x\ln x + (1-x)\ln(1-x))-k_B\sum_nP(n)\ln\left(\binom{N}{n}\right)$. How do I get rid of this last term? Or is the problem that I calculated $S$ rather than $\Delta S$?


title: Can an object reverse its direction of acceleration even though it continues to move in the same direction?

text: Can anyone please explain me on this matter along with day to day examples?


title: Field between the plates of a parallel plate capacitor using Gauss's Law

text: Consider the following parallel plate capacitor made of two plates with equal area $A$ and equal surface charge density $\sigma$: ![enter image description here](http://i.stack.imgur.com/E3Mkn.png) The electric field due to the positive plate is $$\frac{\sigma}{\epsilon_0}$$ And the magnitude of the electric field due to the negative plate is the same. These fields will add in between the capacitor giving a net field of: $$2\frac{\sigma}{\epsilon_0}$$ If we try getting the resultant field using Gauss's Law, enclosing the plate in a Gaussian surface as shown, there is flux only through the face parallel to the positive plate and outside it (since the other face is in the conductor and the electric field skims all other faces). $$\Phi = \oint \vec{E}\cdot\vec{dA} = EA$$ where $E$ is the electric field between the capacitor plates. From Gauss's Law this is equal to the charge $Q$ on the plates divided by $\epsilon_0$ $$\frac{Q}{\epsilon_0}\implies E = \frac{Q}{A\epsilon_0} = \frac{\sigma}{\epsilon_0}$$ I know there is something fundamentally incorrect in my assumptions or understanding, because I frequently get conflicting results when calculating electric fields using Gauss's Law. I am, however, unsuccessful in identifying this. **Edit:** Also, another problem I noticed was that even if we remove the negative plate from the capacitor and then apply Gauss's Law in the same manner, the field still comes out to be $\sigma/\epsilon_0$ which is clearly wrong since the negative plate contributes to the field. So, maybe the problem is in the application of Gauss's Law.


title: Decay width average in the isospin invariant limit

text: Suppose we have the following experimental values for $\eta' \rightarrow \eta \pi \pi$ decay width: $\Gamma_{\eta' \rightarrow \eta \pi^+ \pi^-} = 0.086 \pm 0.004$ $\Gamma_{\eta' \rightarrow \eta \pi^0 \pi^0} = 0.0430 \pm 0.0022$ We are investigating this decay in the isospin invariant limit and want to compare our results with experimental rates. So we should average over these 2 values to find the experimental decay width in this limit. It is written in one paper that we should average $\Gamma_{\eta' \rightarrow \eta \pi^+ \pi^-} = 0.086 \pm 0.004$ **2$\Gamma_{\eta' \rightarrow \eta \pi^0 \pi^0} = 0.0860 \pm 0.0044$** in a specific way (which is not my question). I don't know why we should **twice** the second value and then average?


title: Thermal expansion coefficient times temperature: under which condition is it unity?

text: In Landau's Fluid Mechanics p. 8 (2nd edition) he writes for the thermal expansion coefficient $\beta = (1/V) (\partial V/\partial T)_P$: > For a column of gas in equilibrium which can be taken as a thermodynamically > perfect gas, $\beta T = 1$ [...] This is not not obvious to me. Can somebody provide me with a derivation or an argument under which circumstances I can approximate this expression like that? Or is it as easy as taking Clapeyron's equation for $V$, i.e. $PV = NT \implies V = NT/P$, so that we get $$ \beta T = \frac{TN}{VP} = 1 $$ This leaves me somehow unsatisfied; is there maybe a more physical argument for this?


title: Would computers accelerated to high speeds compute "faster" from our point of view?

text: I woke up to this thought yesterday: Lets say Computers A and B have exactly the same specifications and at time T both are set to process an algorith that would normally take exactly 1 year and exactly at T the A computer is accelerated to 0.5c (or anything c). Both are set to automatically broadcast the result to a central computer.   * Would A finish processing first from my point of view?    * Would we be able to receive the broadcast from A?    * Would it matter if A were travelling at a straight line or orbiting a planet or even a star system?   * Would it be feasible to accelerate a computer to "compress time" on a Machine on the likes of the LHC. Excuse me part: I'm sorry if this question is not fit for this StackExchange , I'm sure someone asked that , but I don't know how to look for this- I'm really new to all things physics.


title: Projectile motion of a charged particle in a homogeneous electric field

text: I'm reading an article about projectile motion, but I'm having some trouble with how the author found the equations of motion when a homogeneous electric field is considered. > To allow an immediate comparison as an example of non-gravitational > acceleration the corresponding motion of a point charge in a static > homogeneous electric field is tackled. We choose the charge $q$ and the > electric field $E$ in the direction of the negative $z$ axis such that > $qE=mg$ and use the same dimensionless quantities as before (time is > measured in units of $c/g$ and space in units of $c^2/g$). Also we use lower > case symbols in the inertial reference frame (t,x,z) to facilitate > comparison. With: > > $t'^2=1+x'^2+z'^2$ > > the equation of motion $x''=0$,$z'' = -t'$ gives immediately: > > $x'=x_0'$, $x=x_0'\tau$ and $z'=z_0'-t$. All terms are derived with respect to the proper time $\tau$. I don't understand how $qE=mg$ implies $t'^2=1+x'^2+z'^2$ and where $z''=-t'$ comes from.


title: Is there a fundamental limit to the temporal resolution of signals from space?

text: In Earth-based experiments, we can measure phenomenon very rapidly in an experiment given appropriate equipment. Clearly if something takes a long exposure to see (due to a weak signal), then the temporal resolution will be filtered by the exposure time. But let's assume a signal was strong enough that exposure time wasn't a concern. When conducting observations of things like transits around stars, tracking orbits of moons in our own solar system, or watching the evolution of supernovae, is there a fundamental limit to the temporal resolution? Does the distance introduce aliasing in the signal, perhaps due to differences in wave speed and packet speeds that only become evident over large distances?


title: A naive look at the band of stability suggests a stable isotope of every element. Why is this intuition false?

text: Looking at the band of stability, my first intuition is to conclude (erroneously) that there is a stable isotope of every element that lies close to the belt of stability. Why is this false? (For example, Uranium has no stable isotopes.) Or in particular, why are some number of protons inherently unstable in the nucleus? Again, based purely on my poor intuition, I would assume that some number of neutrons could be arranged with that number of protons to create stability but that is not true in all cases. Why are these numbers of protons (e.g. 92 for Uranium) unstable?


title: S-Matrix Generating Functional (Problem 4.1 in Weinberg)

text: I'm currently working through Weinberg's QFT book, but I'm somewhat stuck at problem 4.1, which states: > Define generating functionals for the S-matrix and its connected part: > \begin{equation} > F[\nu]=1+\sum_{N=1}^{\infty}\sum_{M=1}^{\infty}\frac{1}{N!M!}\int\nu^*(q_1^{\prime})...\nu^*(q_N^{\prime})\nu(q_1)...\nu(q_M)S_{q_1^{\prime}...q^{\prime}_Nq_1...q_M}dq_1^{\prime}...dq^{\prime}_Ndq_1...dq_M > \end{equation} and \begin{equation} > F^{C}[\nu]=\sum_{N=1}^{\infty}\sum_{M=1}^{\infty}\frac{1}{N!M!}\int\nu^*(q_1^{\prime})...\nu^*(q_N^{\prime})\nu(q_1)...\nu(q_M)S^C_{q_1^{\prime}...q^{\prime}_Nq_1...q_M}dq_1^{\prime}...dq^{\prime}_Ndq_1...dq_M\, > . \end{equation} Derive a formula relating $F[\nu]$ and $F^{C}[\nu]$. He defined the connected S-matrix in order to satisfy the Cluster- Decomposition principle as: \begin{equation} S_{\beta\alpha}=\sum_{PART}S^{C}_{\beta_1\alpha_1}S^C_{\beta_2\alpha_2}....\, , \end{equation} where $\beta$ and $\alpha$ denote the set of initial and final momenta, respectively. For instance, we have: \begin{equation} S^C_{q^{\prime}q}=S_{q^{\prime}q}=\delta(q^{\prime}-q) \end{equation} or \begin{equation} S_{q^{\prime}_1q^{\prime}_2q_1q_2}=S^C_{q^{\prime}_1q^{\prime}_2q_1q_2}+\delta(q_1^{\prime}-q_1)\delta(q_2^{\prime}-q_2)+\delta(q_1^{\prime}-q_2)\delta(q_2^{\prime}-q_1)\, . \end{equation} Now, concerning the problem: I assumed the connection should be something like \begin{equation} F[\nu]=\exp{iF^{C}[\nu]}\, , \end{equation} but I'm missing the way to generate any $i$'s. \begin{equation} F[\nu]=1+F^C[\nu]+\sum_{N=1}^{\infty}\sum_{M=1}^{\infty}\frac{1}{N!M!}\int\nu^*(q_1^{\prime})...\nu^*(q_N^{\prime})\nu(q_1)...\nu(q_M)\sum_{K=1}^{R}\delta(q^{\prime}_1-q_1)...\delta(q^{\prime}_K- q_K)S^C_{q_{K+1}^{\prime}...q^{\prime}_Nq_{K+1}...q_M}\binom{N}{K}\binom{M}{K}K!dq_1^{\prime}...dq^{\prime}_Ndq_1...dq_M\, \end{equation} where $R=\min{(M,N)}-1$, since the sum gives zero otherwise. The problems seems to be entirely combinatoric, but somehow, I don't get it right. I would appreciate it, if you could give me a hint?


