title: Energy formula for separating $O_2$ from mixture of $O_2$, $NH_3$ and $H_2O$

text: I have a physics problem I'd like to make sure I get correct. The practical aspect of this problem is that the photosynthetic efficiency of algae is inhibited with dissolved O2 in the growth medium, and the economics of any algae cultivation system are very sensitive to photosynthetic efficiency. The problem I'm trying to solve is determine the theoretic energy to separate a gas mixture of, O2, NH3 and H2O (in boiling point order) into two gas streams, one of which is 99.99% pure O2 but the other stream can have as much as 50% of the original O2 along with NH3 and H2O. The input and output temperatures and pressures are STP and the input, at least, is at saturated humidity. Presumably, the energy is derived from the equations in this paper. Any approximate ideas of what the formula would look like? PS: The detailed model of the algae system is here.


title: Covariant and contravariant vectors

text: Reading Weinberg's "gravitation and cosmology" I came across the sentence "The partial derivative operator $\partial/\partial x^\mu$ is a covariant vector, or in other words a 1-form,..." (p.115, above equation (4.11.8)) Now, a section of $T^*M$ is called a covector field or a 1-form. Elements of $T^*_pM$ are called tangent covectors. In books about differential geometry I read that tangent COvectors are called covariant vectors. How do I have to understand the quote from Weinberg's book?


title: Reflectance of Round Moon vs Flat Moon

text: Treating the moon as an ideal, bright Lambertian reflector, I was trying to compare its efficiency as a light source to an equally-sized flat sheet of drywall. It's pretty straightforward to set the problem up, but I'm not so sure of my calculus. I got that the flat sheet of drywall would provide 50% more illumination in full moon conditions as compared to the spherical moon, treating both of them as smooth, diffuse lossless reflectors. I wonder if anyone wants to second-guess my math? I talk about the problem on my blogsite here.


title: When does Thevenin's theorem not apply (modelling a power source with a ohmic internal resistance)

text: Most physics text books say that a power source can be modelled as an EMF with a internal resistance. This is also know as Thevenin's theorem or Norton's theorem. However I have read in some sources that this is not always the case. When does this model not apply/break down and for what reasons?


title: Electron recoil after emitting virtual photon

text: Assume that a stationary electron $A$ emits a virtual photon with $4$-momentum $k$ and a stationary electron $B$ absorbs it. Let us assume a description in which time is moving forwards. At the instant that electron $A$ recoils with $4$-momentum $-k$ can we say that the photon has $4$-momentum $k$? I assume not as this would imply that the photon is real with a definite $4$-momentum. It seems to me that, under the assumption that time is going forwards, we can only assert that electron $B$ has $4$-momentum $k$ once the virtual photon has been absorbed. So how can we understand electron $A$'s recoil? Let us assume a description in which time is moving backwards. In that case electron $B$ emits a virtual photon with $4$-momentum $-k$ and electron $A$ absorbs it. Now we can describe electron $A$'s recoil with $4$-momentum $-k$, without implying a real photon, again once the time-reversed virtual photon has been absorbed. So to fully describe the exchange of a photon between electron $A$ and electron $B$ it seems to me that we need to assume time both goes forwards and backwards. Is this right?


title: A terminological question about work and energy

text: Work is force applied over distance. Is it also reasonable to say that work is (the same thing as) the transfer of energy? When work is done, the equivalent energy is transferred. But if energy is transferred, such as by heating something, is that (or could that be) called work? Also, if a black body is subjected to a stream of energy, which it absorbs and radiates away, is any work done? The black body, let's assume, remains at a constant temperature, but energy is transferred to it and then radiated away. Is any work done? So what I'm really asking about is how the term "work" should be used.


