title: Boundary conditions for Couette flow

text: I'm trying to reproduce a result from a paper (T. Thatcher, Boundary Conditions for Grad's 13 moment equations, equation (32), page 6), however, I haven't been able to do so. Hopefully someone can provide some help. The general problem concerns the Couette flow: A flow passes through two infinite parallel plates at distance $L$ move with velocity $v_{w}^{0}$ and $v_{w}^{L}$ ($w$ refers to the wall) relative to each other in their respective planes, each with temperature $\theta_{w}^{0}$ and $\theta_{w}^{L}$. The velocity $v$ of the flow only depends on the $y$ coordinate, that is the coordinate perpendicular to the direction of the flow. After solving the Navier-Stokes-Fourier equations for this problem, you can find the following equations $$\frac{dv}{dy}=a=\text{constant}, \qquad \frac{d^{2}\theta}{dy^{2}}=-\frac{2\text{Pr}}{5}a^{2} \qquad \qquad (1)$$ which are linearized. Using as boundary conditions \begin{array} vv(L)-v_{w}^{L} = -\frac{2-\chi}{\chi}\alpha_{1}\sqrt{\frac{\pi}{2}}\text{Kn}\; a, \qquad \ \ v(0)-v_{w}^{0} = \frac{2-\chi}{\chi}\alpha_{1}\sqrt{\frac{\pi}{2}}\text{Kn}\; a \qquad (2) \end{array} and \begin{array} \theta \theta(L)-\theta_{w}^{L} = -\frac{2-\chi}{\chi}\frac{5\beta_{1}}{4 Pr}\sqrt{\frac{\pi}{2}}\text{Kn}\frac{d\theta}{dy}, \quad \ \ \theta(0)-\theta_{w}^{0} = \frac{2-\chi}{\chi}\frac{5\beta_{1}}{4 Pr}\sqrt{\frac{\pi}{2}}\text{Kn}\frac{d\theta}{dy} \qquad (3) \end{array} the solution for the velocity should be $$v = \frac{v_{w}^{L}}{2}+a\left(\frac{y}{L}-\frac{1}{2}\right) \qquad \text{with} \quad a = \frac{v_{w}^{L}-v_{w}^{0}}{1+\frac{2-\chi}{\chi}\sqrt{2\pi} \alpha_{1}\text{Kn}} \qquad \qquad (4)$$ Don't be concerned about $\alpha_{1}$ or $\beta_{1}$; they are correction terms. $\text{Kn}$ and $\text{Pr}$ are, as usual, the Knudsen number and the Prandtl number, respectively. For now, I'm interested only in the solution for the velocity (equation $(4)$) **Note:** First and second attempts are wrong. **First attempt:** Use equation $(2)$ as $v-v_{w}^{L}=-\frac{2-\chi}{\chi}\alpha_{1}\sqrt{\frac{\pi}{2}}\text{Kn}\;a L$ and $v-v_{w}^{0}=\frac{2-\chi}{\chi}\alpha_{1}\sqrt{\frac{\pi}{2}}\text{Kn}\;a L$. Subtract the first from the second to obtain $a = \displaystyle \frac{v_{w}^{L}-v_{w}^{0}}{\frac{2-\chi}{\chi}\alpha_{1}\sqrt{2\pi}\text{Kn}L}$, and from equation $(1)$ for the velocity, $v = ay+\text{constant}$, evaluating at the plate localized in $L$ we have $v_{w}^{L}=aL+\text{constant}$, which gives us the constant, and _a_ solution but it's not the correct result. **Second attempt:** I think this can be useful: Take equation $(2)$ and solve for $a$: $$a=\frac{-v_{w}^{L}+v_{w}^{0}}{-\frac{2-\chi}{\chi}\alpha_{1}\sqrt{2\pi}\text{Kn}L}$$ Then we use the first part of equation $(1)$, so we have $$\frac{dv}{dy}=\frac{-v_{w}^{L}+v_{w}^{0}}{-\frac{2-\chi}{\chi}\alpha_{1}\sqrt{2\pi}\text{Kn}L}$$ After integrating this equation, we evalute the velocity at the boundary $L$, which allows us to find the constant of integration. Substituting this constant, we obtain the following result $$v-v_{w}^{L}=\frac{v_{w}^{L}-v_{w}^{0}}{\frac{2-\chi}{\chi}\alpha_{1}\sqrt{2\pi}\text{Kn}}\left(\frac{y}{L}-1\right)$$ which is $$v-v_{w}^{L}=a\left(\frac{y}{L}-1\right)$$ Not quite the result I'm looking for, but maybe this is the right path. **Third attempt:** Taking $\displaystyle \frac{dv}{dy}=b$, it follows $v(L)=a L +b$ and $v(0)=b$. Using the boundary conditions to evalute $v(L)$ and $v(0)$ we obtain $$v_{w}^{0}+\displaystyle\frac{2-\chi}{\chi}\alpha_{1}\sqrt{\frac{\pi}{2}}\text{Kn}a = b$$ and $$v_{w}^{L}-\displaystyle \frac{2-\chi}{\chi}\alpha_{1}\sqrt{\frac{\pi}{2}}\text{Kn}a = aL+b$$ After solving this system of equations: $$a = \frac{v_{w}^{L}-v_{w}^{0}}{L+\text{Kn}\sqrt{2\pi}\alpha_{1}(\frac{2-\chi}{\chi})}$$ $$b = v_{w}^{0}-\frac{\left(v_{w}^{0}-v_{w}^{L}\right)\text{Kn}\sqrt{\frac{\pi}{2}}\alpha_{1}\left(\frac{2-\chi}{\chi}\right)}{L+\text{Kn}\sqrt{2\pi}\alpha_{1}\left(\frac{2-\chi}{\chi}\right)}$$ Substituting back in the original equation $$v(y)-v_{w}^{0}=a\left(y-\text{Kn}\sqrt{\frac{\pi}{2}}\alpha_{1}\left(\frac{2-\chi}{\chi}\right)\right)$$


title: Double slit experiment near event horizon

text: What happens if you perform a double slit experiment near an event horizon, if one of the slits is outside, one is inside the event horizon?


title: Temperature; Why A Fundamental Quantity?

text: Temperature is just an indication of the combined property of mass of the molecules and their random motion. We can explain no effective energy transfer between two conducting solid bodies in contact by the condition in terms of mass of molecules and their speeds such that due to collisions of molecules of two bodies net energy transfer between two bodies is zero. But it would be a complex calculative work to derive this condition so we use the temperature scale just as a parametre to easily determine the condition of no net energy transfer between conducting solids. So it does not denote any new property of body. So why do we say it a fundamental quantity?


title: What is the shape of a clamped bent bar?

text: How would I figure out the Cartesian graph that describes a bar clamped flat for a length on one end with downward force being applied to the other? I have an idea that the bar will try to average out the stress over the free length. Does it in fact evenly distribute stress over the whole free length? If so, can I use that fact to calculate curvature at each point to derive a Cartesian graph?


title: "Quantum version" of Maxwell-Boltzmann speed distribution

text: Is there a "quantum mechanical version" of the classical Maxwell-Boltzmann speed distribution for monatomic ideal gases? I'm quite new to thermodynamics, but I just wanted to know. If there is such a distribution, can it be used to calculate the average speeds (such as the most-probable speed, or rms speed classically) of particles, given a mass? Or is it more subtle, dealing with quantum states and such?


title: Electrolytes and electric field

text: Let me assume, that I have an arbitrary electric field. Is there any way to determine what happens to this electric field if it is appolied to a charge in let me say water with natrium chloride in it? I guess there is some kind of response to the electric field by the ions, so that it will just "die out", as the electrolytes will screen it. Is there an equation that would confirm this idea?


title: What is an intuitive explanation using forces for the equatorial bulge?

text: The earth is not a sphere, because it bulges at the equator. I tried fiddling with centripetal force equations and gravity, but I couldn't derive why this bulge occurs. Is there (a) a mathematical explanation using forces (not energies) and (b) a simple intuitive explanation to explain to others why the bulge occurs?


title: Does the Milky Way have dark matter satellite galaxies?

text: This recent paper by Weinberg et al. discusses that one potential problem with our current model of Cold Dark Matter (CDM) is that is predicts a greater number of satellite galaxies for the Milky Way than are actually observed, and satellite galaxies with larger masses than those the Milky Way has (although the paper does say that many of the Milky Way's satellite galaxies may be too dim for us to currently observe). I know that at least dark matter galaxies are believed to have been observed (see this article). Is there any observational or other evidence for the existence of dark matter satellite galaxies to the Milky Way? And how have they been found? Edit: As has been pointed out, this is a question we likely don't have the answer to yet (otherwise there wouldn't be recent papers posing the question still), so perhaps a better question would be how could/would you find dark matter satellite galaxies of the Milky Way?


title: Violation of the Normalization Constraint?

text: Say we have two qubits $|a\rangle$ and $|b\rangle$ both initialized to $|0\rangle$. We then apply the rotation gate $R_{x}(\frac{\pi}{2})$ of matrix representation $\left( \begin{array}{} \frac{1}{\sqrt2} & \frac{-i}{\sqrt2} \\\ \frac{-i}{\sqrt2} & \frac{1}{\sqrt2} \end{array} \right)$ to get $|a\rangle = |b\rangle = \left( \begin{array}{} \frac{1}{\sqrt2}\\\ \frac{-i}{\sqrt2}\end{array} \right)$ $|a\rangle|b\rangle$ is then $\left( \begin{array}{} \frac{1}{2}\\\ \frac{-i}{2}\\\ \frac{-i}{2}\\\ \frac{1}{2}\end{array} \right)$ A controlled rotation gate $R_{x}(\frac{\pi}{2})$ of matrix representation $\left( \begin{array}{} 1 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 &\frac{1}{\sqrt2} & \frac{-i}{\sqrt2} \\\ 0 & 0 &\frac{-i}{\sqrt2} & \frac{1}{\sqrt2} \end{array} \right)$ is then applied to $|a\rangle|b\rangle$ to get $|a\rangle|b\rangle = \left(\begin{array}{} \frac{1}{2}\\\ \frac{-i}{2}\\\ \frac{-i}{\sqrt2}\\\ \frac{1}{\sqrt2}\end{array} \right)$ But the sum of the squares of the amplitudes is $\frac{3}{2}$, violating the normalization constraint. Where did I go wrong?


title: Significance of Poles of Correlation Function in QFT

text: In QFT, specifically in scattering processes, what is the physical significance of the poles / residues of the $N$-point correlation function? And why?


title: Can the weak force create a bound state?

text: My understanding is that the exchange of Z bosons could yield an attractive or repulsive "force" between two fermions. For most combinations the electromagnetic or strong interactions will take place. However, if we have two neutrinos, or a neutrino and a charged lepton, only the weak interaction comes into play. Are these possible bound states?


title: Speed of Magnetic Signal over Large Distance

text: If I had a very strong magnet on Earth and a very sensitive compass on Mars (just using planets to illustrate large distance), how long would the compass take to notice if I turned the magnet 180deg? I assume it can't beat the speed of light. Do the outer reaches of the magnetic field move as a rigid body with the turning magnet or do they lag in motion?


title: How does one detect a single photon?