This has started to bug me more and more… it involves:   * colour theory   * the trichromatic properties of our eyes through cone cells   * and light. Is there a one-to-one relationship between colour theories and our trichromatic vision? Are colour theories — the subtractive and additive properties of colour — strictly a by-product of our trichromatic vision?! Could our colour theories also hold some truth in, say tetrachromatic vision (if both vision had an identical electromagnetic radiation range)? Imagining we had perfect RGB screen technology, capable of reproducing all the colours within our visible spectrum range (400nm—700nm). Could you, if you had tetrachromatic vision, use the same RGB screen and still see all the colours in your "visible spectrum" of the same range? Could you use this perfect RGB screen technology if, while trichromatic, your cone cells were "tuned" to different frequencies (but still covered the identical "visible light" range)?! _Math can be applied regardless of the base you're using (binary, octal, decimal, hexadecimal, etc.). I would like to think colour has a similar beauty to it._


title: Scalar-fermion bound state

text: Is it possible to have a bound state between a scalar and a fermion? For example, a squark--anti-squark bound state, provided that the decay width is sufficiently small compared to the binding energy?


title: Electron in a covalent bond: what happens when it moves out?

text: Let's look at a covalent bond in, say, a water molecule: H : O : H H is bonded to O by a pair of electrons. But electrons are in constant motion. What happens when the electron leaves? Why does not the bond break? Both nuclei on both sides of it cannot move, obviously. Also, if the electron has to orbit around it's respective atom, does it mean that it would actually spend only 1/10 of the time in the "sweetspot" position (right in between the two cores), and will spend 9/10 or most of its time "orbiting" (ie. out of that position)? I am rather puzzled by this. Any ideas will be greatly appreciated.


title: Geodesics in a point mass universe

text: This question may reflect my (lack of) knowledge about general relativity, please ask for any clarifications or note any corrections in the comments and I'll try to address them. The Schwarzschild solution to the Einstein's field equations gives the solution to a a non-rotating, spherically symmetric uncharged mass distribution in a static vacuum. From this solution, we can compute geodesics from one point to another. Is it possible (analytically or numerically) to compute the geodesics from one point to another under the influence of a set of static point masses? The point masses can assume arbitrary geometry, but like the Schwarzschild problem they are uncharged and non-rotating.


title: Problem with relativity of acceleration

text: In this answer http://physics.stackexchange.com/a/92833/36977 John said that acceleration is not relative in the general theory of relativity. But this is a problem: as we all know, accelerating charges emit electromagnetic waves. So let's build up the components of our thought experiment: me, a charged body and point X. Now the problem is: if acceleration is relative, and if I accelerate, from my reference frame it is the electric body that is accelerating, and thus would emit electromagnetic radiation. But if it is at rest and then start accelerating at say 9m/s^2 relative to X and I'm also at rest and start accelerating at 9m/s^2 relative to X then by definition it will starts emiting electromagnetic radiation, but from my perspective it wont since it's acceleration is the same as mine. so where is the problem?


title: what is the mechanism of water conductivity?

text: I'm trying to wrap my head around the kelvin water drop generator and more specifically what happens in the water before it drops. If on one side the drops of water are positive and on the other side negative, that means there is current flowing through the water _before it drops_. What I dont understand even more specifically is how electricity is carried by water, and why water conductivity increases with ions content (and other impurities perhaps). Are there electrons flowing ? Or are there ions flowing ? Let's imagine that we are below the potentials at whitch electrolysis occurs (about 1 volt). What carries the charges ? I first thought that ions were responsible for the charges in the kelvin generator, but when a spark occurs, because the water has dropped into separate boxes it's impossible for them to move, so there must be some kind of redox reaction occuring. But if table salt is in the water to raise it's conductivity, would that mean that Na+ is reduced to sodium and Cl- oxydized to chlorine ?


title: Field created by a wire, known voltage

text: I am having some mental troubles with this problem I came across reading about transmission lines (I mean, it is not a problem from a book, I just thought about it). Say I have a couple of wires (parallel infinite straight lines) with known electric potential (in free space). They model a transmission line and they carry some AC current (the relation between voltage and current is given by some constant, the characteristic impedance). I want to know the Electric field (the magnetic part doesn't bother me) created by it outside the wire. Is the problem well posed with just this information? If no, exactly why? If yes, how can I find the solution? In the latter case I don't expect a closed form solution, but the procedure to find the equations, or the equations themselves. I think it would require numerical computation to solve them. Edit: how about the case with constant voltage? Would it be just zero?


title: plasma pressure initial condition

text: The authors in the paper below did a simulation of plasma plume evolution from a laser heated metal target. They used compressible transport equations and ideal gas EOS for Ar. I am having difficulty understanding their initial condition. They say ", the initial internal pressure of plasma is assumed constant, 210 MPa.". How is this value determined or assumed. It is based on the temperature of the surface? So, im thinking about the problem some more, and it seems the density is what should be known because when ablation occurs, phase transition occurs in the metal surface which generates a vapor. Thanks S. Harilal, G. V. Miloshevsky, P. K. Diwakar, N. L. LaHaye, and A. Hassanein, Phys. Plasmas 19(8), 083504 (2012).


title: calculating electrodynamic momentum of a dumbbell (consisting of two point charges) in longitudinal motion

text: I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure how. $$ \int_{0}^{\infty}\int_{0}^{\pi} {r^{4}\sin^{3}\left(\theta\right) \over \left\\{(r^{2} + d^{2}/4)^2 - \left[rd\cos\left(\theta\right)\right]^{2}\right\\}^{3/2}} \,{\rm d}r\,{\rm d}\theta \space-[1] $$ If I focus on the integration w.r.t. r first, I get [2]. If I focus on the integration w.r.t. $\theta$ first, I get [3]. A,B,C are just constants w.r.t. the variable of integration. I believe I can make a series of substitutions but they don't seem to get me closer to an answer. $$ \int_{0}^{\infty} {Ar^{4} \over \left[(r^{2} + B)^2 -\left(rC\right)^{2}\right]^{3/2}}\,{\rm d}r \space-[2] $$ $$ \int_{0}^{\pi} {A\sin^{3}\left(\theta\right) \over \left\\{(C)^2 - \left[B\cos\left(\theta\right)\right]^{2}\right\\}^{3/2}} \,{\rm \rm d}\theta \space-[3] $$


title: Is there a surface charge density?

text: Consider a dielectric sphere placed within a dielctric medium. There is a uniform electric field $E_0$ present throughout in the medium. Would there be surface charge on the sphere?


title: Why can a killing vector field be determined globally by its value and first derivative at one point?

text: It is said in Weinberg's Book, _Gravitation and Cosmology,_ page 377, that a killing vector field (which we a priori assume exists globally) can be uniquely determined by its value and first derivative at one point. The argument uses Taylor expansion at the given point. My question is, as also stated in the book, the Taylor expansion only holds in a **finite neighborhood** of that given point. Then how can we talk globally? Are we assuming the space-time is compact?


title: Why can't you hear music well over a telephone line?

text: Why can't you hear music well well over a telephone line? I was asked this question in an interview for a university study placement and I unfortunately had no idea. I was given the hint that the telephone sampling rate is 8000 samples per second.


title: Does measuring the exact position of a ball destroy the ball

text: If you have a macroscopic ball (say, a tennis ball) and you (hypothetically) try to measure the exact position of the center of that ball by measuring the exact positions of the atoms making up the ball, will this measurement destroy the ball (i.e. is there a point where, as a result of the uncertainty principle, the more accurately you measure the position of each of the atoms, the more likely the atoms are to scatter off in some random direction, destroying the ball)?