title: Relativity and Entanglement

text: Say we have two particles which are entangled so that they have opposite spins. If one is up, the other is down. They are sent off to two spatially separated observers A and B. Both observers can claim the following: The probability of measuring their particle in the spin up state is 1/2 = probability of measuring spin down state. Now, suppose observer A measures his particle in the spin up state. Now, the probability of observer B measuring spin down = 1, and probability of measuring spin up = 0. I think it is fair to say that the Observation made by A influenced the outcome measured by B. He could have gotten spin up if he had measured his particle before A. However, there is no transfer of information due to the random outcomes. B doesn't know whether he measured spin down due to A measuring up, or if he was the first to measure his spin. Observations made by A are random too, and so if we repeat this experiment many times, B will measure spin up 1/2 the time and spin down 1/2 the time. However, the ordering of which measurements report spin up and spin down will be different than if B always measured before A. The probability would still be spin up 1/2 the time, and spin down 1/2 the time. Are we left to say that relativity is about information and not this subtle influence? Einstein certainly was not of this opinion, "The following idea characterises the relative independence of objects far apart in space (A and B): external influence on A has no direct influence on B; this is known as the Principle of Local Action, which is used consistently only in field theory. If this axiom were to be completely abolished, the idea of the existence of quasienclosed systems, and thereby the postulation of laws which can be checked empirically in the accepted sense, would become impossible." -- A. Einstein


title: Field content and symmetry groups of Minimal Composite Higgs Models

text: I'm trying to teach myself the Composite Higgs Model, both its theory and its LHC phenomenology (particularly the 4DCHM). Unfortunately, I'm struggling; the literature is contradictory and/or omits basic details (?!). I have a few related questions, that should help me get started understanding the models,   1. What are the extra symmetry groups in the 4DCHM/minimal CHM? Is the $SO(5)/SO(4)$ approximate symmetry global or local? If it is local (as this source suggests hep-ph/1302.2371v2), why aren't the Goldstone bosons (the four Higgs dof) eaten by the broken generators?   2. What is the extra field content in the 4DCHM/minimal CHM? What extra gauge or fermion fields are present? Are they composite from the strong dynamics or fundamental?   3. How are the SM, new fundamental, new composite and new gauge fields arranged into representations of the extra symmetry groups?  Any help or sources of information would be appreciated.


title: Confused on Calculating Resistance Distance Matrix

text: I am trying to create a computer program to compute the equivalent resistance over any points on any rectangular set of resistors (all with a resistance of 1 ohm). It seems that the resistance distance matrix is exactly what I need. I understand how to compute the inverse and what a Laplacian matrix is, but what is the difference between $\Gamma_{ii}$, $\Gamma_{jj}$, and $\Gamma_{ij}$ in $\Omega_{ij} = \Gamma_{ii}$ + $\Gamma_{jj} - 2\Gamma_{ij}$?


title: What does it mean for objects to follow the curvature of space?

text: In science documentaries that touch on general relativity, it is often said that gravitational pull isn't an actual a pull (as described by classical physics), but rather one body travelling in a straight line on the curved space around another massive body. I don't know if I misunderstood that or if it's just an oversimplification meant to help the uninitiated understand the concept better, but something doesn't seem to follow. If the Earth travelling around the Sun is just moving along a straight line in the curved space, shouldn't light also be trapped in orbit around the Sun? I expect the actual equations in the theory also take into account a body's velocity, not just the curvature.


title: Tokamaks and the reason they are not efficient

text: I'm curious about why tokamaks are inefficient as generators. In laymans terms, what is the main reason(s) tokamaks still cannot be used as generators? My limited understanding of tokamaks tells me that the magnetic field required to keep the plasma in place and moving demands a vast amount of energy, much more than the tokamak itself can produce. Is there other ways we could create strong magnetic fields to contain the plasma? And just how small could a tokamak be?


title: How to create frame of reference?

text: Is this possible to create a inertial frame of reference in the earth?   How it is possible?