text: I understand the double slit experiment up until the point that we begin "detecting" single photons. What does it mean to detect. You cant place a camera in the slit because that would capture the photon just like the photosensitive plates would. I've also heard that you can do the double slit experiment with electrons. Perhaps you can use the attribute of charge to detect the electron from a distance. I have read many times of how the double slit experiment works, but never the specific details of how the experimenters manage to shoot single particles and then detect them moving through the slit. Any links describing the equipment used in these experiments would be appreciated


title: What are the limits of validity for the magnetic field of a solenoid?

text: 1) The field outside a solenoid is approximated to be zero, because of opposites points "cancelling out". Does this approximation of the field being almost zero become worse as the diameter of the solenoid increases? (since the opposite sides are further apart) I'm looking for an answer for two cases: 1) Only the diameter increases, but all other dimensions remain the same. 2) All dimensions increase proportionally, keeping the aspect ratio the same. 2) Are solenoids approximated as a series of perfect circles (since opposite sides cancel only in a circle)? And consequently shouldn't a stretched out solenoid's field be completely different since its nothing like a series of circles? **EDIT:** ![enter image description here](http://i.stack.imgur.com/IHvMQ.png) The field at P is the superposition of the fields from the opposite "points". If the diameter of the solenoid increases, the bottom "points" are further away from $P$, so their contribution to the field decreases. Therefore the assumption that the fields from the top points cancel with the bottom points becomes falser/less valid.


title: Does charge of a metal charged by induction determine by which ends of the metal is grounded to?

text: Does charge of a metal charged by induction determine by which part of the metal is ground to? I draw a diagram to make it simple to understand:![enter image description here](http://i.stack.imgur.com/dyAHZ.png) Right diagram: When ground is touched with the positive side of the metal, electrons flow to the negative and the metal become negatively charged. Left diagram: When ground is touched with the negative side of the metal, electrons flow to the ground and the metal become positively charged. Is this true? Because opposite charges repel.


title: How do inductors produce current?

text: This is how I understand batteries and capacitors. The terminals produce an electric field which causes the electrons to move in random directions. The surface charge builds up causing new electric fields to be created, which forces the electrons to move in the direction of the wire, creating current. But with inductors, there is no negative or positive terminal (as in an end with a surplus or deficit of charge), so, on a microscopic level, how does it produce current?


title: Are there forces which do not involve a change in momentum?

text: I am familiar with the equation $$\vec{F}=m \vec{a}$$ I am wondering as to whether it is possible for something to exert a force on another object without changing the momentum of said object. My question is: is force defined as a change of momentum? If that is the case, my question is pointless. But if it is not. What is the underlying framework? Also, when some phenomena causes the spins or angular momentum of subatomic particles to change, does it mean that the phenomena caused the particle to accelerate or decelerate? What is force in the framework of quantum mechanics? Does the definition of force only apply in classical mechanics (Sidenote: I am not very familiar with quantum mechanics. I apologize beforehand...)


title: Neutrino annihilation and bosons

text: Would it be theoretically possible for a high energy neutrino and another high energy anti-neutrino to annihilate into a boson?   Which boson(s) would be possible theoretically?   \- one or more photons ?   \- the Higgs boson ?   \- Z boson ?


title: If the wave function can be collapsed, can we collapse the particle function?

text: No doubt it's been questioned before so what is the theory or experiment? If there is an experiment.


title: Expression of kinetic energy in polar coordinates

text: Expression for kinetic energy in Cartesian coordinate: ![](http://chart.apis.google.com/chart?cht=tx&chl=T%3D%20%5Cfrac%7B1%7D%7B2%7Dm%20%5Cdot%7Bx%7D%5E2%20%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20m%20%5Cdot%7By%7D%20%5E2%0A%0A) Expression for kinetic energy in polar coordinate (applying the transformation of coordinates): ![](http://chart.apis.google.com/chart?cht=tx&chl=T%3D%20%5Cfrac%7B1%7D%7B2%7D%20m%5Cbig\(%5Cfrac%7Bd%7D%7Bdt%7D\(r%5Ccos%20%5Calpha\)%5Cbig\)%20%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20m%20%5Cbig\(%5Cfrac%7Bd%7D%7Bdt%7D\(r%5Csin%20%5Calpha\)%5Cbig\)%5E2) Why can't we express it in the following terms by taking the time derivative of each degree of freedom: ![](http://chart.apis.google.com/chart?cht=tx&chl=T%3D%20%5Cfrac%7B1%7D%7B2%7D%20m%20%5Cbig\(%20%5Cdot%7Br%7D%20%5Cbig\)%20%5E2%2B%20%5Cfrac%7B1%7D%7B2%7Dm%20%5Cbig\(%20%5Cdot%7B%20%5Calpha%20%7D%20%5Cbig\)%20%20%5E2)


title: Why are there no particles in conformal theories?

text: In Matt Strassler's recent post (here) he makes the statement that scale invariant (I assume he means conformally invariant, more generally) theories have no particles in them. What's the reason for this? What are the technical and heuristic explanations, apart from what he mentions in his post?


title: How does the Fraunhofer irradiance distribution look for a double slit aperture of different lengths?

text: How does the Fraunhofer irradiance distribution look for a double slit aperture with $d$ = integer multiples of $b$?


title: Is the index of refraction dependent on the wavelength of light?

text: I just read in some class notes for a crystallography class that there are no refractive lenses for X-rays because the index of refraction of most materials is close to 1. Is the index of refraction dependent on the wavelength of the light then? Like, a material has different indices for different types of electromagnetic waves? Thank you.


title: Symmetry breaking

text: What is a good place to learn the details of symmetry breaking? What I am looking for is a more serious exposition than the wiki-article, which explains the details, especially the mathematical part, but at the same time less detail than the typical quantum field theory book. Preferably I would like to see a clean made up example with all the details, but not necessarily a real example, what one can see in most books, which naturally has a lot of details that are there because that's what the world is, and not to illustrate the idea. I hope it is clear what I am looking for. EDIT: Just to clarify. I am looking for a reference, but if somebody is willing to write an example here, I would be more than happy.


title: Modern and complete references for the $k\cdot p$ method?

text: I've recently started studying the $k\cdot p$ method for describing electronic bandstructures near the centre of the Brillouin zone and I've been finding it hard to find any pedagogical references on the subject. In every paper I read there's simply a statement of the Hamiltonian and it is followed by the definition of the matrix elements. However, nowhere have I been able to find a source that derives (in full) the actual matrix. I've been reading the original 1955 paper by Luttinger and Kohn, as well as Physics of Photonic Devices by S.L. Chaung and the book on the $k\cdot p$ method by Van Voon, but in all cases, only a run up is given and the calculation of the actual matrix elements is not given. This may not seem like such a big deal, but it is to a first timer when you have to try and work through the calculations yourself. With no guide, I have been able to derive some of the matrix elements, but that's using my own intuition as to which matrix elements of the momentum operator vanish and/or are equal to others and I'm not sure that I'm right. A single pedagogical reference would be useful to confirm or dispel some of the things I believe to be true and would be very helpful for me to continue.


title: Textbook disagreement on geodesic deviation on a 2-sphere

text: Apologies if I have this completely wrong (and for the general long- windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0$$ to find the geodesic deviation on the surface of a unit 2-sphere. My question is are the following calculations correct? I start with the Riemann tensor components: $R_{\phantom{\theta}\phi\theta\phi}^{\theta}=\sin^{2}\theta$, $R_{\phantom{\theta}\phi\phi\theta}^{\theta}=-\sin^{2}\theta$, $R_{\phantom{\theta}\theta\theta\phi}^{\phi}=-1$, $R_{\phantom{\theta}\theta\phi\theta}^{\phi}=1$. Let $u^{\sigma}\equiv\frac{dx^{\sigma}}{d\lambda}$. Then expand out the Riemann components to get: $$\left(R_{\phantom{\mu}\theta\theta\theta}^{\mu}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\theta\phi}^{\mu}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\theta\theta}^{\mu}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\theta\phi}^{\mu}u^{\phi}u^{\phi}\right)\xi^{\theta}+\left(R_{\phantom{\mu}\theta\phi\theta}^{\mu}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\phi\phi}^{\mu}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\phi\theta}^{\mu}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\phi\phi}^{\mu}u^{\phi}u^{\phi}\right)\xi^{\phi}.$$ Set $\mu=\theta$ to give $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}+\left(R_{\phantom{\mu}\theta\theta\theta}^{\theta}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\theta\phi}^{\theta}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\theta\theta}^{\theta}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\theta\phi}^{\theta}u^{\phi}u^{\phi}\right)\xi^{\theta}+\left(R_{\phantom{\mu}\theta\phi\theta}^{\theta}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\phi\phi}^{\theta}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\phi\theta}^{\theta}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\phi\phi}^{\theta}u^{\phi}u^{\phi}\right)\xi^{\phi}=0$$ $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}+\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\phi}\right)\xi^{\theta}-\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\theta}\right)\xi^{\phi}=0$$ $$\frac{D^{2}\xi^{\theta}}{D\lambda^{2}}=\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\theta}\right)\xi^{\phi}-\left(\sin^{2}\theta\right)\left(u^{\phi}u^{\phi}\right)\xi^{\theta}$$ Set $\mu=\phi$ to give $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}+\left(R_{\phantom{\mu}\theta\theta\theta}^{\phi}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\theta\phi}^{\phi}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\theta\theta}^{\phi}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\theta\phi}^{\phi}u^{\phi}u^{\phi}\right)\xi^{\theta}+\left(R_{\phantom{\mu}\theta\phi\theta}^{\phi}u^{\theta}u^{\theta}+R_{\phantom{\mu}\theta\phi\phi}^{\phi}u^{\theta}u^{\phi}+R_{\phantom{\mu}\phi\phi\theta}^{\phi}u^{\phi}u^{\theta}+R_{\phantom{\mu}\phi\phi\phi}^{\phi}u^{\phi}u^{\phi}\right)\xi^{\phi}=0$$ $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}+\left(-1\right)\left(u^{\theta}u^{\phi}\right)\xi^{\theta}+\left(1\right)\left(u^{\theta}u^{\theta}\right)\xi^{\phi}=0$$ $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}-\xi^{\theta}\left(u^{\theta}u^{\phi}\right)+\xi^{\phi}\left(u^{\theta}u^{\theta}\right)=0$$ $$\frac{D^{2}\xi^{\phi}}{D\lambda^{2}}=\xi^{\theta}\left(u^{\theta}u^{\phi}\right)-\xi^{\phi}\left(u^{\theta}u^{\theta}\right).$$ My reason for asking is that Misner et al give the geodesic deviation equation (the one I use in my question) as $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\beta\alpha\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0,$$ whilst in another of my textbooks it's (spot the difference!) $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\phantom{\mu}\alpha\beta\gamma}^{\mu}\xi^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{d\lambda}=0,$$ with $\alpha$ being the _first_ lower index on the Riemann tensor and the upper index on the right-hand-side $\xi$. When I use the second equation in my 2-sphere calculations everything seems to come out to zero, which doesn't seem right. So I'm wondering if the second equation is wrong.


title: Alternative to Cooper pairing in superconductivity

text: BCS theory, in which electrons form Cooper pairs, has been successful in explaining conventional superconductivity. Physicists have adopted similar pairing mechanism in an attempt to explain high-temperature superconductivity (HTS). Is there any HTS theory that is not based on pairing? If not, what experiments preclude the existence of such a theory?