title: Hamiltonian mechanics and special relativity?

text: Is there a relativistic version of Hamiltonian mechanics? If so, how is it formulated (what are the main equations and the form of Hamiltonian)? Is it a common framework, if not then why? It would be nice also to provide an example --- a simple system with its Hamiltonian. As far as I remember, in relativistic mechanics we were only taught to use conservation laws, that is integral invariants and thus I have a vague perception of relativistic dynamics.


title: Shape of the Higgs branching ratio to ZZ

text: I've been looking at the, now very popular, graph of the SM Higgs decay branching ratios: ![](http://www- cdf.fnal.gov/physics/exotic/r2a/20050623.lmetbj_wh_tc/higgs_branch.jpg) You see that the ZZ branching ratio has a funny dip around the $170\, GeV$, very different from the WW counterpart. It's true that graphs with Z will always have the $\gamma$ interference and, perhaps, this is the cause for this funny shape. However, the value of the mass for which this happens is **not** something that I can intuitively explain. Given that I have no idea of the right answer, let me ask: What causes this dip in the $H\rightarrow ZZ$ branching ratio around $M_H=170\, GeV$?


title: Measurement of kaluza-klein radion field gradient?

text: I've been very impressed to learn about kaluza-klein theory and compactification strategies. I would like to read more about this but in the meantime i'm curious about 2 different points. I have the feeling that there are no precise answers to these questions at the current moment but nonetheless i feel obligued to ask:   1. Does a compactification strategy has to impose a fixed thickness in the compactification or this arises as a dynamic consequence of the evolution equations? and if i start with a small thickness, does the dynamic in all cases keeps the thickness in a bounded value? are there results that addresses this?  **EDIT** answer appears to be No, it is not assumed to be fixed. There is an effective scalar field that precisely describes this.   1. Given that the argument is that vibrations in the compactified dimensions determine the internal states, any significant gradient ratio in the compactification length would have to be probably measurable as either spontaneous decay or spontaneous stimulation events of particles. My question is, would experimental effects of any thickness gradients would be as difficult to measure as expected effects from a fixed but non-zero compactified dimension?  **EDIT** a better rephrase of this question is that; _since energy of non- gravitational fields like electromagnetism, etc. is stored in vibrations along the compactified dimensions, it would seem that even a small gradient in the radion field (even 0.1%) should be measurable as a effective diffraction index (since a 0.1% variation in the length of the scale would affect each non- gravitational mode by that amount)_ I guess even another way to frame the question is: is the low-energy phenomena we are able to see currently, invariant/non-dependent under small changes of scale (from point to point in space-time) of the compactification? isn't scale of compactification itself a vibrating degree of freedom? **So bottom line:** since we haven't see any low-energy consequences from small gradients in the radion field, can't we infer that the radion field, is, by whatever reasons (dynamics, broken symmetry, etc.) _effectively fixed_? can't we currently estimate a bound in the radion gradient given the negative results mentioned above it would have? I hope i've made my questions clear and interesting enough.


title: Informational capacity of qubits and photons

text: ### How much information is contained in one qubit? A qubit is defined in Wikipedia as $a\left|0\right> +b\left|1\right>$, where a and b are complex numbers subject to $a^2 + b^2 = 1$.   One complex number is equivalent to two real numbers, which suggests that a qubit is equivalent to four real numbers worth of information.   Somehow I doubt this is true. How much information is contained in one mathematical qubit, exactly transmitted? ### How much information is contained in one physical photon? Alice and Bob share an arbitrarily but not infinitely accurate clock. They also have a shared cordinate frame (arbitrarily accurate relative location, no relative motion, no acceleration) Alice and Bob can also have an arbitrarily complex shared coding system. Alice is allowed to send one and only one photon to Bob between 12:00 and 1:00 oclock, and Bob receives it. Alice sends the photon directly to Bob through an intervening vacuum. If it makes a difference, assume the wavelength is exactly 500 nm.   How much information can Alice send Bob?   I assume it is not infinite, but only limited by the precision of the clocks, the availability of bandwidth, Alices range of transverse motion, the accuracy of Bob's telescope, etc. Now assume the photon is delivered by a single mode optical fiber, so Bob can gain no information by measuring the direction of the incoming photon. Also assume that Eve puts in a variable delay of two to three hours, so Bob receives the photon between 2:00 and 4:00 and thus cannot gain any information from the time of receipt. How much information can Alice now send Bob?   Is it exactly one qubit, or something else? We all know the plan where Alice transmits vertical for 0 and horizontal for 1, Bob measures exactly vertical or exactly horizontal and one bit per photon is transmitted.   Is there any improvement possible?


title: Multiple channels of information in single electromagnetic wave?

text: I'm trying to understand how can multiple radio stations transmit information just by transmitting using different frequency. The way I understand it all those different frequency waves add up to a single wave? If so how can that not corrupt information on other frequencies? Say if I transmit: $\sin(2\pi x)$ And separately: $\sin(2\pi x\times 2)$ Does it end up as a single wave of: $\sin(2\pi x)+\sin(2\pi x\times 2)$? Or does it work in a different way completely?


title: Where's the energy in coupled magnetic fields?

text: Suppose I have two identical current loops superimposed upon one another, but with their currents in opposite directions. The coefficient of coupling $K=1$, and therefore the mutual inductance $M =\sqrt {L_1L_2} = L$, giving the total magnetic energy as $$E_m = 1/2(L_1i_1^2 + L_1i_1^2) + Mi_1i_2 = 2Li^2$$ But since the total magnetic field is zero, the total magnetic energy is zero, so where has the apparent excess energy come from?


title: Path of an electron through an electric circuit

text: When a potential difference is applied across a conductor, and if an electron moves from the negative terminal of the battery and reaches the positive terminal, then I want to know if the electron will remain at the positive terminal or will it again move toward the negative terminal through the battery?


title: Conservation of energy in electromagnetism?

text: How is energy conserved in electromagnetism? Is it hinted only from Lenz's law or from Maxwell? Also, if a solenoid's flow of current produces a magnetic field, and from the magnetic field forces are applied on dipoles, how is energy conserved? I know that energy is always conserved, but in this case when electric & magnetic fields interact to do work I don't know how to picture conservation.


title: how to model the exponential growth of luminosity when you flick a light on

text: My original thought was: $$ L = L_0 \left( 1 - e^{-t/\tau} \right) $$ where $L$ = luminosity and $L_0$ = initial luminosity. But when I tested this I found that the time constant $\tau$ was higher for smaller values of Voltage, so I have no idea and all I can find is the luminosity of planets etc. Any help would be good.


title: Non-linear Dirac equation in Einstein Cartan theory

text: Can someone explain this Wikipedia article, specifically the section on Einstein-Cartan theory? I have no idea how the equation \begin{equation}(i\gamma^{\mu}D_{\mu}-m)\psi=i\gamma^{\mu}\nabla_{\mu}\psi+\frac{3\kappa}{8}(\bar\psi\gamma_{\mu}\gamma^{5}\psi)\gamma^{\mu}\gamma^{5}\psi -m\psi\end{equation} is justified. They state that $D_{\mu}$ is a covariant derivative acting on spinors, and then later state that $\nabla_{\mu}$ is a general-relativistic covariant derivative acting on spinors. I don't know what the difference between the two would be. Lastly, where does the cubic term come from? That's what confuses me the most.


title: Confusion about time shift in special relativity

text: I have never really found a way to comfortably comprehend the idea of time shift even though I know its not the hard part of relativity theory. In that light, can someone point out what is wrong or right about the following logic I'm thinking of here: The scenario is: There are four things in my universe: a wallclock, a flashlight, my wrist watch and myself. I shine the flashlight at my wrist which is in the same line of sight as the wallclock and see they two clocks are synchronized. Now my wristwatch, my flashlight and myself begin traveling away from the wallclock along the line of sight. It seems to me that logically the light I'm running away from will take longer to reach my eye, so that means I would see the wall clock run _slower_ , no? But according to what I think I know, when a minute passes for me, _more than a minute_ passes for the wall clock. I just feel like I must be looking at it wrong. Any pointers are appreciated. This is just recreational thinking...


title: How do the "tidal forces warming moons" theories hold when apart from heating from expansion, there may be also cooling from contraction?

text: I can understand a temporary heating, from the tital forces exerted on the moon but wouldn't there be cooling as well eventually when particles "give in" to contraction? Wouldn't they eventually net a unchanging whole body temperature? i.e. How can Europa's oceans be warmed by that and how can Io's crust be melted by that?