title: The nature of theoretical models

text: Mathematics is exact. It is a beautiful language that allows us to express quantities that aren't possible to be represented physically. We build theoretical models of physical systems that work out great on paper, yet there are cases where we cannot solve these models exactly. We need these tools, such as perturbation theory, to get almost exact answers. For example, in Quantum Mechanics we use Perturbation Theory to solve the Schrodinger equation for the hydrogen atom. **Shouldn't our theoretical models be refined if they cannot be solved exactly?** Let us imagine we needed the exact answer to whatever our problem is; an approximation, no matter how good, just won't do. **Shouldn't this be a fundamental rule of accepted theories; it needs to work out exactly?**


title: Defects in 3+1 TFTs/2+1 CFTs

text: I would like to know of good pedagogic references to learn about the notion of "defects" in TFTs and CFTs. I am specially interested in 3+1 TFTs (.and probably about their relation to 2+1 CFTs..) In the same strain I would like to know of references which explain whatever is known about being able to translate defects and any other such non-local structures for TFTs into CFTs in one lower dimension. I guess such a dictionary exists and I would like to know of references about it, especially for 3+1 TFTs and their "related" 2+1 CFTs. (..I guess there is some intuition about defects in TFTs being writable as boundary condition changing operators in CFTs in lower dimension..it would be great if someone can provide some references in that direction..)


title: Water doesn't flow above the rim, one reason is surface tension. Is another reason viscosity?

text: ![enter image description here](http://i.stack.imgur.com/kPKaJ.jpg) According to Surface tension, water molecule don't get the force from outside and get little bit outward. Is one reason viscosity? Let's look at the **water in a fully filled glass**. No part is outside the surface. Now I pour more water, due to friction from lower level, water doesn't flow towards the side and stays there, the more water we add the more surface is build above _0 level_. As the level increases, force of friction decreases, and at certain point, force of friction is overcome by water flow and water flows out. So there are some water left above the surface of fully filled glass and one of the reason reason is viscosity?


title: Potential function and angular momentum of a quantum particle

text: Is there any functional relation between the orbital angular momentum $L$ of a microscopic particle moving in a potential $V(x,y,z)=x^2+y^2+(z^2/2)$.


title: Will my hard-drive wear/damage when I tilt my laptop?

text: I have an old laptop that I turn off every 2 months or so, it's always on, and I use it whenever I need to do something away from my desk, where I have my real machine. Anyway, every time I grab the laptop (which was laying flat on my desk), I tilt it 90 degrees and hold it under my arm for moving. I noticed that every single time I tilt it, I hear a sound which closely resembles the word "tuck". I've finally had an idea about what could it be. My best guess is the torque on the hard disk wants to keep its orientation, so the disk moves slightly and touches something when I tilt the laptop quickly. Could the torque really do such a thing?   I also know that hard-drives are pretty darn solid, so, could the torque have worn out whatever thingie holds the rotating disk to allow it to move as such?


title: Why do objects fall at the same acceleration?

text: I read these two posts and now I am more confused. Do heavier objects fall faster? Don't heavier objects actually fall faster because they exert their own gravity? I was going to ask: if mass is an objects tendency to resist acceleration then why do two objects of different masses fall to the Earth at the same acceleration? Then I read those posts and it seems that even though it is very small, the more massive object falls faster. Okay I understand, both objects attract each other. If two cars of different masses collide doesn't the car with less mass accelerate more even though both cars received the same Force. Then that implies you need more force to accelerate a large mass than to accelerate a small mass. Because that is how I see it, the Moon attracts the Earth with the same Force as the Earth attracts the Moon but the Earth accelerates less due to its larger mass. So then how is mass an object's tendency to resist acceleration? I am aware of $F_1 = F_2 = GMm/r^2$. So should we not really be able to see the difference in acceleration when dropping a massive object?


title: Wavefunction of an electron

text: Electron is a spin $\frac{1}{2}$ particle, so needs 2-component wave function but the Dirac theory of electron is based on 4-component wave function, are all components of it independent or only two of them are independent?