title: Ground states of Chiral Boson Theory with tunneling

text: I am reading this paper(pdf) and on page 11, the chiral boson theory on a cylinder is studied when both edges of the cylinder are brought in close proximity so that electron is allowed. Why is it that for large tunneling amplitudes, $g$, the field $\phi$ can be treated 'classically' in the sense that it gets pinned to the classical minima of the cosine to obtain the ground state energy? Could someone give a more detailed explanation?


title: Orbifold CFT of SU(2)/G and SO(3)/G

text: In this paper by Dijkgraaf, Vafa, Verlinde, Verlinde, orbifold CFT is discussed. In that paper, it outlined that orbifold CFT provides a way to generate the new theories from the old known ones. (i.e. "... it would be very interesting to devise operations that act on the space of RCFT's and generate new theories from a given one. As an example one can think of the construction of nondiagonal modular invariant combinations of characters, coset models, etc. Another operation of that kind is the concept of **an orbifold**.") In p.521, Table 6 of this paper, they provide an example of SU(2)/H$_8$ $\simeq$ SU(2)/($Z_2 \times Z_2$) orbifold. (here H$_8$ is a quaternion group.) **Question** : $\bullet$ Here I am interested in **orbifold CFT** of **SU(2)/G and SO(3)/G** kinds (with G as its normal subgroup) . Can someone provide some more examples and data (i.e. S and T matrix, fusions of quasi-particles) of **SU(2)/G and SO(3)/G** **orbifold CFT**? $\bullet$ What are their **central charges** $c$ of this 1+1D CFT? (e.g. what is the central charge of this SU(2)/D$_2$ model?) What is their **ground state degeneracy** of 2+1D bulk TQFT on the spatial $T^2$ torus? $\bullet$ If there is also information on its corresponding **twisted quantum doubles** , in the form of $D^\omega(G')$ of some certain group $G'$. It will be nicer. Reference and books are welcome, but please **offer summaries** instead of just giving Reference. (ps. Francesco yellow-book CFT Sec 17.8.4 has some discussions, but I could not find orbifold CFT of $SO(3)/D_n$ of $D_n$ dihedral group or $SU(2)/H$ of $H$ as a quaternion.)


title: Renormalization Group for non-equilibrium

text: For equilibrium/ground state systems, a (Wilson) renormalization group transformation produces a series of systems (flow of Hamiltonians/couplings $H_{\Lambda}$ where $\Lambda$ is the cut-off) such that long-wave/asymptotic behaviour of $H_{\Lambda}$ is the same as of $H_{\Lambda'}$ after rescaling by $\Lambda/\Lambda'$. The idea of this definition implies an exact starting point for RG formalisms, with technical details varying between the fields and approximation methods. (For examples, see arXiv:1012.5604 and Wikipedia article). Now, for **non-equilibrium condensed-matter systems** there is research direction aiming at generalization of the RG approach to a steady state, e.g., a voltage-biased strongly interacting quatum dot (or Kondo impuryity). For examples, see arXiv:0902.1446 and related references. I would like to understand the conceptional foundations for the non-equlibrium RG. > What is the definition of an RG transofrmation in a non-equilibirum, steady > state ensemble? I see a problem in the fact that the non-equilibirum desnity matrix which is used to define the problem is not a function of the Hamiltonian alone, thus it is not clear to me how is the effect of the change in the cut-off is split between the Hamiltonian (running couplings) and the density matrix (renormaltizaton of the boundary/external conditions?)


title: Physics in high lepton chemical potential

text: I consider zero temperature and high lepton number chemical potential $\mu$. This results in a neutrino (or antineutrino, depending on the sign of the potential) "sea" filling a Fermi sphere in momentum space. There are profound effects on particle stability In the $\mu << 0$ case, the neutron becomes stable. This is because it would produce an antineutrino upon decay but it would have to carry prohibitively large energy because of Pauli's exclusion principle. The negative pion becomes stable for the same reason In the $\mu >> 0$ case, the positive pion becomes stable It is then possible to consider bound states of pions and nucleons: neutrons and negative pions in the negative potential case, protons and positive pions in the positive potential case*. This is interesting since they have a mass ratio of about 1 : 7. In molecular physics, the high mass ratio between the electron and the nuclei is the reason for the immense structural richness**. Here the ratio is way smaller but maybe it leads to interesting effects anyway > Is this a correct analysis? What is known / can be said about physics in > these conditions? In particular about particle and bound state spectra? *You can't mix negative pions with protons, since it would produce neutrons. Similarily, mixing positive pions with neutrons yields protons **At least I think it is. This mass ratio leads to the Born-Oppenheimer approximation which gives rise to a complicated effective potential for the nuclei which posses many local minima: the molecules EDIT: Actually, I don't want the lepton number chemical potential to be _too_ high, since then pairs of $e^- + \pi^+$ (or $e^+ + \pi^-$, depending on the potential sign) will start forming which introduces additional complications EDIT: Let's make it a bit more quantitative. What is the potential needed to stabilize a charged pion? W.l.o.g. let's use a negative pion. Under normal condition it mostly decays into $\mu^- + \bar{\nu}$. If this decay is forbidden it still has the $e^-+ \bar{\nu}$ channel (albeit much slower). Since the electron mass is about 0.4% of the pion mass, the resulting electron is ultrarelativistic. Hence the energy splits roughly 50-50 between the electron and the antineutrino and each gets $m_\pi / 2 = 70 MeV$. Thus if the antineutrino Fermi level $-\mu$ is above 70 MeV, the pion is stabilized. The neutron is stabilized under much milder conditions, since $$m_n - m_p - m_e = 780 KeV$$ which is an upper bound on the required chemical potential. Now, to form $e^+ + \pi^-$ pairs we need the chemical potential to reach pion $m_\pi = 140 MeV$ (as above, position mass is relatively negligible). Thus the range of interest is 70 MeV - 140 MeV. Trouble is, we can also have $2 \pi^- \rightarrow 2 e^- + 2 \bar{\nu}$ processes. Here momentum conservation doesn't constrain us hence to rule this out we are left with the very narrow range of 139 MeV - 140 MeV (the size of this range is $2m_e$). And we _do_ need to rule this out to get multipion bound states EDIT: There's another aspect to this thing. Sufficiently high negative chemical potential destabilizes the proton due to $p + \bar{\nu} \rightarrow n + e^+$ processes, where the excess energy comes from the antineutrino. Once this destabilization becomes greater than nuclear binding energy, protons cannot appear as constituents of nuclei. In similar fashion, high positive $\mu$ makes the neutron even less stable and at some point neutrons cannot appear as constituents of nuclei. For lower $\mu$ proton-neutron nuclei exist but the beta stability order might be modified


title: Connections of iterative solvers for large systems of equation in Physics?

text: I am trying to find the domains in physics where solving large systems of equations is computationally expensive. The sparse systems are of my particular interest, where the input matrix A is in GBs (up to 100 GBs).


title: Which microscopy technique is for wet condition?

text: I am searching for a microscopy technique in wet conditions. It must be possible to see through the specimen. ESEM is good but can't see through. TEM needs high vacuum which will boil away water. The specimen is embedded inside a gel. The size of the specimen is between 0.5 µm and 20 µm.


title: Why is there no theta-angle (topological term) for the weak interactions?

text: Why is there no analog for $\Theta_\text{QCD}$ for the weak interaction? Is this topological term generated? If not, why not? Is this related to the fact that $SU(2)_L$ is broken?


title: Is it possible for bubbles to exist in vacuum?

text: In the case of a bubble, the outside pressure is less then the inside pressure. If that is the case can bubbles exist in vacuum? I am not sure but this should be true if vacuum has zero pressure


title: Does spin of the earth vary at different locations?

text: Does the spin of the earth around its axis vary at different distance from the sun?,if so how does it correspond to the length of day and night,?


title: How could we see microwave radiation with our eyes?

text: A few years ago I read a short little article about how big our eyes would have to be to observe microwaves (or any long-wave radiation for that matter). I don't remember enough about the article, or know enough about observing EM waves to reconstruct the equations, and I can't find the article again. So my question is simply this: is there some equation or rule of thumb that says if you want to observe ('see') a wave of length x, then your observing tool ('eyes') have to be such and such a size?


title: Velocity in a turning reference frame

text: I often see the relation that $\vec v=\vec v_0+ \vec \omega \times \vec r$ in a turning reference frame, but where does it actually come from and how do I arrive at the acceleration being $$\vec a=\vec a_0+ 2\vec \omega \times\vec v+ \vec \omega \times(\vec \omega \times \vec r)+\dot{\vec \omega} \times \vec r\,\,\text{?}$$ Is there a simple method to see this? All approaches that I saw use some non- intuitive change of differential operators and so on ($\frac{d}{dt} \rightarrow \frac{d}{dt}+\vec \omega \times$) and so on.


title: Low rank entangled states

text: In some recent works, I got the information that low rank mixed states need not be bound entangled. In particular, for the system $3\otimes 3$ there is no bound entangled states. Can anyone tell me how to prove it or give a proper reference?


title: Variation of Electrostatic Potential with Distance in a Uniform Field

text: I came across this question in a problem book: If a proton is released from rest in a uniform electric field, does the electric potential at those points where the electron moves increase or decrease. What about its electrostatic potential energy? The answer given was: The electric potential increases while the electrostatic potential energy decreases. Now, I can agree that the electrostatic potential energy decreases due to the conservation of energy. But, shouldn't the electric potential also decrease (keeping sign in mind) since it it defined as the negative integral of electric field with respect to displacement? Apparently the answer is the same for an electron too.


title: What is the metric of Vaidya black-hole horizon?

text: The metric of a Vaidya black hole in outgoing/retarted null coordinates are $$ds^2=-\left(1-\frac{2m(u)}{r^2}\right)du^2-2dudr+r^2\Big(d\theta^2+\sin^2\theta d\phi^2 \Big)$$ The eveolving horizon locates at $r=2m(u)$ and thus $dr=2\dot{m}du$ with dot denoting $\frac{d}{du}$. Hence, I conclude that the degenerate metric of the Vaidya black-hole horizon should be $$ds^2=-4\dot{m}du^2+4m(u)^2\Big(d\theta^2+\sin^2\theta d\phi^2 \Big)$$ Am I right? Thank you very much! I am unsure because the null time coordinate $u$ is NOT orthogonal to $r$.


title: During reflection does the emitted photon have same properties?

text: When light (photon) is reflected the the original photon is absorbed by an electron and then emitted again. Does this "new" photon have the same wavelength, frequency etc. as the original?