title: Physical significance of Killing vector field along geodesic

text: Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter. What physical significance do the scalar quantity $X_iu^i$ and its conservation hold? If any...? I have seen this in may books and exam questions. I wonder what it means...


title: how find resonance in extrasolar planet?

text: I make this program, i have a 2xN matrix in which the columns are the ID of planets and their period, the rows are the number of planets, for istance something like that:               1 0.44     1 0.8     1 0.9     2 0.9     2 1.2     3 2.0     3 3.0      The trick to change from one system of planet to the other is to rename all the planets of a system with the same number and the planets of other system with another number so, i can be able to change the resonance condition from one system to another one. The program is simple: 1) read the file and save the columns and rows numbers, 2) create and save a matrix of col*row objects, 3) save as a vector the name and period of planets, 4) start the cycle: `for r=1,row` <\--- THIS MUST READ all the file `if (difference in name = 0.)` then start the resonance find criterion `for l = 0,4` (number of planet in each system: THIS MUST BE MODIFIED !!)               for i = 1,5      for j =1,5      `if (i*period(l)-j*period(l+1) eq 0)` then write on file <\- RESONANCE CONDITION !!!               end if        end for        end for        else write a separation between the first set and second set of planets !        end if      This is the IDL code i wrote: pro resfind               file = "data.dat"     rows =File_Lines(file) ; per le righe     openr,lun,file,/Get_lun ; per le colonne     line=""     readf,lun,line     cols = n_elements(StrSplit(line, /RegEx, /extract))          openr,1,"data.dat"       data = dblarr(cols,rows)      readf,1,data     close,1               name = data(0,*)     period = data(1,*)          openw,2,"find.dat"      for r = 0, rows-2 DO BEGIN ;             if (name(r)-name(r+1) EQ 0) then begin                      for l = 0,rows-2 do begin                               for j = 1,4 do begin                                      for i = 1,4 do begin                                                     if (abs(i*period(l)-j*period(l+1)) EQ 0.) then begin                                                 printf,2, 'i resonance:', i , ' j resonance:',j,' planet ID:',l,' planet ID:',l+1                                                endif                                      endfor                               endfor                     endfor             endif else begin             printf,2, '                                                    '      endfor               close,2          end      PROBLEMS: 1) i can't understand how to eliminate the multiply of resonance (2:4, 3:6 and so on); 2) in the second for loop (the one with the planet) the number of planets must be change every time but i don't understand how to change this. And this is the f90 version of the code:               program resfind         implicit none         integer::i,j,k,s,n         real*8,allocatable::ID(:),period(:)              open(10,file='data.dat')         n=0             DO               READ(10,*,END=100)               n=n+1             END DO           100     continue             rewind(10)              allocate(ID(n),period(n))     s=0         do s=1, n             read(10,*) ID(s),period(s)         end do         open(20,file='find90.dat')          ! do r = 0, rows-2 DO BEGIN ;     !        if (name(r)-name(r+1) EQ 0) then begin                i=0     j=0     k=0      do i = 1,n-1        do j = 1,5         do k = 1,5            if (abs(j*period(i)-k*period(i+1)).EQ.0.0) then             print*, i,j,k         write(20,*) 'j resonance:', j , ' k resonance:',k,' planet ID:',i,' planet ID:',i+1         end if        end do       end do      end do           close(20)          end program resfind      Thanks a lot.


How are degrees of freedom and energy related in classical theory? How do we come to know that each quadratic degree of freedom classically contributes a factor of $\frac{k_{B}T}{2}$.


What is the capacitance of parallel plate capacitor with different electrode material? The capacitance of parallel plate capacitor depends on dielectric material, surface area, and separation between two plates. $$C = ε_r * ε_0 * \frac{A}{D} $$ But how about electrode materials between dielectric, will the capacitance be different? Thanks.


title: Relation between Casimir and hydrophobic effects

text: ## Background Some years ago I was studying "Fundamentos de biología" (Biology fundamentals) and learned how the lipids create a bilayer due to the water repulsion. Some time later I learned that this was called the hydrophobic effect and occurred in some other cellular materials. But something else was in my mind back then. I had already read in wikipedia about a particular effect, the Casimir effect and _felt_ like they were similar, but in completely different fields and medium. ## Similarity When there's a material immersed in the medium, the entropy of the medium is reduced. Therefore, if there are two materials, there's a force that bonds them together as the total amount of reduced entropy also depends upon the total surface [and closed systems tend to increase the entropy, not reduce it]. Note: In the English Casimir Effect's wikipedia page there's no allusion to entropy; nor there's in the Spanish one, but only in the Spanish one there's a couple of notes that strongly suggest that the entropy is reduced with the two plates. ## Questions 1.a Are they actually the same "kind" of effect but in completely different areas or I'm interpreting them wrong? 1.b Is there any other similar effect to these? 2.a If the material affecting the Casimir effect was liquid, wouldn't it act similar to the lipid bilayer, forming an sphere? 2.b If so, wouldn't be a good way to measure the Casimir effect to rotate this small sphere at certain velocity and see the expected deformation vs the one that occurs taking into account the Casimir effect?


title: Independence of frequency in sound waves?

text: Why does the frequency of sound wave depend only on the source? Why is the frequency and not any other "quality" independent of everything but the source? And that said, why is velocity and wavelength of the wave dependent only on the medium? And finally - in case of the doppler effect in case of sound, does or doesn't the independence of these quantities hold true?


title: linear response for a simple harmonic oscillator

text: Really sorry for this simple question, but I think it will be useful/interesting in general. Consider a quantum simple harmonic oscillator. Add a perturbation $H_I = -\lambda \hat{x}$ Calculate $\langle \psi| \hat{x} |\psi \rangle$ in the new ground state, to first order in perturbation theory in $\lambda$, and relate it to $\langle 0| x^2 |0 \rangle $ where $| 0 \rangle$ is the unperturbed ground state. Can someone help me on this


title: Feynman Diagram for Interaction of Deuteron with Oxygen-16

text: I have to draw a Feynman diagram for the interaction of a deuteron with Oxygen-16. The interaction is as follows: $ d + {}^{16}O \rightarrow p + {}^{17}O . $ I am generally ok with Feynman diagrams when I know what particles are actually interacting - proton, neutron etc. but I am not sure how to proceed when an actual element such as oxygen in given. Any help would be much appreciated.


title: Signal strength drop-off from a geosynchronous satellite as a function of angular pointing error

text: If we have a compact antenna used to communicate with a satellite in geosynchronous orbit somewhere between the $K_u$ and $K_a$ bands of the spectrum (for high broadband applications where we can't use something like the Iridium network), we presumably need to properly orient the dipole of the antenna with respect to the position of the geosynchronous satellite. Let $\alpha$ be the angular pointing error of the satellite (some fraction of a degree). How does the signal strength fall off as a function of $\alpha$? Or, if it makes a difference, we can uniquely define an azimuth angle $\theta$ and elevation angle $\phi$. Now, I'm guessing that the answer to this question has something to do with the satellite beam configuration / geometry. So is it possible for there to be a "typical" answer to this question? Also, to be clear, here I'm asking about the "form" of the expression for the drop-off in signal intensity as a function of $\alpha$ (or azimuth and elevation angles $\theta$ and $\phi$), as the exact parameters will likely depend on the precise antenna being used. Where I'm coming from with respect to this question: I heard recently that is indeed possible possible to send data packets back and forth to a geosynchronous satellite from a sufficiently large boat (or aircraft) with proper antenna equipment (indeed to establish a continuous connection), and I was wondering how in the world this worked? In other words, in the context of something like a small vessel or an airplane, to what extent is sophisticated inertial navigation equipment required for real-time compensation of rotational (pitch / roll / etc.) and translational transformations on the orientation and position of the antenna? [3-11-2014] Is it safe to assume that these vessel-to-GEO antennas use circular polarization to avoid issues associated with the relative orientation of the transmitting and (primarily) receiving antenna? It seems then safe to assume that only the angular error $\alpha$ should be sufficient to estimate signal strength?


title: Can Event Horizon be created without actively involving Gravity (Spacetime curvature)?

text: The definition of Event Horizon I took from Wikipedia: > _An Event Horizon is a boundary in spacetime beyond which events cannot > affect an outside observer._ From the definition, I can abstractly define such boundary without involving Gravity. Now, I want to prove it by construction. Is it possible without involving Gravity?


title: Why doesn't the electric field inside a wire in a circuit fall off with distance from the battery?