title: Conjugate of an operator applied to a function

text: In section 6.3.1 of the following MIT Open Course Ware (PDF) (22-02, Introduction to Applied Nuclear Engineering), the author, Prof Paola Cappellaro, derives Heisenberg's equation using the definition of expectation values, $$ \frac{d}{dt}\langle\hat{A}\rangle=\frac{d}{dt}\int d^3x\psi^*(x,t)\hat{A}\psi(x,t)\\\ =\int d^3x\left(\frac{\partial\psi^*}{\partial t}\hat{A}\psi+\psi^*\frac{\partial\hat{A}}{\partial t}\psi+\psi^*\hat{A}\frac{\partial\psi}{\partial t}\right) $$ and Schrodinger's equation, $$ \frac{\partial\psi}{\partial t}=-\frac{i}{\hbar}\hat{H}\psi,\qquad \frac{\partial\psi^*}{\partial t}=\frac{i}{\hbar}\left(\hat{H}\psi\right)^* $$ Then the author uses $\left(\hat{H}\psi\right)^*=\psi^*\hat{H}^*=\psi^*\hat{H}$ to get $$ \frac{d}{dt}\langle\hat{A}\rangle=\int d^3x\left(\frac{i}{\hbar}\psi^*\left[\hat{H}\hat{A}-\hat{A}\hat{H}\right]\psi+\psi^*\frac{\partial\hat{A}}{\partial t}\psi\right)\\\ =\frac{i}{\hbar}\langle\left[\hat{H},\,\hat{A}\right]\rangle+\langle\frac{\partial\hat{A}}{\partial t}\rangle $$ There are two things I don't understand in this derivation. The first is that the total time derivative $\frac{d}{dt}$ turns into a partial time derivative $\frac{\partial}{\partial t}$ when inside the integral. The second is that the author states $$(\hat{H}\psi)^*=\psi^*\hat{H}$$ This confuses me as $\hat H$ is the Hamiltonian operator, and contains an $x$ derivative. Thus the left hand side of the above equation is another state $\phi^*$ such that $\phi=\hat H\psi$, but the right hand side is an operator: the derivative is yet to be applied to anything. This equation by itself just makes no sense to me. Any help with either of these two issues would be greatly appreciated!


title: What is the simplest way to prove the Earth is round?

text: Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin. How would you prove that the Earth is, in fact, round?


title: Proofs that Earth is not flat?

text: Well, I do not believe that Earth is flat, but I met some conspirationistas believing that all physics we learn in school is twisted, all the Earth's pictures from space are photoshopped by NASA, etc, etc. So those guys wanted to put a satellite into orbit, take some pics, and see for themselves. Of course, a satellite won't stay above a flat Earth, but what other very simple means to prove/check that earth is round (well, geoid) do we have? I am looking for simple means, available to anyone, that **do not need any external tool or knowledge** that a naysayer would reject. Some things I've considered:   * Meteorological balloon with camera and a phone with GPS, with a parachute to recover it. It has been done by amateurs, and in good conditions, you can see the Earth's curvature.   * Measure the Sun's height above horizon at noon at two different latitudes. The difference, along with some simple geometry would give you Earth's diameter.


title: What if Correspondence principle in QM happening in classical sense?

text: I assume a simple set up (hope to generalize it later)... suppose there is a position pdf (need not be position but any) (probability density function) which is the magnitude of a $\Psi(x)$ predicted by QM (lets call it P_quantum) and let there be a classical prediction pdf (P_classic), according to CP, for large quantum numbers, P_quantum agree with P_classic in an average sense, but my question is for suppose, what if it agress to converge to P_classic(x) in pointwise convergence, while still explaining all the microscopic things like hydrogen atom..blah blah, what would that mean philosophically? for example see Particle in a 1-D box and the correspondence principle For supose What if classical mechanics originate from QM in a classical sense ( it currently does in only statistical sense) but still explaining all the microscopic things like Hydrogen atoms, atomic spectra etc..., what would that mean? Right now they converge only in statistical sense, hence they say you need not converge pointwise as anyway you lack measuring device at that microscopic level for macroscopic things. What if they did converge pointwise, would that mean the lack of measuring device argument disappear from the QM scene?


title: How exactly does WiTricity's wireless power transfer using resonance work?

text: The WiTricity group has made a lot of buzz but no internet article explains the exact method that they are using. Does any one know how it works?


title: How would you connect a destination to a wormhole from your starting point to travel through it?

text: Lets say it was possible to create as massive traversable wormholes as we want. Say we started from earth and wanted to cross to alpha centauri. How would I create the wormhole so that it leads directly to alpha centauri? So we start from earth and want the wormhole to start from their and end at alpha centauri, how would you extend the wormhole there from your starting point?


title: Fraunhofer Diffraction using lenses

text: I've come across a question that I don't know how to tackle: > An alternative way of observing Fraunhofer diffraction uses lenses to > provide appropriate conditions. Sketch an optical conﬁguration for observing > Fraunhofer diﬀraction using a point source of light and two lenses. > > A parallel beam of light is incident normally on a diﬀraction grating which > consists of a regular array of 200 narrow slits per mm, each slit being 1 µm > wide. Light emerging from the grating is focussed by a lens of focal length > 300 mm onto a screen. The lens and screen are centred on a line > perpendicular to the centre of the grating, and are parallel to the grating. > It is intended to observe the 3rd diffraction order with light at a > wavelength of around 460 nm. Where on the screen does this diffraction order > appear? I've done the first part of the question (sketching the optical configuration, with the aperture between two lenses, and the screen and source at a distance f away from each other lenses). Everything I've learnt about lenses is used to magnify images, and I have no idea how to deal with a ray coming in at an angle. Surely you need to know at what distance the aperture is from the lens? Any help/hints would be greatly appreciated.


title: Is the singlet state $\eta '$ stable under the strong nuclear force?

text: According to group theory, the $SU(3)$ flavor symmetry for two quarks decomposes into an octet and a singlet. Is the prediction of this decomposition that the particles in the octet can only transform into each other under the strong nuclear force, while the singlet state $\eta '$ does not feel the strong nuclear force ?


title: Hamilton-Jacobi equation with time dependent Hamiltonian

text: I was struggling with this exercise about Hamilton-Jacobi equation. I have to solve by menas of Hamilton's principal function the system with Hamiltonian: $$\tag{1} H=\frac{p^2}{2m}-mAtx $$ with $A$ constant and initial conditions $t=0$, $x=0$, $p=mv_0$. Before attacking the Hamilton-Jacobi (HJ) equation, I've observed that the equation of motions are easily integrable and I get $x(t)=\frac{1}{6}At^3+v_0t$ and $p(t)=\frac{1}{2}mAt^2+mv_0$. Following the HJ's approach and named the Hamilton's principal function as $S=S(x,\alpha,t)$ I set $p=\frac{\partial S}{\partial x}$ in the Hamiltonian, and the HJ'sequation reads: $$\tag{2} \frac{1}{2m}\left(\frac{\partial S}{\partial x}\right)^2-mAtx+\frac{\partial S}{\partial t}=0 $$ From here on I'm not sure how to proceed. The separation of variables is not feasible. I've observed that a function of the type $$\tag{3} S=\frac{1}{2}mAxt^2-\frac{1}{40}mA^2t^5 $$ satisfies the HJ's equation, but how can I set the constant of integration $\alpha$ to procced and solve the equation? * * * **[EDIT]**After some research I've found the right way of solving equation (2) and indeed my previous equation (3) is incomplete. I add here the right solution for $S$ from (2) and a short derivaton of the solution for $x$ and $p$: $$ \tag{4} S(x,\alpha,t)=\left(\frac{1}{2}mAt^2+\alpha\right)x-\left(\frac{1}{40}mA^2t^5+\frac{1}{6}A\alpha t^3+\frac{\alpha^2}{2m}\right) $$ With (4) the equation for the transformed canonical coordinate $X$ can be found as: $$\tag{5} X=\frac{\partial S}{\partial\alpha}=x-\frac{1}{6}At^3-\frac{\alpha}{m}t $$ while the expression for the momentum $p$ is $$\tag{6} p=\frac{\partial S}{\partial x}=\frac{1}{2}mAt^2+\alpha $$ By imposing the initial conditions the solutions for $x$ from (5) and for $p$ from (6) are identincal to the ones obtained by solving directly the Hamilton equations.


title: How to calculate the charge on a capacitor

text: ![enter image description here](http://i.stack.imgur.com/YiGA9.png)I have a network in which $C_1$ ($8.73 \;\rm \mu F$) and $C_2$ ($2.46 \;\rm \mu F$) are connected in parallel and $C_3$ ($7.45 \;\rm \mu F$) and $C_4$ ($2.22 \;\rm \mu F$) are connected in parallel. $C_{12}$ and $C_{34}$ are connected in series. The network is connected to a $21.9\;\rm V$ battery. I calculated the total capacitance to be $5.187 \rm \;\mu F$. How do I find the charge on $C_1$? I use the equation $Q=CV$ right? So I did $5.187\;\mu F$ multiplied by $21.9\;\rm V$ and got $113.6\;\rm\mu C$ but that is not the right answer? What am I doing wrong?


title: Why is the sun brighter in Australia compared to parts of Asia?

text: Background: I've lived in Philippines for several years, and visited other parts of Asia occasionally (Singapore, Indonesia, Hongkong). I just moved to Western Australia a few months ago and I realized that the sun is brighter here, in the sense that every after sunrise and every nearing sunset, the sun shines too bright that it is blinding. This happens almost everyday, so this isn't just some one off thing. In Asia, this never occurred to me. The sun was always bearable to the eyes. Why is this so?


title: What is the relationship between string net theory and string / M-theory?

text: I've just learned from this one of Prof. Wen's answers that there exists a theory called string net theory. Since I've never heard about this before it picks my curiosity, so I`d like to ask some questions: How is string net theory related to the "usual" string /-M-theory framework? I mean _if_ there is a relationship between them ... Are the strings in string net theory and in string / M-theory the same? What are the differences in the goals one wants to achieve or phenomena in nature one can describe with those two theories?


title: Thermodynamics of cooking rice

text: The question is whether (1) cooking rice in a pot of boiling water for an hour and then leaving the pot (of water and rice) to cool to room temp heats my house the same amount as compared to (2) just boiling a pot of water for an hour and leaving it to cool as compared to (3) simply leaving the stove flame on for an hour. Assume an identical stove setting for all three scenarios of course. My inclination is that when cooking rice, much of the energy is absorbed by changes to the bond structures of the rice (just as a sweet potato converts to simpler sugars when cooking) and this energy will not be radiated back when you allow it to cool, and thus it will heat house less than (2) or (3). As for (2) versus (3), my inclination is to say that if all of the water vapor that boils off stays in the house to condense, the two scenarios will heat the house identically.


title: Validity Of the Drake Equation

text: The Drake Equation is designed to estimate the number of Planets possibly having intelligent life forms existing in our Galaxy. But I don't see how it's justified. As far as we know, Earth is the only planet with (intelligent) life forms existing on it, apart from Jupiter's moon Europa and Neptune which have oceans that may or may not contain life. I wonder what Dr Frank Drake set as the criteria to obtain the estimate value. Can I get help with that?


title: Loudspeaker frequency response dip

text: I'm trying to determine the reason for a dip in the frequency response curve of my loudspeaker. I have a loudspeaker mounted on the inside of an enclosure, and a 100mm ID tube mounted in-front of the speaker on the outside of the enclosure. The measured frequency response is: ![enter image description here](http://i.stack.imgur.com/AsZbD.png) At roughly 4100 Hz, a large dip in the response is noted. This dip corresponds to the cut-off frequency of the (0, 1) cross mode. This leads me to believe that it has something to do with the higher-order mode propagation, but I don't understand what. Could someone help me understand this problem please.


title: What particles carry various forms of energy?

text: If I didn't get this wrong, light or heat energy consists of photons and they in turn effect electrons' behavior and thus responsible for chemical and electrical energy. What kind of similar particle like photons are kinetic energy of objects? What about other forms of energy, like potential energy especially in springs (gravity remains tricky I understand)? I'm not specialized in physics so please simplify your answer as much as you can...


title: Problem regarding quantum mechanical notation of photons

text: I have recently been reading about spontaneous parametric down conversion(SPDC). I do clearly understand the process. What has been intriguing lately is the notation. For those of you who are unfamiliar, SPDC is a process which converts one vertically polarized photon to two horizontally polarized photon. (Yeah, it's not 100% accurate, but just to get the idea across.) Many paper wrote: $ \left|V\right\rangle \rightarrow \left|H\right\rangle\left|H\right\rangle$ I was just wondering about the meaning of $\left|H\right\rangle\left|H\right\rangle$ since it's neither the inner product nor the outer product. Additionally, if $\left|H\right\rangle$ could be represented by the vector $(1,0)$, what could be the implications? Thanks!


title: How can there be a quantum field theory that predicts all particle masses?

text: Say I have a theory with only one (energy) scale, e.g. one given by the fundamental constants $$\epsilon=\sqrt{\dfrac{\hbar c^5}{G}}.$$ > In this case, where I can't compare to something else, is there a way to > argue that > > $$\epsilon<\epsilon^2<\epsilon^3<\dots\ ?$$ By that reasoning, can there be a (field?) theory, where values are obtained from some expansion like in a path integral (which needs a hierarchy of that sort)? If you really only need/have a theory with $\hbar, c, G$, how can energies like particle masses be _deduced_ from the theory (instead of being experimental input)? And then if, at best, the theory predicts some mass of a particle $\phi$ to be $m_\phi=a_\phi\dfrac{\epsilon\ }{\ c^2}$, then the number $a_\phi$ must have some geometrical meaning, right?


title: Interpretation of the wave function in quantum mechanics

text: I just started watching the coursera lectures on the basics of quantum mechanics and one of the first lectures were on deriving Schrodinger's equation and its interpretation it under Born's interpretation. What I want to ask is what the wave function \begin{equation} \psi({\bf r},t) \end{equation} return and represent. Now I know that \begin{equation} |\psi({\bf r},t)|^2dxdydz \end{equation} is the probability of finding the quantum particle described by \begin{equation} \psi({\bf r},t) \end{equation} in the volument element \begin{equation} dV = dxdydz \end{equation} at time T. But im not sure what the wave function $\psi$ returns. Could someone please explain in laymans term the return type of the $\psi$ function and what the lone $\psi$ function represents.


title: How to interpret temporal coherence in Young's double slit experiment with single photons?

text: I have a problem with understanding what is the role of coherence in such experiment. Taking the Dirac's statement that photon interferes only with itself, it's fairly understandable, that single photon virtually splits to pass trough different slits and interferes with itself, therefore the optical path difference (if I can even talk about it in this example) is pretty much 0, because it can't actually split. It might expand or something, but definitely not split. But my interpretation must be wrong, because in such situation coherence wouldn't matter. Interference would always occur, because every photon would interfere with itself, not with the photons from delayed beam, and OPD would always be 0. I think I missed something really simple...?


title: How does an atom look like in momentum space?

text: I am reading an article in a popular science magazine about atoms in extremely low temperature (condensation?). It's interesting and I've got few questions regarding this phenomenon. 1) In the article, the author show an figure of how the atom looks like in momentum space (like the top figure here). I'm pretty confused at how people measure the momentum of the atom? By light? I remember in the text, it introduced in one chapter how people shine a line on a material, and collect the emitted light, compare how the light intensity is reduced so as to be able to estimate the number of atoms there are. But does this method work for momentum estimation too? 2) I remember in the lecture of quantum mechanics, even at absolute-zero temperatures, the atoms won't be still, it has something called zero-point energy, so does it mean that the atom will be vibrating even at zero kelvins? So in the top figure here, we can observe that the momentum distribution has width about the peak, is that width telling us there will be some atoms moving at different momenta or the width is due to the uncertainty? So what does the distribution of momentum look like in absolute zero temperature? 3) The last question is in the article, the author said the momentum distribution is a Gaussian. But I searched in the library, they said for Boson, the condensation satisfies Bose-Einstein distribution. So what is the physical origin to use Gaussian distribution instead?


Why does water have several different solid phase but only one liquid and gas phase? Is there any meaning? or any reason behind it? Or is it just the way the nature behaves?


title: Difference between Poynting vector and energy flux density?

text: Are those two terms the same, or...? My book says that the Poynting vector is an energy flux density given by: $$\mathbf{S} = \frac{1}{\mu_{0}}(\mathbf{E} \times \mathbf{B})$$ So that alone should indicate that it's not the same thing. But I've been looking through my book, and every time I look up energy density, not flux, it points me in the direction of the Poynting vector. So, are they the same, and if not, what is the energy density given by?


title: Matrix integral in multi-matrix model

text: Though it is a mathematical problem, maybe more physicists know it well. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} e^{-\frac{1}{g_s} \rm{Tr} W(\Phi,Q)}$$ where $Q_{ab}$ are complex $N_a\times N_b$ matrces satisfying $Q_{ab}^\dagger=Q_{ba}$, and $\Phi_i$ are $N_i\times N_i$ matrices. $W$ is given by $$W(\Phi,Q) =\sum_{i,j=1}^{r} s_{ij}Q_{ij} \Phi_jQ_{ji} +\sum_{i=1}^{r} W_i{\Phi_i} $$ where the constants $s_{ij}$ are antisymmetric, $s_{ij}=-s_{ji}$, they obey $s_{ij} = 1$ if $i < j$ and the nodes in the quiver diagram are linked, and $s_{ij} = 0$ otherwise. The notation $<a,b>$ for the range of the product denotes all pairs $(a, b)$ with $1 ≤ a, b ≤ r$ s.t. $s_{ab}\neq 0$. In order to calculate the path integral, first integrate out $Q_{ab}$: $$ \int \prod_{(a,b)\in E}[dQ_{ab}dQ_{ba}] \exp \sum_{(a,b)\in E}-\frac{s_{ab}}{g_s}\rm{Tr} (Q_{ab}\Phi_b Q_{ba}-Q_{ba}\Phi_a Q_{ab})$$ where$E=\\{(a,b)|1 ≤ a< b ≤ r\\}$ **My question is** how to integrat $Q$ out to obtain something like $$ \det (\Phi_a\otimes 1-1\otimes \Phi_b)^{-1}$$ Thanks in advance.


title: Probability density of Klein-Gordon equation

text: This may, perhaps, stir some healthy debate; at least I am having some "fun" thinking about it, hopefully I can solicit some outside views too. It is often regarded that the Klein-Gordon equation does not have a positive definate probability density. In short the argument is that the density in the Schrodinger equation is $\rho=\psi^*\psi$. And the continuity relation is $\nabla\cdot\vec{J}-\partial_t \rho=0$, where $J\equiv$ probabilty current density. Fair enough. As we go to the relativistic version "it is said" that the $\rho$ in the above must be extended to $\rho\rightarrow \psi^*\partial_t\psi-\psi\partial_t\psi^*$, and is no longer positive definate because we are free to pick initial values for each $\psi$ and $\partial \psi$ in the above. Here is why the above argument drives me nuts. There is the viewpoint that: They have simply transformed the expression for the density in error. Sure, it is true that the thing they have decided to label as the new $\rho$ is clearly not positive definate. I feel it is appropriate to entertain the view that this is because the thing they have chosen to reinvent $\rho$ as is simply NOT the probablity density anymore. Who decided to call it hat in the first place? The continuity equation is really just a statement that ${D\over d{\lambda}}\rho=0$...conservation of probablity density. This is identical to the equation $"v"\cdot\nabla \rho=0$ or if you choose $\nabla\cdot\rho"v"=0$ , where $\nabla$ now pertains to a four-operator. The 'v' in quotation marks corresponds with (though perhaps not literally) proper 4 velocity ala $dx\over d\lambda$. In short DON'T redefine $\rho$. We continue to have$\rho=\psi*\psi$ to be the case. No need to not call the ford a ferrari when it gets us around just fine. Now consider $"v^0"(\rho)\rightarrow \psi^*v^0\psi+\psi v^0\psi^*\leftrightarrow \psi^*\partial_t\psi-\psi\partial_t\psi^* $. the quantity they were attempting to call $\rho$ earlier. The continuity equation remains satisfied here. And, there appears to be no obvious reason why $\psi^*\psi$ cannot be considered positive definite under the previous rational. $"v^0"\psi*\psi=J^0$ is not positive definite, but $\rho\neq J^0 $ anymore; $\rho=\psi^*\psi$. Please comment, tell me why you disagree, agree, indifferent, etc. I won't argue much, may add comment, clarify if wanted. I will be interested


title: Reflection positivity in general

text: In the Euclidean QFT obtained by "Wick-rotating" a unitary QFT, the correlation functions satisfy a property called reflection positivity, see e.g. this Wikipedia article for the case of a scalar field. What's the correct formulation if you have chiral fermions and/or terms like the QCD theta angle? Could someone give the references?


title: What makes *electric* charge special (wrt. CPT theorem)?

text: I'm wondering why the 'C' in CPT - charge conjugation - refers specifically to electric charge. Of course you could say that C is just defined as $e^+ \leftrightarrow e^-$... but there has to be something that makes electric charge special. For example, why doesn't e.g. weak hypercharge or any other quantum number take it's role? After all, in the derivation of the CPT theorem you use Lorentz invariance and the fact that the Hamiltonian is bounded from below, but not specific knowledge about QED. This should be closely related to the question why we choose the particle with flipped electric charge as the antiparticle, and not ones with other flipped quantum numbers. Again, besides historical reasons, there must be a reason electric charge is special. I guess this is somehow related to the fact that (electrically) charged Fermions are represented by spinors, but I can't really make the connection explicit. _(The reason that I came to this question is that I was thinking about some of the implications of CP variance. If CPT is conserved, T must be violated. While certainly not at odds with observation - I mean we do have an arrow of time - it seems at least to me unintuitive that the laws of nature are not time reversal invariant. I was wondering if there was a loophole - maybe the measured CP violation is really a CPX violation, or the fundamental symmetry is CPTY (X and Y being some other transformations). This would probably warrant a question of its own, but I thought I'd provide some context.)_


Is it possible to reproduce the energy spectrum of quantum chaos using classical cellular automata? It's hardly impressive to reproduce harmonic oscillators.


title: Is there a simulator for observing the force between two charges comprehensively?

text: I basically want to observe the electrostatic force between two charges. I want to know if is there a simulator which could both help me to observe and calculate the force between two charges. I found this simulator which is pretty good I think. However, that would be great if there is a more comprehensive simulator with greater distances, more parameters etc. I know this is a duplicate but nobody answered the other question and I really want to know if there is a simulator like that. I am sorry if I broke the rules. Thank you.


title: Can all quantum superpositions be realized experimentally?

text: When textbooks in QM give example of finite dimensional Hilbert spaces they give examples of photon polarizations or of 2-states systems and sometimes they mention how one can achieve superposition in such cases experimentally. On the other hand when they talk about simple potentials like particle in an infinite potential well, and talk about superposition of the stationary states of this problem they never mention how such superposition can be achieved experimentally (despite the fact that we are in the era of nanotechnology and scientists can "engineer" effectively such potentials and many others). Typical example that may appear in textbooks can be as simple as $\Psi(x)=\frac{4}{5}\phi_1(x)+\frac{3}{5}\phi_5(x)$, where {$\phi_n(x)$} are the normalized eigenfunctions of the particle in a box Hamiltonian. Other more general superpositions could be between infinite numbers of stationary states via $\Psi=\sum a_n\phi_n(x)$. That makes me wonder, is there a fundamental reason that prevents us from engineering such superposition in case of particle in a box and the like, or that we just do not know how to do it yet? why is it possible with spin and seem to be hard with particle in a box? or is it something related to the energy eignestates in the position representation? It is kind of frustrating to study for long hours/read/solve problems/HW on all kinds of potentials and on superpositions without knowing how/if they can be realized in experiment. If someone knows references in which this issue is discussed it would be greatly appreciated.