text: We studied electric fields due to point charges. The magnitude of these fields decreases with the square of the distance from the point charge. It seems to me that we could treat the positive terminal of a battery as a point charge. So, I would conclude that the magnitude of the electric field set up by the positive terminal inside a wire in a circuit should fall off with the square of the distance from the end of the battery. However, this does not happen. Instead, the electric field inside a wire in a circuit is constant. Why is this? Is it that the positive terminal can't be modeled as a point charge? Or is it perhaps some special property of the wire, or the fact that there is moving charge in the system?


title: Shear Flow corresponding to Eccentric Shear Force of a Closed-section Beam (Structural Analysis - Mechanics)

text: Been stumped with this question for way too long... its a beam with a thin- walled rectangular cross-section, and a shear force is acting at a distance from the shear center. I know my decomposition of that force into the torsional moment and shear-center acting shear force is correct, but for some reason the shear flows don't add up.... rather than causing a higher shear to the left of the cross-section like I'd expect from my diagram, they cause a higher shear on the right side of the cross-section, sort of the mirror image of what I would expect. I've tried finding my mistake looking at books, websites, notes but none of them address this topic directly, leaving me with no other choice. :( Could anyone please tell me what I'm doing wrong? Also sorry for the blurry image and messy diagrams... I'll draw a new one if it's useless. ![enter image description here](http://i.stack.imgur.com/U31Qw.jpg) ![enter image description here](http://i.stack.imgur.com/zxXJH.jpg)


title: Dark matter and tides

text: As I understand, "dark matter" is what they call a theoretical substance which is only known by it's influence upon velocity curve of the galaxy. If indeed the gravity of "dark matter" is so strong that it keeps our solar system in orbit, then it's force effects on Earth should be comparable with that of our Moon. However, we can observe the moon tides, and combined Moon-solar tides, but we never observe the tides resulting from pull towards the center of the galaxy. Why would that be? Could you please share your thoughts? ![enter image description here](http://i.stack.imgur.com/ZGLAh.gif)


title: Multi-Fermion interactions induced by integrating-out Yukawa-Higgs terms?

text: Suppose one considers a **multi-component free fermions field theory** with field $\psi_{q_i}$ with a give global symmetry (such as U(1)). We can say that every component of fermions carry some U(1) charge $q_i$. Now, let us introduce Yukawa-Higgs term to mediate fermion-fermion interactions. Explicit Higgs charge are assigned to maintain the U(1) symmetry in Yukawa-Higgs term. We also consider the Higgs field strongly-disordered so do NOT break global symmetry (here U(1)) via the normal Higgs mechanism. This can be done by adding kinetic terms to Higgs field, and in principle $\langle \Phi \rangle=0$ but $ (\Phi-\langle \Phi \rangle)^2\neq0$, i.e. the Higgs field does not condense, but Higgs field strongly fluctuates. > Here is the **Question** : > > whether **including all symmetry-allowed Yukawa-Higgs term of a single Higgs > field ($\Phi$)** can induce **any symmetry-allowed multi-fermion > interactions**? I can imagine this may be true if we have: > including **all symmetry-allowed Yukawa-Higgs term of (multi-component and > multi-species) Higgs fields ($\Phi_1$,$\Phi_2$,$\Phi_3$,$\Phi_4$,$\dots$)** > > $\Leftrightarrow$ (iff: correct?) > > induces **all symmetry-allowed multi-fermion interactions**. But let us say if we have only **ONE single component scalar Higgs field** , say all symmetry-allowed Yukawa-Higgs term of $\psi_{q_i}^{n_i} \Phi^{m} \psi_{q_j}^{n_j}$ (with $\psi$ are fermions field and $\Phi$ is the only Higgs, with some power ${n_i},{n_j},m$ perhaps with complex conjugate $(\psi_{q_i}^{\dagger})^{|n_i|}$ if ${n_i}<0$.) can one **prove or disprove** this is true or no that it can generate all multi-fermion interactions, such as $$V_{interaction}=\big( ({\psi}_{q_1} \nabla^1_x {\psi}_{q_1} \nabla^2_x {\psi}_{q_1} \dots \nabla^{N_{q_1}}_x {\psi}_{q_1}) \dots ({\psi}_{q_n} \nabla^1_x {\psi}_{q_n} \nabla^2_x {\psi}_{q_n} \dots \nabla^{N_{q_n}}_x {\psi}_{q_n}) \dots ) \big) + \text{hermitian conjugate}$$ with potential higher derivative terms?


title: Why there is electromagnetic resistivity in vacuum?

text: I cant understand that. If there isn't a material that makes it hard to pass, why there is a resistivity in vacuum?


title: total intensity measurement (of the whole visible light spectrum; from 400 - 800 nm) with a powermeter PM100?

text: I was using a power meter PM100, with a Si-photodiode sensor head, to measure the intensity of the white light source (250 W quartz halogen lamp). I have recorded the power, shown by the sensor, by setting the wavelength equal to 633 nm (i.e. the maximum intensity is supposed to be at this wavelength, in the spectrum); however, this value for the intensity (power divided by area of the sensor) is only true for the monochromatic light (supposed to use for laser light?) and want to estimate the intensity (of the white light source) from the measured value. I have the responsivity curve for the senor, I used, and the intensity at a wavelength. How can I estimate the total power/intensity of the light source? Your help will be much appreciated.


title: Explain Momentum

text: It's a shame for me to be in Year 12 and still haven't understood the concept of Momentum. This is what I think, and I know I'm wrong. But, it is a good place to start for you to explain: "Since the unit of the momentum is kg/ms^-1, it is a measure how much mass is being moved at the speed of 1ms^-1. By the way, this is not a homework or anything. I'm really eager to learn about this concept. Thanks In Advance.


title: Volts, Amps, and Practical Electrical Knowledge

text: I've heard people say that high amperage or high voltage is dangerous to humans-aren't they both? And what are the advantages to high amperage versus high voltage? And how do you generate one or the other? They are related via Ohm's Law, so I always run into problems trying to think of how to generate one, because I feel like it comes at the cost of the other. For a simple example, if you have two batteries-you can either put them in series or parallel, parallel gives a lot of current, but the same voltage while series would give a higher voltage? Why would you want a higher voltage? Also, power is proportional to current squared-so putting batteries in parallel drains them faster?


title: Relation between voltage and current

text: What I know is that voltage is the electrical pressure which is required to drive the electrons in a circuit. According to that we have higher currents when there is a high voltage. Ohm's law confirms that. What I don't understand is that some books say that it is the current that is more dangerous to the human body and not the voltage. What I mean is higher currents are dangerous but high voltages aren't. But if we have a high voltage aren't we indirectly meaning that there is high current since they are directly proportional to each other.


title: What kills you: Voltage or Electric current?

text: > **Possible Duplicate:**   >  What causes an electric shock - Current or Voltage? When someone gets electrocuted, what kills them; a high Voltage or a high Electric current, and why?


title: Pendulum: Deduce proportionality from experiment

text: _I know, very easy for all of you, but I'm a beginner in physics ... ;)_ I have to work with the mathematical pendulum. After some experiments (changing mass, chaning pendulum's length etc.), I could see that the changes in mass have no effects on the cycle duration ($T$) but changing the length of the pendulum ($\ell$) has. So I have to find out how exactly they correlate, right? What I thought about: We know (from Huygen's law) that the formula is: $$T = 2\pi\sqrt{\frac{\ell}{g}}$$ So I have to find out that $T$ is proportional to $\sqrt \ell$. How do I do that mathematically? Just guessing and seeing that it's correct is probably not the right solution, is it? And how do I then find the constant of proportionality? Just dividing $T$ by $\sqrt\ell$? Thanks a lot in advance!


title: what is the proper way to connect two light bulbs in a circuit? in series or parallel?

text: What is the proper way to connect two light bulbs in a circuit? in series or in parallel and why? My thought is that it's better to hook them in parallel, since if we take into account Ohm's Law, the sum of their resistance will be less than when connected in series therefore the battery will last longer. What do you think?


title: Hawking radiation: direct matter -> energy conversion?

text: When a black hole evaporates, does it turn all the matter that has fallen in directly to energy, or will it somehow throw back out the same kind of matter (normal or anti) that went in?


title: Wormhole related experiments?

text: Are there any simple "experiments" that can be done in a high school science lab that could demonstrate some sort of basic principals of wormholes or spacetime? Or sort of proving how long something would take to get through a wormhole or why you wouldn't be able to travel through them etc.


title: How do you explain the fact that when air expands freely into an evacuated chamber from a constant pressure atmosphere, its temperature increases?