title: The Fifth Gamma Matrix

text: This is regarding $\gamma^5$, the fifth gamma matrix in quantum field theory. I know its defining properties, namely, $$\gamma^5= -i\gamma^0 \gamma^1 \gamma^2 \gamma^3 $$ with $\\{\gamma^5,\gamma^{\mu}\\}=0$ and $(\gamma^5)^2=1$, but so far all I've used these for is to help prove some gamma matrix identities. It seems obvious to me that $\gamma^5$ is dependent on our representation (e.g. Weyl or Dirac), so my original idea that it's useful because of invariance on choice of basis seems wrong. What do we like about $\gamma^5$?


title: Radiation pressure at high temperature

text: I am reading a book on the history of the hydrogen bomb. The author describes the Teller-Ulam design and explains that the liquid deuterium charge was compressed by the radiation pressure generated by a vaporized plasma of polyethylene. This would allow the generation of pressures orders of magnitude higher than with high explosives. Is it possible to quickly estimate this pressure? If I assume a temperature of $T=1,000,000K$ and use $P=\frac{4\sigma}{3c}T^4$ I calculate a pressure of about $2.5$ million bars. Is this realistic? How does it compare with explosives?


title: Taking signal travel time into account in Special Relativity

text: I am having problems taking the time it takes for a light signal from an event to reach an observer into account: For instance, if we have two observers $A$ and $B$ who synchronize their clocks when they pass each other at the origin $O$ and $A$ then emits a light pulse at a time $t$ as measured on their clock, we have the event: $$X_{1}^\mu=\left(\frac{t}{c},0,0,0\right) \implies X_{1}'^\mu=\left(\frac{\gamma t}{c},-\frac{\beta \gamma t}{c},0,0\right)$$ However, if I'm correct, then $X_{1}'^{\mu}$ is not the time that $B$ observes the signal, it is merely the time which the event takes place in the reference frame at which $B$ is at rest. Therefore we have the actual time that $X_{1}$ is observed in $B$ as: $$t_{1}'=\gamma t + \frac{\beta \gamma t}{c^{2}}=t\left(\gamma+\frac{\beta\gamma}{c^{2}}\right)$$ Therefore, if $B$ emits a light signal in response $2t$ after they receive the light signal, we have: $$X_{2}'^{\mu}=\left(\frac{t}{c}\left(2+\gamma + \frac{\beta \gamma}{c^{2}}\right),0,0,0\right) \\\\\implies X_{2}^\mu = \left(\frac{\gamma\left(2+\gamma+\frac{\beta \gamma}{c^{2}}\right)t}{c},\frac{\beta\gamma\left(2+\gamma+\frac{\beta \gamma}{c^{2}}\right)t}{c},0,0\right)$$ And therefore, the time that $A$ receives the signal would be: $$t_{2}=\gamma\left(2+\gamma+\frac{\beta\gamma}{c^{2}}\right)t+\frac{\beta\gamma\left(2+\gamma+\frac{\beta\gamma}{c^{2}}\right)}{c^{2}}$$ However, this expression doesn't look right to me as it doesn't simplify and has factors of $c^{2}$, but I'm not sure where I've erred.


title: Calculating the coherence length from a spectrum

text: I have measured the spectrum of the LED in my interferometry set-up, and now I want to calculate the coherence length from it. A commonly found formula is $l_{coh} = \frac{c}{\Delta f}$, sometimes with an additional factor for the shape of the spectrum. However, I have only ever found factors for Gaussian and Lorentzian lineshapes, and mine is neither. I am looking for either a way to determine this factor, or derive the coherence length from the spectrum in a different way. Thanks in advance.


title: How is gravity consistent when you split an object into multiple pieces?

text: Here's the picture: ![figure of gravity acting on an object in one and two parts](http://i.stack.imgur.com/J9Rxw.png) **In picture 1** : Assume that we have 2 objects, namely _A_ , and _B_. Object _A_ with mass $2M$, and object _B_ with mass $m$; and their distance is $r$. Then the gravitational force of _A_ acting on _B_ ($\vec{v}$)'s magnitude will be: $$G\times\frac{2Mm}{r^2}.$$ **In picture 2** : Now divide object _A_ into **2 equal parts** of mass $M$ each. The distance from the centroid of each part of _A_ to _B_ is $\dfrac{r}{\cos \alpha}$. The gravitational forces of 2 parts of _A_ acting on _B_ ($\vec{v}_1; \vec{v}_2$)'s magnitudes are: $$G\times \frac{Mm\cos^2\alpha}{r^2}.$$ Now, I'm pretty sure that if I take the sum $\vec{v}_1 + \vec{v}_2$, I wouldn't get $\vec{v}$. The direction is the same, but the magnitude isn't. They are off a factor of $\cos ^ 3 \alpha$. :( What's going on here?


title: How can the Lorentz transformation cause two objects' masses to both increase?

text: Suppose there are two people **A** and **B**. A is standing on the earth and **B** is on a rocket in space. Now **B** passes by the earth at a speed of 0.9c and **A** sees him. According to **A** , **B's** mass would have increased. But since the lorentz factor considers only relative velocity, we can say that even **A** is moving at a speed of 0.9c relative to **B**. So will **A's** mass also increase according to **B** after applying the the lorentz transformation formula? If yes then my doubt is how can the mass of both increase?


title: How strong is electron degeneracy pressure?

text: I'm trying to get some specific numbers for electron degeneracy that I can understand, using a concrete example. Take for example this portion of carbon crystal: ![carbon crystal lattice](http://i.stack.imgur.com/WODZS.gif)   1. Exactly how much energy would be required to overcome the electron degeneracy pressure of these 18 carbon atoms, in a theoretical laboratory capable of such things.   2. I believe that neutrinos would be emitted as a byproduct of the collapse, is that correct? What other reactions if any would be needed to fully account for conservation of energy/momentum etc?


title: Group rings in Physics

text: In my institute a few mathematicians work on Group Ring. Since it has close connection with representation theory, I thought that there must be some interesting connections of it with Physics. However I did not manage to find any information from our local experts; perhaps for one reason, they are more interested on finite groups and its structure. Hence I am asking the same question to the community. Does there exists any connection between group rings and physics? (I am having a feeling, that it can be used as a tool for solving representation theoretic problems. But does its usefulness (if there is any) end there only?) Advanced thanks for any reply. Also please provide some references (paper or books), which describes such connection(s), if exists,


title: How to calculate specific heat capacity of an alloy

text: I have done an experiment in which I measured the specific heat capacity of a coin. I would like to compare the results to the actual value. I have found out that the coin is made of Cupronickel alloy (for simplicity let's assume it's just 75% copper and 25% nickel). Knowing the SHC values of the two elements (copper and nickel) how would I go about calculating the theoretical value of the SHC of the Cupronickel alloy, so that I can compare it to my experiment's results?


title: How to optically rotate images in small increments (for eyeglasses)?

text: As the result of an accident, one or more nerves that control the rotation of my left eye were damaged. The result is that my left eye views the world rotated clockwise several degrees, compared to my normal (and dominant) right eye. I'm told that it is not possible to construct a lens that rotates images, other than a camera lens that inverts an image a fixed 180 degrees. Has anyone created a portable optical system that rotates images in small adjustable increments? As shown in the attached image, the yellow line represents the plane of vision in my normal right eye, and the red line represents the rotated, or tilted, vision in my left eye. ![As shown in the attached image, the yellow line represents the plane of vision in my normal right eye, and the red line represents the rotated, or tilted, vision in my left eye.](http://i.stack.imgur.com/89hLl.jpg)


title: Would the horizon of a black hole be different for a tachyon than for subluminal matter or photons?

text: One of the most useful black hole analogies I've seen imagines that space is "flowing" like a river into a black hole, and the point at which it flows in faster than c is the horizon. This analogy leads to the idea that a tachyon could escape from the "c"-based horizon and perhaps penetrate a bit further. Given that the tachyon has finite speed there would presumably be a horizon closer to the singularity that applies to the tachyon. However, could a tachyon device send us information about what's going on beyond the c-horizon? Let's assume tachyons exist and we have a device that can generate an interpretable message transmitted w tachyons (the Apple iTach, if you will).


title: Equation for Hubble Value as a function of time

text: I am trying to write the equation for the situation where the Hubble parameter $H$ would be changing over time. In other words, it would represent an accelerated expansion of the Universe. That is, $H$ can no longer be the simple $H=1/t$. In the new equation, I should be able to plug a future time and see what the Hubble Value will be in that future. I think I got most of the concepts right. First of all I understand that the key to the problem is in $H=a˙(t)/a(t)$. Where $a(t)$ is the scale factor from the Friedman equations. I also understand that if $H$ is changing then $a¨(t)> 0$ and also $H'(a)>0$. But I'm still very uncomfortable when my pencil meets the paper. The Friedman equations are not stated as a function of $t$, but as a function of $a$ where $a$ is the time scale, and frankly, I don't know how to use the time scale factor. In any event, here is my poor attempt to do it. According to Wikipedia, one of the solutions of Friedman equation is (assume flat space k=0): $a(t) = a_0 t ^{2/3(w+1)}$ Therefore: $a'(t) = d(a_0 t ^{2/3(w+1)}) / dt$ $a'(t) = (2a_0/3(w+1)) t^{ -(1+3w)/3(w+1)}$ And I suppose that we could now substitute: $H=a˙(t)/a(t)$ with the above: $H = (2a_0/3(w+1)) t^{ -(1+3w)/3(w+1)} / a_0 t ^{2/3(w+1)}$ Simplified: $H = (2/3(w+1)) t^{-(3w+2)/3(w+1)}$ And $w$ is typically known from observation. I will appreciate if someone can let me know if I am in the right path or totally derailed. I have a feeling that $a(t) = a_0 t ^{2/3(w+1)}$ was not the right place to start because if $w=-1$, then everything goes down the drain. But then again, in an accelerated expansion, $w$ would not equal -1. It would always be less than -1. Also, in the final equation, if $w<-1$ then H<0, which could not be right. So I'm not sure what to think. Many thanks in advance, Luis


title: Reversible gates

text: Is it possible to make any gate reversible merely by retaining the input bits in the output and introducing ancilla bits as necessary? That is, given an irreversible gate with $k$ inputs and $l$ outputs, can we always find a reversible gate with $k + l$ inputs and outputs? Why or why not?


title: Can an observer be the observed?

text: As a supplement to this question as to whether particles can be observers, supposing that the answer is yes. One could suppose a setup where particle A is observing particle B, but what to stop us switching viewpoints around here and supposing particle B is observing particle A? I find this is an intriguing possibility considering the importance of symmetry in Physics.


title: How to calculate how weak does a magnet get when you get an other magnet closer to it?

text: I heard that when you take two magnets and get them closer together so they reject each other (north pole to north pole or south pole to south pole) they weakens. Does anybody knows how to calculate how much it weakens depending on their distance? And does it work the other way around? If you take two magnets the attract each other (north pole to south pole) will they get stronger? If so, how to calculate how stronger they'll get depending on their distance


title: Is it possible to condense heat?

text: Suppose that I have a warm room. Is it possible to condense the heat of the room into a small object so that the room turns cold and the small object very hot? If it's possible: How? If it's not: Why?


title: How can a green laser be reflected as orange?

text: My friend recently bought a high-powerd pocket laser pen (from China). It is green, and reflects off most surfaces as green. On some orange surfaces, it reflects orange instead. That is to say, we see an orange dot instead of a green one. It seems to be certain hard, shiny plastics (for example a Kryptonite bike lock). Orange fruit peel, for example, reflects green, and most orange fabrics reflect green. I suspect it's some kind of pigment that is reflecting this way. When the reflection is orange, the intensity of the reflection is subjectively about the same, i.e. it doesn't seem to be losing much energy. If one shined light from a green incandescent bulb onto an orange surface, one would expect that surface to be lit less intensely than a white surface. Not so the laser. So it seems to be that the photons are being re-emitted at a different wavelength than they were absorbed at. How can this happen? And why is it only orange?


title: uncertainty deviations for vacuum astronomy

text: Since i've done this question, i've been trying to improve and make more precise the statements regarding cosmic squeezed states and how different uncertainties affect the vacuum energy, but as it turns out, it's more difficult than it seems, as i hope to make clear in this question Intuitively, i can play with some algebra of the uncertainties and write a "minimum" of energy in function of them: $$ H = E^2 + B^2 $$ $$ H + \Delta H = (E+ \Delta E)^2 + (B + \Delta B)^2 $$ $$ \langle \Delta H \rangle = 2 ( \langle E \rangle \langle \Delta E \rangle + \langle B \rangle \langle \Delta B \rangle ) + \langle \Delta E^2 \rangle + \langle \Delta B^2 \rangle $$ In the vacuum, $\langle E \rangle = \langle B \rangle = \langle \Delta E \rangle = \langle \Delta B \rangle = 0$ so we can simplify: $$ \langle \Delta H \rangle = \langle \Delta E^2 \rangle + \langle \Delta B^2 \rangle $$ Now, if we accept the uncertainty principle applicability to both the total fields and its modes ( a tall proposition as we'll see) and assume a minimum uncertainty mode: $$ \Delta E = \frac{ \hbar }{ \Delta B} $$ then we are left with $$ \langle \Delta H \rangle = \langle \Delta E^2 \rangle + \frac{ \hbar^2}{\langle \Delta E^2 \rangle} $$ which has a minimum when $ \Delta E = \Delta B = \sqrt{ \hbar}$. This is the equation that defines "true" electromagnetic vacuum in terms of the uncertainties of the vacuum. Now, all of this is cute and nice, but all this applies to a single mode of radiation. As it becomes clear from this other question of mine stuff gets really hairy when we consider a real quantum field. In any case, i would hope that one can ask in experimental, absolutely concrete terms is: given one region of the cosmos, what deviation from this "true" vacuum, as defined above, can we expect? note that since the electromagnetic spectrum that is relevant for astrophysical phenomena is huge (from gamma ray to microwaves) there might be different deviations at different wavelengths Now, my question is: are there concrete, existing expressions for the effective field uncertainties in QFT in terms of the uncertainties of the modes that can be used for the purposes of this "vacuum astronomy"? Can we in practice, think in individual EM modes and their uncertainties as "isolated" from the rest of modes?


title: Boltzman distribution for chemical potentials

text: I read that if we have a system with two co-existing phases with chemical potentials respectively $\mu_1$ and $\mu_2$ then, at the equilibrium, the concentrations $X_1$ and $X_2$ are related by the relation: $X_1=X_2e^{[\frac{-(\mu_1-\mu_2)}{kT}]}$ Is that true? How can I derivate this expression?


title: Could I see the quantum states as representations of the Galilei algebra (or Galilei group)?

text: In somes references of Relativistic Quantum Mechanics, the one-particle states are given by representation theory of Poincaré algebra. Could I mimics this for the non-relativistic case? States in non-relativistic quantum mechanics could be seen as representation of Galilei algebra (or Galilei group)?


title: Rotation, cats landing on their feet, and conservation of angular momentum

text: Let _θ_ be the orientation (angle) of a body (such as a cat), and let _ω_ be its angular velocity. It is well-known that _θ_ can change even when the body is not rotating, using the conservation of angular momentum; that is, even when _ω_ = d _θ_ /d _t_ = 0. That's how cats land on their feet so well. But how can _θ_ possibly ever change, when its derivative is zero?! What's wrong with the math?


title: How does mass relate to resistance (Ohm's Law)

text: How can I relate $mg$ to $R$? Just to give you a scenario: If a free hanging weight was tied to a motor... I know the voltage ($V$) and current ($I$) so I can use $V = IR$ to find $R$. What if I reduce the weight by say half? Will that be the same thing as $.5R$?


title: Proof that a force applied to the center of mass is the same as force applied off-center

text: There is a similar question that gives a bit of an explanation, but little mathematical proof here: force applied not on the center of mass I would like **mathematical proof** that shows that the velocity of a rigid body when a force is applied to the center of mass is equal to the velocity of the same rigid body when the same force is applied to a point on the body other than the center of mass. Thanks in advance!


title: Is there any simple quantum model by Gerard 't Hooft which can explain the double slit experiment?

text: This question is directed to Prof. 't Hooft and anybody who is familiar with his papers. It is a reaction to Prof. 't Hooft's question why nobody is excited about his classical models for quantum mechnanics. My reasoning behind the question is simple. Humans have no good intution for quantum mechanics. But they normally have a very good intution for any kind of classical model. Now one of the simplest and in fact the canonical example for quantum weirdness is the double slit experiment. Combining these two things I think it is a very natural question to ask.


title: functional determinant question

text: why when we want to evaluate a functional determinant we use the expansion near $ t=0$ for the partition function $ \sum_{n}e^{-tE_{n}} \sim \sum_{n} a_{n} (t)t^{n} $ instead of just using the semiclassical partition function ?? $ \iint _{\Gamma}dx dpe^{-tp^{2}-tV(x)} $ which is just easier to handle with ? fo rexamplelin almost all papers i have seen this asymptotic approximation for the partition function near $ t=0$ although coefficients are hard to evaluate.


title: Fluid Dynamics - Water Bottle Drink Mix: Air or No Air?

text: If you take an ordinary sized plastic water bottle full of water and pour a packet of powdered (or liquid) drink mix into it, Will shaking the bottle with the cap screwed on to dissolve the mix into the water work better if there is a small amount of air inside the bottle? Or does it not make a difference at all and will shaking work just as well with no air bubbles at all in the bottle?


title: Non-stationary spacetime

text: What is an example for a spacetime that is non-stationary that is considered as a description of something in nature? So far all the spacetimes I encounted have always been stationary (Schwartzschild, FRW, Kerr, etc.).


title: What do we know of superconductivity in thin layers?

text: motivated by another question, i wonder if there are special properties of superconductivity when restricted on 2D or very thin layers related to the effective permittivity in function of the frequency $\epsilon ( \omega )$ near the first-order transitions between the superconducting and normal phases , any references about the main state of the art would be appreciated **Edit:** I've noticed that most research papers on superconductivity report on the current transport properties, but they usually don't talk much about the permittivity. Is this because its harder to measure?


title: Why does air remain a mixture?

text: As we all know, air consists of many gases including oxygen and carbon dioxide. I found that carbon dioxide is heavier than O2. Does the volume difference neglect the mass difference? Is it same for all other gases in air or is there another force that keeps all of these gases together? If I take a breath of a fresh air, will the exhaled air be heavier because of its higher CO2 content? Will it fall on the floor?


title: Quantum wave interference in 3d space

text: Why, in the double slit experiment, is there an interference pattern only along the horizontal axis and not along both axes?


title: Determining $y$ coordinate of charged particle motion

text: I've tried several different ways, with a tutor, and with other colleagues. One of my colleagues has even gone to the professor with whom he worked out an equation for the component, and it was later confirmed to be incorrect. Here is the question: > A proton ($q = 1.6 \times 10^{-19}$ C, $m = 1.67 \times 10^{-27}$ kg) moving > with constant velocity enters a region containing a constant magnetic field > that is directed along the $z$-axis at $(x,y) = (0,0)$ as shown. The > magnetic field extends for a distance $D = 0.55 m$ in the $x$-direction. The > proton leaves the field having a velocity vector $(v_x, v_y) = (4.8 \times > 10^5, 1.8 \times 10^5)m/s$. This is the diagram: ![enter image description here](http://i.stack.imgur.com/EDPe3.png) The question we have is how to determine $h$, the $y$ coordinate of the proton as it leaves the magnetic field. I've already determined that the velocity $v$ is equal to 512640.2247 m/s, and that the radius $R$ is 1.533 m. The trajectory of the proton is curved on the inside, so we can't use anything involving any derivation of similar triangles. I've tried solving for $v$ as a function of time, but then realized that $v_y$ is always changing anyway, which lead me to calculate the acceleration so the time term that I used incorporated the change, but that didn't work either. Edit: Really sorry folks, I wrote this in a hurry to get to work on time. Yes, the radius I mentioned is indeed the radius of the circular trajectory of the proton after it enters the magnetic field. Also, I do understand to keep things variable, and I mentioned the radius and velocity only as to sort of show what I have available to work with. I've drawn the imagined circle suggested below, and I've made the realization that it would be helpful to relate the radius, the angle, and the velocity, but I'm not sure how to do so accurately.


title: Renormalizing QED with on-shell fermions

text: When renormalizing QED, we calculate the 1 loop correction to the fermion- fermion-photon vertex using the diagram, $\hskip2in$ ![enter image description here](http://i.stack.imgur.com/1VHVt.png) When doing the calculation we typically let the photon go off-shell but demand that the fermions be on-shell. In other words, \begin{equation} q ^2 >0, \,(q-p)^2 = p ^2 = m_e^2 \end{equation} We then go on to calculate the diagram using its Lorentz structure and then splitting the contribution into a part contributing to the $g$ factor ($F_2(q^2)$) and another which does not ($F_1(q^2)$) giving a vertex factor of, \begin{equation} \Gamma ^\nu ( p , p ' ) = ( - i e ) \left[ \gamma ^\nu F _1 ( q ^2 ) + \frac{ i \sigma ^{ \mu \nu } q _\mu }{ 2 m } F _2 ( q ^2 ) \right] \end{equation} However, keeping the fermions on-shell seems like a very strange requirement as in general this vertex may appear in the middle of a diagram and so could have off-shell incoming fermions. In fact, we don't make this requirement when renormalizing the propagator, I believe for this exact reason. So what makes it justified when discussing vertex renormalization?


title: Propagating higher-order Hermite-Gauss modes using the Complex Beam Parameter?

text: A Gaussian laser beam can be propagated through an optical system (consisting of free space, thin lenses, curved and flat interfaces, etc) by using the "ABCD" ray-transfer matrices, and the complex beam parameter $\tilde{q}$. A higher-order Hermite-Gauss or Laguerre-Gauss laser beam will gain Gouy phase more quickly than the fundamental Gaussian mode. Is there a simple modification to the complex beam parameter propagation that will also work for these higher order modes?