text: I came across this paper: > Baker, B. (1999). _An easy to perform but often counterintuitive > demonstration of gas expansion._ American Journal of Physics, 67(8), > 712-713. > http://scitation.aip.org/content/aapt/journal/ajp/67/8/10.1119/1.19357 And I didn't fully understand the explanation. I can't see how this is any different from the Joule free expansion experiment with no temperature change. Essentially the author of the article says that the atmosphere does work on the gas and increases its internal energy. This is the reason for the temperature increase. However wouldn't that logic also lead to a temperature increase in the joule free expansion? Any insight would be much appreciated.


title: If the universe could emerge from nothing, what about physical laws?

text: I've watched Lawrence Krauss's lecture and read his book. I think I got what he was saying, and I don't have any problems with that; however, what I can't get is how the laws of physics that makes this phenomenon possible came to be in the first place?


title: From where randomness comes from and why it exists?

text: I recently began to study statistics and probability and I have two questions:   1. Where does randomness come from? What is the source of randomness?    2. Why does the randomness exist? Is it possible to answer my question that non mathematician or non physicist can understand it? I searched Internet for the answer but there is very little information that non mathematician or the information is for professional physicists and mathematicians. Also please suggest books concerning my questions that is written so that non mathematician or non physicist can understand it.


title: Concept of separation of charges in lightning clouds

text: I have read in an article that when lightning strikes in the clouds the cloud disintegrates into two parts of which the negatively charged part is bigger than the positively charged one. Why does this happen? Here is the screenshot of that article which is a textbook paragraph. ![textbook](http://i.stack.imgur.com/nrwc9.png)


title: "Pressure - Average Energy" ratio of ideal quantum gas?

text: Classically, one can easily show the following relation using a straightforward canonical ensemble computation for a non-interacting gas: $$\frac{PV}{\langle E\rangle}=\frac{2}{3}$$ Now, apparently the same relation is valid also for a quantum ideal gas. Trying to reproduce it I obtained the grand canonical (bosonic) partition function: $$Z_\alpha=\prod_{i}\left(\frac{1}{1-e^{-\beta(\alpha_i\epsilon_i-\mu)}}\right)$$ where $\alpha_i$ are auxiliary parameters I introduced for convenience (normally one would set $\alpha_i=1$). Per definition we now have: $$PV=\left(\frac{1}{\beta}\ln Z_\alpha\right)|_{\alpha_i=1}$$ And $$\langle E\rangle=\left(-\frac{1}{\beta}\sum_j \frac{\partial}{\partial\alpha_j}\ln Z_\alpha\right)|_{\alpha_i=1}$$ For both expressions I do not see how one should proceed with the evaluation. Any suggestions? The sums appear to be too difficult to solve (especially since it is not clear which values $\epsilon_i$ actually take on). Also, it might very well be that I did something wrong on the way there. A solution would be much appreciated.


title: can radiocarbon dating be used on living things?

text: I have been going through a wiki article about worlds oldest living creature. As a matter of fact its a plant, a shrub to be precise. Wiki says that the plant age was determined by carbon dating. But what i know is that carbon dating can only be done for dead tissues i.e which no longer assimilates atmospheric carbon onto itself. Can someone clarify this?? regards,


title: Optics: finding total power of a syste, including wavelength

text: Specifically I need to create an ABCD matrix given 2 radii of curvatures, the thickness of the lens, type of glass (BK7) and a given wavelength. I will have to figure this out eventually, so help would be appreciated with this as well, but my main question is, how do I correlate the wavelength of the incident light to the total power? If I have the wavelength of light, how do I decide what the power of the lens is if nothing in the lens changes, just the wavelength of the beam of light (for example, how would the power differ if a yellow helium line was shone, or a blue hydrogen line)? Thank you!


title: Boundary terms involving fields at infinity

text: In trying to prove that $[P^\mu,P^\nu]=0$ for a real quantized scalar field, where $P^\mu$ is the 4-momentum operator obtained from $T^{\mu\nu}$, I had to have my fields and/or their derivatives vanish at infinity, since I got boundary terms after integration by parts. For plane wave solutions, the fields clearly do not vanish at infinity. Is it possible to prove this commutator is zero without having to require that fields vanish at infinity?


title: Particle Accelerator Energy and Luminosity

text: Both beam energy and luminosity are important for succesful particle accelerator experiments. The LHC's nominal design is for 7 TeV beams and 1E34 cm^-2 sec^-1 luminosity, while the SSC's nominal design was for 20 TeV beams and 1E33 cm^-2 sec^-1 luminosity. The LHC has 10x greater luminosity while the SSC would have had almost 3x greater energy. If the SSC was alive and both machines were operating at nominal levels, which would have the greater advantage for discovery of: i. the Higgs boson, ii. supersymmetry, iii. extra dimensions? I'm asking on the assumption that all three of these exist.


title: Quantum entropy in term of density matrix

text: Why in von Neumann expression of quantum entropy we have trace of density matrix expression? Why don't off diagonal term play a role?


title: If energy can neither be created nor be destroyed,what is the ultimate source of energy?

text: We usually say that the ultimate source of energy for earth is the Sun. So this means that sun creates energy but according to the law of the conservation of energy energy can neither be created nor destroyed, so how does the energy come into the sun? That is just an example. We know that energy is transformed from one form to another but is not created and destroyed. But that is confusing because there must be an ultimate source of energy. How is energy created in this source? If it is related to the atomic level, then how can atoms possess energy? Is the law of the conservation of energy untrue?


title: Simple question regarding the Green's function for the diffusion equation

text: The differential operator for diffusion in three dimensions is given by $\partial_t - k \nabla^2$ where $k$ is a constant. The Green's function is (according to Wikipedia) $$\theta(t)\left( \frac{1}{4\pi k t} \right)^{3/2} e^{-r^2/4kt}$$ where $\theta$ is the Heaviside step function and $r = |\mathbf{r}|$. Hence, we should have $$ (\partial_t - k \nabla^2)\theta(t)\left( \frac{1}{4\pi k t} \right)^{3/2} e^{-r^2/4kt} = \delta(t)\delta^3(\mathbf{r}).$$ I've played around with this for a while, but I simply can't see how evaluating the left-hand side in the above expression gives the right-hand side. Where do the delta functions even come from?


title: Describing a motion of gyroscope with gimbal

text: Can you tell be how to set the equations to describe the motion of this machine in movie "Contact": https://www.youtube.com/watch?v=TSaO9VGjLXc This is gyroscope with gimbal, am I right?


title: Enthalpic Excess - Regular Solution Upon Mixing

text: My professor, in his handout, says the following about this diagram : > For $\beta>+2$ there are two minima and phase separation occurs driven by > unfavourable enthalpic interactions. I disagree with him partly. For $\Delta_{mix} G/nRT<0$ mixing is spontaneous and hence there solutions would be miscible. Hence at $\beta=2.5$ should we not expect the components to be fully miscible. ? Following on from this logic; for $\beta=3$ the components are only miscible for $\Delta_{mix} G/nRT<0$ so the maxima represents a phase separation between the two components. Who is correct ? ![enter image description here](http://i.stack.imgur.com/ruQbE.jpg)


title: Descent equation and anomaly polynomial

text: I am just reading Ryu, Moore and Ludwig's paper on classifications of topological insulators and quantum anomaly. They are trying to relate the quantum anomaly as a signal of the presence of a topological phase in higher dimension. However, the descent equation they used in their paper has the sign different from the usual descent equations. I am wondering why do they need to change the sign?


title: Reducibility of tensor products of Lorentz group representations

text: Consider the statement: (34.29 in Srednicki's QFT text) $$\tag{34.29} (2,1)\otimes(1,2)\otimes(2,2)~=~(1,1)\oplus\ldots$$ Where of course, $(a,b)$ label representations of Lorentz group in the usual way. In particular, it says this tensor product of the group representions is completely reducible, into the direct sum of $(1,1)$ with some other representations. Why is this so? It would be helpful if someone could point me to good literature or lecture notes on this- my understanding of representation theory and Lie groups & algebras is at the level of arXiv:math-ph/0005032.


title: Infinitely charged wire and Differential form of Gauss' Law

text: I have tried calculating the potential of a charged wire the direct way. If lambda is the charge density of the wire, then I get $$\phi(r) = \frac{\lambda}{4 \pi \epsilon_0 r} \int_{-\infty}^\infty \frac{1}{(1+z^2/r^2)^{1/2}} dz.$$ But this comes to $+\infty$ unless I am doing the calculation wrong. Why doesn't this work the direct way? Also, is it possible to calculate the potential of a charged wire using Gauss' differential law? What about in the case of an infinite charged sheet? Or does Gauss' differential law only apply to charged volumes?


title: Why/How is this Wick's theorem?

text: Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$. $$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$ $$ ~+~ 4<0|T(\phi(x)\phi(0))|0>:\phi(x)\phi(0): ~+~ :\phi^2(x)\phi^2(0):$$ It would be great if someone can help derive the above expression - may be from scratch - and without outsourcing to Wick's theorem - and may be help connect as to why the above is related (equal?) to the Wick's theorem?   * Isn't the above also known as OPE (Operator Product Expansion)? If yes, then is there at all any difference between OPE and Wick's theorem? Is there a systematic way to derive such OPEs?    * Can one help extend this to Fermions?


title: As the universe expands, why do some things stretch but not others?

text: I got into watching a video on Olbers' Paradox a few days ago, and from there read about the origins of the universe, its expansion, and so on... it's always fascinated me, but now something about it bothers me. I've heard many analogies about it (the dot-balloon, the raisin bread loaf, and others), but none really seem to explain this question. (This comes close, but dances around the answer more than explain it.) At the beginning of the universe as we know it, the universe itself was very small, so all the stars giving off light would have made it very bright (16:29 in this video). Since that time, the wavelength of that light has been stretched (17:01, same video). I found a few explanations saying that space itself stretched (here; described as "ether" in the article), which would stretch out the wavelengths. But here's what bothers me: If space is stretching out, redshifting all the light soaring around our universe, why are we not stretching? Theoretically, the universe is expanding an incredible amount faster than the speed of light, and the edge of the universe is an unimaginably large number of megaparsecs away from us. But should we not notice some of the stretch here, too? That is to say, if the light in space (the "ether", though I'm not fond of that term) is stretching out, why is everything on Earth still the same size as it was a hundred years ago? Is it stretching uniformly, but we are just unable to notice such a small stretch? Or does mass have some property that space and light do not, that prevents it from stretching out? I've also heard about time stretching, too; does this have an impact on it?


title: Flux through a conduting cylinder?

text: > _A point charge of magnitude $Q$ is placed inside a conducting cylinder of > length $L$ and radius $R$ at its centre. What is the flux through the > cylinder?_ I know that I have to use Gauss Law here but my concern is: > _Since it is a conducting cylinder, there would be equal and opposite > charges induced on its surface. How do I account for these charges?_ Help would be appreciated.


title: Bass and acute tones. Perceiving their direction

text: In a modern sound system, you put the sub-woofer somewhere on the floor, not worrying much about its position, but putting the loud speaker around my ears, to generate a sound-surround effect. I wonder why we can perceive the direction an acute tone is coming, but not so well the direction of a bass tone. Is our perception tuned like that? Or is something in the waves that make them different?


title: Bending of a spline

text: The introduction section on this page states: "the spline will take a shape that minimizes the bending", while showing the formula for curvature of a real curve. Question is, what is the formula for the bending that is to be minimized, the curvature is a function rather than a number?! Is it some integral or something?


title: The impossible possibility - weightless mass

text: I have a hypothetical question about inertia. Let's say I have an object with inertial mass a ton (2,000 lbs.), and it is sitting in my front yard, for instance. If it would suddenly become immune to gravity, i.e its gravitational mass becomes zero, what would happen to it? Now forget about whether this is actually possible, because science at its best tries to disprove itself. I just need to know what would happen to this block of iron sitting in my front yard if it would lose gravity. **Edit:** By immune to gravity, I mean that it no longer is affected by gravity. It still has the same mass and everything else, it just isn't affected by gravity anymore. I do have a reason for asking this. I said "ignore whether it's actually possible" because I know that everyone accepts it as fact that gravity can't be messed with (at least to my knowledge). However, I am the type that ask what if questions, so I asked "What if we could make an object not affected by gravity: What would happen to it?" Translated into technical terms (from a comment): In the context of Newtonian mechanics, let there be a test particle with inertial mass $X$ but no gravitational mass, i.e., the test particle does not gravitate. The test particle is released from my front yard at t = 0. Describe the motion of the test particle. Also, if you give technical answers (a good idea), please explain them in layman's terms, if that isn't asking too much. I more or less understand technical answer according to how they read in layman's terms.


title: How to calculate frequency in the following question

text: A police car with a siren of frequency $8$ kHz is moving with uniform velocity $36 \mathrm{km}\, \mathrm{hr}^{-1}$ towards a tall building which reflects the sound waves. The speed of sound in air is $320 \mathrm{m}\, \mathrm{s}^{-1}$. The frequency of the siren heard by the car driver is?   * A: $8.50$ kHz   * B: $8.24$ kHz   * C: $7.75$ kHz   * D: $7.50$ kHz


title: Why do electrons make a Fermi sphere?

text: In Sommerfeld theory for metals, after determining all of the possible levels for a single electron, one says that we build up a state for a system with $N$ electrons by filling up those levels, beginning from that which has the lower energy to those which have more energy, in a crescent way. In fact, one call this configuration a Fermi sphere. My question is: once a level is an eigenstate for the Hamiltonian operator of the problem, why can not the electrons be each one at any random level (respecting, of course, Pauli Exclusion's Principle), not necessarily making a Fermi sphere?


title: Ghosts in Pauli Villars Regularization

text: I'm trying to understand how Pauli Villars Regularization works. I know we add ghost particles, but I want to see more precisely. To do this, we'll work with $\phi^3$ theory. The Lagrangian is $$ {\cal L} = \frac{1}{2}(\partial_\mu\phi)^2 - \frac{1}{2} m^2 \phi^2 + \frac{\lambda}{3!}\phi^3 $$ We wish to calculate the correction to the scalar propagator. We know that this integral diverges. We then introduce a Pauli Villars ghost in the Lagrangian $$ {\cal L}' = - \frac{1}{2} (\partial_\mu \phi')^2 + \frac{1}{2}M^2 \phi'^2 + \frac{\lambda}{3!}\phi'^2 \phi $$ This Lagrangian has a negative norm. We can see this by calculating the contribution of the free part to the Hamiltonian $$ {\cal H}' = -\frac{1}{2} {\dot \phi'}^2 -\frac{1}{2} \pi'^2 - \frac{1}{2} M^2 \phi'^2 $$ Because it has a negative norm, such a particle is called a GHOST PARTICLE. Now, from what I understand from current text, this field has a propagator $$ D_F'(x-y) = \int \frac{d^4p}{(2\pi)^4} \frac{e^{ i p (x-y) } } { p^2 - M^2 + i \epsilon} $$ This is where I'm having so much trouble. How can we prove that this is the propagator? I'm trying to use the usual method to find the propagator, and I seem to be stuck. Any help?


title: Question about resolving forces

text: Say, in the example below, the weight $mg$ of the object is $800N$. To find $R$, the conventional method is to use $R\sin(28^\circ) = 800$. But why isn't it possible to use instead the component of the weight in the direction of $R$ to find $R$? That component would be $ 800\cos(62^\circ)$, but equating it to $R$ arrives at a small, incorrect answer for $R$. ![enter image description here](http://i.stack.imgur.com/au2RP.png)


title: What prevents bosons from occupying the same location?

text: The Pauli exclusion principle states that no two fermions can share identical quantum states. Bosons, one the other hand, face no such prohibition. This allows multiple bosons to essentially occupy the same space, a phenomenon that has been theorized responsible for superconductivity. Bosons do not, however, occupy exactly the same space as can be readily observed by the fact that a Bose-Einstein condensate does not collapse into a singularity. Both of the rather unusual examples cited above are inherent to low-energy systems. A large collection of $^{12}$C (e.g. in a diamond) does not exhibit particularly unusual behavior. This leads me to hypothesize that the energy distribution of the system is largely responsible for keeping bosons apart. Given the rather basic nature of the question, however, I figured someone here would likely know the "correct" answer. So, what keeps bosons from occupying the same location?


title: Density operator in second quantization

text: I would want to understand why the density operator in second quantization takes the form: $$\rho_\sigma(\mathbf{r})=\Psi_\sigma^\dagger(\mathbf{r})\Psi_\sigma(\mathbf{r})?$$ Is this a definition or can we derive it from some formula?


title: Which limit for matsubara frequency sum?