title: Is subspace a "real" theory?

text: Is subspace a "real" theory? by "real" l mean can it be derived from GR or some other stringy theory and did Einstein ever toy with subspace?


title: The physics of sound boards

text: As a kid I was bemused at why soundboards worked. A small sound could be demonstrably amplified simply by attaching the source to a surface that is rigid and not too thick. How could the volume increase so much given that there was no extra energy added? As an adult I kind-of-think I know, but there are still many nagging questions. I assume it has to do with the waves propogating from a vibrating object actually being a compression on one side of the object just as they are a decompression on the other side, and something about that lack of coherence limits the volume. Exactly why remains a mystery to me. Is separating the pocket of compression and decompression so that the boundary along which they meet is quite small part of the issue? My question is what are the physics that make a soundboard work? Interesting specifics that would be nice-to-knows would be why does a hollow one (like a violin) work better than a solid one (imagine a filled in violin)? How important are the harmonics of the solid? But the real question is what are the physics that make a soundboard work? P.S. I am a mathematician, so feel free to wax very mathematical if it is necessary to give a good explanation.


title: Critical Dimension of Bosonic Strings and Regularization of $\sum_{n=1}^\infty n$

text: If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at $$ \frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0. $$ Now mathematically this is clearly a divergent series, but using zeta function regularization here we are taking $$ \sum_{n=1}^\infty n = \zeta(-1) = -\frac{1}{12}. $$ And obtain $ D = 26 $ where $\zeta $ is the analytic continuation of the zeta function we know. But it makes no sense in putting $ s = -1 $ in the formulae $$ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}. $$ As the above is only valid for $ Re(s) > 1 $. So what is going on in here? Can anyone give me a reasonable explanation about obtaining $ -1/12 $?


title: What areas of physics depend on the sum $1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12$?

text: This youtube video from Numberphile, http://youtu.be/w-I6XTVZXww shows how the value is derived. In the video, one interviewee claims that "this result is used in many areas of physics". In the video, only string theory is mentioned. Which areas of physics use or depend on the sum $$1 + 2 + 3 + 4 + 5 + 6+ 7+\ldots= -1/12?$$


title: The sum of positive integers equals minus one twelfth

text: I was watching a lecture online from the american physicist Lawrence Krauss, when he made an off the cuff remark about the sum of all the positive integers being equal to one twelfth. My question is in what mathematical sense does minus one twelfth relate to the positive integers, and does it have any use in physical theories? Thanks.


title: What would happen if charged plates are placed horizontally?

text: My idea is placing charged conducting plates in such a way that they won't see each others' surfaces unlikely to the typical design of parallel plates. If they are placed like this, would be the force that one plate exerting to other same as in the typical design? If we say the charges on the plates are q1 and q2 how could be the force calculated? Since the electric field is perpendicular to surface of a conductor would this placement affect the electrostatic force and field? Here is how will the plates are placed: ![Charged plates sketch](http://i.stack.imgur.com/Ih2ad.png)


title: If MOND theory doesn't explain gravitational lensing, couldn't well placed black holes explain it?

text: When I first heard about star velocity as a function of its distance from the center of the galaxy and the difference between prediction and observation I immediately thought "there must be a threshold where gravity reacts differently". Then I discovered MOND theory and that it can't explain gravitational lensing. But what if observed gravitational lens is only well placed black holes ? After all we detect black holes with their gravitational influence. So, why dark matter instead of MOND + more black holes ?


title: Generalizing Heisenberg Uncertainty Priniciple

text: Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$ $$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$ Then using the Lagrangian density for classical electrodynamics $$\mathcal{L} = \frac{1}{2}\left(\epsilon _0E^2- \frac{1}{\mu _0}B^2\right)- \phi \rho _{\text{free}} + A\cdot J_{\text{free}} + \mathbb{E}\cdot \mathbb{P} + \mathbb{B}\cdot \mathbb{M}$$ Q1: Does it make sense to substitute the canonical scalar field $\phi$ (or electric scalar potential) for $\pi _i$ ; and $\phi \rho _{\text{free}}$ for canonical coordinates $x_i$. Such that these satisfy Heisenberg's Uncertainty relation? Q2: Is there a generalized method for taking a Lagrangian and deriving conjugate variables, which then satisfy Heisenberg's Uncertainty relation?


title: Question on Projectile Motion equation

text: > A golf ball is shot into the air from the ground. If the initial horizontal > velocity is 20m/s and the initial vertical velocity is 30m/s, what is the > horizontal distance the ball will travel before it hits the ground? Ans: The golf ball will continue to travel through air until gravity brings it back down to the ground. Under the gravity force, the vertical location of the ball as a function of time is given by the following equation:- $$ z = z_0 + v_{z_0} t - \frac{1}{2} g t^2 $$ The time it takes the golf ball to return the ground is equal to $$ T = \frac{2v_{z_0}}{g} $$ The horizontal distance the ball will travel before it hits the ground can be computed using the following equation:- $$ \Delta x = x-x_0 = v_{x_0} t = 2 v_{x_0} v_{z_0} / g = 122.4\text{ m} $$ The problems are: 1) what is the parameter $v$ represent? 2) In the example above $ \Delta x = x-x_0 = v_{x_0} t = 2 v_{x_0} v_{z_0} / g = 122.4\text{ m} $ What is the actual value in this equation? So that $x=?$ $x_0=?$ $v=30?$ $g=10?$ $z0=?$


title: Is it really true that slow-roll inflation is incompatible with string theory?

text: In this blog post, Motl claimed that slow-roll inflation is incompatible with string theory. Is that really true, and why do people think so?


title: Conserved charge from conserved current associated with translational invariance

text: (c.f Di Francesco, 'Conformal Field Theory' P.45) Di Francesco calls the conserved charge arising from the conserved current associated with a translation invariant theory the 'four momentum'. While I see why such a result makes sense, what was the motivation for calling it so? Also, why is it called the four momentum? Di Francesco's treatment seems to be valid in not only four space time dimensions but in a space time with $d$ space time dimensions. This is evident in his portrayal of the conserved charge $P^{\nu}$ defined like $$P^{\nu} = \int \text{d}^{d-1} x T^{0 \nu},$$ where $T^{\mu \nu}$ is the stress energy tensor. So should the conserved charge not be called something like a $d$ momentum so that we define $P^{\nu}$ analogously to the familiar 4D Minkowksi setting of special relativity with $P^{\nu} = (E, \mathbf{P})$, where $\mathbf{P}$ is the $d-1$ momenta in the same vein that in SR, we refer to the $3$ momenta?


title: What happens to heat waste produced by energy generation?

text: What happens to heat waste produced by energy generation on earth that must be there according to the laws of thermodynamics? So, it never dissipates and remains on earth?


title: Work Done to click a mouse?

text: Is there any good research done to find out the work done in clicking a mouse button. any link to that would be greatly appreciated. P.S. i am not too sure whether this question belongs here or not, so please let me know, if it doesnt, i will remove it, I have already googled "work done to click a mouse" "mouse click research" and oher relevant queries on google and google scholar, but only in vain!


title: Pair production - mathematically?

text: Allover the web i am only seeing a statement similar to this: > Pair production is not possible in vaccum, 3rd particle is needed so that > conservation of momentum holds. Well noone out of many writers shows, how to prove this matematically. So this is what interests me here. * * * **First i wanted to know if pair production really cannot happen in vacuum** , so i drew a picture and used equations for conservation of energy and conservation of momentum to calculate energy of a photon $(h \nu)$ needed for pair production. It turns out $h\nu$ is different if i calculate it out of conservation of energy or conservation of momentum. And even more! It can never be the same because equallity would mean parts $v_1 \cos \alpha$ and $v_2 \cos \beta$ should equall speed of light $c$. Well that cannot happen. Below is my derivation. * * * ![pair production](http://i.stack.imgur.com/wGF1v.png) CONSERVATION OF ENERGY: $$ \scriptsize \begin{split} W_{1} &= W_{2}\\\ W_f &= W_{e^-} + W_{e^+}\\\ h\nu &= W_{ke^-} + W_{0e^-} + W_{ke^+} + W_{0e^+}\\\ h\nu &=\left[m_ec^2 \gamma(v_1) - m_ec^2\right] + m_ec^2 + \left[m_ec^2 \gamma(v_2) - m_ec^2\right] + m_ec^2\\\ h\nu &=m_ec^2 \gamma(v_1) + m_ec^2 \gamma(v_2)\\\ h\nu &=m_ec^2 \left[\gamma(v_1) + \gamma(v_2) \right]\\\ \end{split} $$ * * * CONSERVATION OF MOMENTUM: $y$ direction: $$ \scriptsize \begin{split} p_{1} &= p_{2}\\\ 0 &= p_{e^-} \sin \alpha - p_{e^+} \sin \beta \\\ 0 &= m_e v_{1} \gamma(v_{1}) \sin \alpha - m_e v_{2} \gamma(v_{2}) \sin \beta\\\ &\text{If $\boxed{\alpha = \beta} \Longrightarrow \boxed{v_1 = v_2}$ and:} \end{split} $$ $$ \begin{split} \scriptsize 0 = 0 \end{split} $$ $x$ direction: $$ \scriptsize \begin{split} p_{1} &= p_{2}\\\ \frac{h}{\lambda} &= p_{e^-} \cos \alpha + p_{e^+} \cos \beta \\\ \frac{h \nu}{c} &= m_e v_{1} \gamma(v_{1}) \cos \alpha + m_e v_{2} \gamma(v_{2}) \cos \beta\\\ h \nu &= m_e c \Big[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Big] \end{split} $$ Alltogether: Because momentum in $y$ direction equals 0 (holds for some combinations of $\alpha, \beta, v_1, v_2$) whole momentum equals just the momentum in $x$ direction. So if i add them i get: $$ \scriptsize h \nu = m_e c \Big[ \gamma(v_1) \underbrace{v_{1} \cos \alpha}_{\neq c} + \gamma(v_{2}) \underbrace{v_{2} \cos \beta}_{\neq c} \Big] $$ * * * From this i can conclude only that i cannot sucessfully apply conservation of energy and conservation of momentum at the same time and therefore pair production in vacuum cannot happen. **QUESTION1:** Why do writers state that 3rd particle is needed so that conservation of momentum holds? What if conservation of momentum holds and conservation of energy doesn't? How can we say which one holds and which one doesnt? **QUESTION2:** Do writters actually mean that if a 3rd particle is included we can achieve $h \nu$ to match in both cases? **QUESTION3:** Can someone show me mathematically how this is done? I mean it should right?


title: Path Integrals Page Peskin

text: Hi this problem relates directly to path integrals but I imagine it is a maths trick that I am missing. One has an expression such as $$\int dx \exp\left[i\frac{(p)^{2}}{2}-iV(\frac{f}{4})\right] $$ it says simply expand the expression in powers of $(x-y)$ to get $$\int dx \left(i\frac{(y-x)^{2}}{2a}\right)\left[1-iaV(y)+...\right]\left[1+(x-y)\frac{\partial}{\partial y}+\frac{1}{2}(x-y)^{2}\frac{\partial^2}{\partial y^2}\right] $$ I thought it was some sort of Taylor expansion by I can't explain the change in exponential and I don't really understand the derivatives. Can anyone explain why these are equal?