text: in the context of a simple toy problem for Feynman path integrals, I consider a two-site Hubbard model for spinless fermions. I expand the path integral to first order in the interaction $V$, which means that I have to compute an "average" of the type $$\left\langle \bar \psi_1(\omega_n) \bar \psi_2(\omega_m) \psi_2(\omega_p) \psi_1(\omega_q) \right\rangle$$ with respect to the non-interacting action. Since the Grassmann numbers for sites $1$ and $2$ are different numbers, this average immediately factors, and I'm left with two-variable averages, which give me just the non-interacting Matsubara Green's functions. But now I have to calculate a Matsubara sum of the type $$\frac{1}{\beta} \sum_n \frac{1}{i\omega_n - \epsilon}$$ where $\epsilon$ is just a constant. From searching in the literature, I know that contour integration is the way to go, but unfortunately this sum is ambiguous and I have to introduce a factor $e^{-i\omega_n \tau}$ with $\tau \rightarrow 0^+$ or $\tau \rightarrow 0^-$, and the result of the contour integration will be quite different depending on which limit I take. My question now is, how do I know what limit to take? And is there an easier way of carrying out the sum over frequency?


title: Is there anything to prevent paired-up neutrons from a complete overlap

text: The reason "neutrons don't overlap", as DarenW explained it, has to do with intricate forces at play that take into account the spins, iso-spins and symmetry of the wavefunctions. However, assume I have two paired-up neutrons. I assume they should behave as bosons now. What prevents them from overlapping? Are the same forces (that are beyond my current knowledge) in play?


title: What's best shape for linear concentrated solar panel?

text: I found 3 shapes which are suitable for concentrated solar panel.      1\. **Semicircle** \- first thing that comes to ones mind.   It's a simple shape, the position of the focal point is obvious.      2\. **Parabolic mirror**   A bit more difficult shape.   ![Parabolic mirror](http://i.stack.imgur.com/qWq30.png)      3\. **Compound parabolic mirror**   ![Compound Parabolic mirror](http://i.stack.imgur.com/VUupy.jpg)   Compound parabolic mirror has an advantage over the parabolic mirror in acceptance angle. Collector doesn't necessarily need to track sun, because it's able to receive sunlight from a range of angles.   **Is there any other shape, that might have interesting properties for the purpose of solar collector?**   I found many DIY designs with semicircle mirrors. And many scientific papers with parabolic mirrors.   **What is the advantage of parabolic mirrors are over semicircle mirrors?**


title: POVM advantage in state discrimination

text: Suppose we are given the task of discriminating, with minimum error, between a set of states $\\{|\psi_1\rangle,|\psi_2\rangle,\ldots,|\psi_N\rangle\\}$. In other words, we are given an unknown state form this set and our task is to output which of the states we were given. Let $P_{err}$ be the probability of incorrectly identifying the state. Under what circumstances is this probability minimized by using a projective measurement?


title: The most general procedure for quantization

text: I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me): > _A physical system is usually described in terms of states and observable. > In the Hamiltonian framework of classical mechanics, the states form a > symplectic manifold $(M,\omega)$ and the observables are functions on $M$. > The dynamics of a (time invariant) system is a one parameter group of > symplectic diffeomorphisms; the generating function is the energy or > Hamiltonian. The system is said to be free if $(M,\omega)$ is an affine > symplectic space and the motion is by a one-parameter group of symplectic > transformations. This general descriptions applies to any system that > includes classical particles, fields, strings and other types of objects._ The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:   1. Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?    2. What is the general procedure for such a construction, given $M$ and $\omega$?    3. For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)   4. Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?   5. I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)   6. What is meant by a 'free' system as given above?   7. By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?   8. In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system? I have asked a lot of long questions, so please answer as many as you can and link relevant articles.


title: Coadjoint orbits in physics

text: I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.


title: Relation between Radiance and Irradiance

text: I know the radiance is expressed as $$\text{[Radiance]} = \frac{W \cdot m^2}{\text{sterad}}$$ and $$\text{[Irradiance]} = W \cdot m^2$$ But what's the relation between theese two quantities? Is the irradiance commonly used to referring to power reflected on a surface? And the radiance from a direct source?


title: Potentials in Helmoholtz Theorem

text: I read about Helmholtz theorem on the Griffiths (Introduction to electrodynamics) and in the appendix B it shows how to use the Helmholtz theorem to determine the field **F** starting from the potential. The book says that: Suppose we are told that the divergence of a vector function **F** (r) is a specified scalar function D(r): $\nabla \cdot \textbf{F}=D(r)$ and the curl of **F** (r) is a specified vector function **C** (r): $\nabla \times \textbf{F}=\textbf{C}$ For consistency, **C** must be divergenceless, $\nabla \cdot \textbf{C}=0$ because the divergence of a curl is always zero. Question: can we, on the basis of this information, determine the function **F**? If D(r) and **C** (r) go to zero sufficiently rapidly at infinity, the answer is yes, as I will show by explicit construction. I claim that $\textbf{F}=-\nabla U + \nabla \times \textbf{W}$ where $U(\textbf{r})=\frac{1}{4\pi}\int \frac{D(\textbf{r'})}{\textbf{r-r'}}d\tau'$ and $\textbf{W(r)}=\frac{1}{4\pi}\int \frac{\textbf{C(r'})}{\textbf{r-r'}}d\tau'$ Can someone show me how to obtain the potentials formulas written above?


title: General expression of the redshift: explanation?

text: In some papers, authors put the following formula for the cosmological redshift $z$ : $1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$ where :   * $S$ is the source and $O$ the observer   * $g_{\mu\nu}$ is the metric   * $k^{\mu}=\frac{dx^{\mu}}{d\lambda}$ is the coordinate derivative regarding the affine parameter $\lambda$   * $u^{\nu}$ is the 4-velocity of the cosmic fluid My first question is : according to the Einstein summation, is it a fraction of sums $\left(\frac{\sum_{\mu\nu}X}{\sum_{\mu\nu}Y}\right)$ or a sum of fractions $\left(\sum_{\mu\nu}\frac{X}{Y}\right)$ ? My second question (and more important) is : where does this formula come from ? Where can I find a "demonstration"/"derivation"/"explanation" of this ?


title: Wick rotation and special relativity

text: CMIIW, but as I understand it, Wick rotation replaces the Minkowski basis (t,x,y,z) with the Euclidean basis (it,x,y,z). Suppose that $t_2=t_1 \cosh \beta+x_1 \sinh \beta$ and $x_2=t_1 \sinh \beta+x_1 \cosh \beta$. If a Euclidean basis is defined with $\tau_1=i t_1$, then a coordinate vector with a real value $\tau_1 \neq 0$ will have an imaginary component $t_1$ and a real component $x_1$. However, converted to $t_2$ and $x_2$, this separation of real and imaginary coordinates doesn't apply. So even if two coordinate frames have the same origin and the same tangent spacetime, their Wick-rotations span different spaces. Is there a Cauchy-like criterion that would make this redundant? If so, how does it follow from known physics? If not, does using Wick rotation imply postulating complex space?


title: Is a sonic boom a one-time bang or a continuous noise?

text: Is a sonic boom a one-time bang, caused when an object initially goes supersonic, or is it a continuous noise emitted by the object as it's travelling? To be more specific, will an aircraft which is continuously accelerating cause a boom only over the point where it first transitions to supersonic, or does it cause a 'boom' to be heard over the whole length of it's supersonic journey by all of the observers along it's route (the same way that a fast boat will create a bow wave which follows it wherever it goes)?


title: Correlation between speed, distance, mass, etc between a planet and a star (or moon and planet)?

text: While playing around with some little scripts, generating solar systems, I started to wonder about the correlation between the different objects. Up until now, distance, mass, size and speed are randomly generated. Are there any 'common rules' on how some of theese objects correlate? Are planets close to a star always smaller and faster, and planets far away usually larger and slower? Is the rotation speed affected? And how about the correlations between moons and their planet? At the end of the day, I'm triyng to make the the generation a bit more accurate / realistic. Thanks in advance! Sorry for the english, it's a second language for me.


title: mode-locking a laser on prime-powers' frequencies

text: Being rather a theoretician than an experimental physicist, I have a question to the community: Is it experimentally possible to mode-lock a laser (fixed phase relationships between the modes of the laser's resonant cavity) in a way that the longitudinal modes of the cavity would be exclusively from a discrete set of frequency $\\{p^m\\}$, where $p$ a prime number and $m$ a positive integer? If yes, how? If not, why? For instance: $\\{2^m\\}$ or $\\{2^m,3^m\\}$ or ... Thanks


title: How is everything a field

text: I've heard before that everything in physics can be thought of as either a field, or its excitation. Is there some intuitive explanation of how I can look at gravity, light, electromagnetism, etc as a field/excitation?