title: Cause of radiation resistance

text: I was reading Radiation Damping and light scattering from Feynman's lectures on physics. The following excerpts are from chapter 32, Vol I > What is this radiation resistance due to? Let us take a simple example: let > us say that currents are driven up and down in an antenna. We find that we > have to put work in, if the antenna is to radiate energy.....Against what > force are we doing this work? Then he goes on to say that this problem is resolved in case of such antennas(field of charges in one part of the antenna produce a force on charges in other part), but not for "point" charges or electrons. He provides an obsolete explanation based on the finite size and internal structure of an electron but goes on to say that it is false, and hence the question is unanswered. **Now knowing that this excellent book was published in 1964, has the modern quantum mechanical ideas solved this problem for an electron?**


title: Does the Lorentz force applied to a current carring wire by a magnetic field act in the negative or positive direction of the right hand rule?

text: If I say that I am calculating the Lorentz force $F$ applied to a wire carrying a current $i$ at a point $P$ in a magnetic field $B$, would the actual force be opposite of that given by the right hand rule since electrons are actually flowing rather than the positive charge suggested by conventional current (i.e. positive charge doesn't actually flow through a wire)?


title: Proof of Connected Diagrams

text: If $Z[J]$ is the generating functional for the path-integral, could any prove (or more reasonably, refer me to a proof) that $$W[J]\equiv\frac{\hbar}{i}\log\left(Z[J]\right)$$ "generates" only connected diagrams? So far I've only seen theory-dependent "examples" (basically showing how in $\phi^4$ theory the two-point function from $W$ gives only connected contributions). I'm looking for a generic systematic proof for a general field theory.


title: Average force and pressure in a 3D box due to quantum non-interacting particles

text: A gas of non-interacting quantum particles occupies a box with lengths $L_1, L_2, L_3$. Calculate its energy and thus the average force and pressure exerted by the gas on the walls of the box. I have this as a solved example but the steps aren't explained and I'm confused. I tried looking online for a similar derivation but couldn't find anything. The idea is to express the energy and calculate the force as its gradient but it does so taking the derivatives with respect to the lengths $L_1, L_2, L_3$ which I find puzzling. I mean if we had an expression for a potential at any points in the space in the box than I would understand this approach but as it is, I don't see it! Here's the expression for energy which is then differentiated w.r.t to the $L$'s to get the force: $$E = \frac{\hbar ^2}{2m} \left( \left( \frac{n_1}{L_1} \right) ^2 + \left( \frac{n_2}{L_2} \right) ^2 + \left( \frac{n_3}{L_3} \right) ^2 \right) $$


title: What's the domain of usability of these two expressions for entropy

text: $$dS=\frac{dQ}{T}$$ $$S=k\ln \Omega$$ What assumptions are being made about the system/process that allow for using those expressions?


title: How to use Ising Spin model for prediction of time series "Phase"

text: I am investigating a 2d Ising Spin Lattice. I have been able to generate a Monte Carlo app that gives me the changing spin matrix through my iterations - like the many examples on the web. However, I am trying to predict an independent variable (atmospheric pressure). I believe it is possible to measure the pseudo "Magnetism" and "Energy" of the timeseries using the Ising approach - but I am not sure how to do that. I assume it entails flipping the "spin" based on the timeseries changes, but I am not really sure weather that is the right approach. If anyone could give me some advice that would be great. EDIT 1 I have a series of pressure readings:                Xt :1..N      Instead of using Metropolis flip where               deltaE = 2 * J * Si * sumOf(Sneighbours)     if (deltaE < 0 || exp(-deltaE/TEMP) > rand())            Si+1 = -Si      I need to set               Si+1 = someFunctionOf(Xt)      But I don't know what "someFunctionOf" should be. If I did I could update the Lattice from Xt, and workout a "live" Magnetism and I am hoping that might tell something about the current phase of pressure changes.


title: Quantum Collapse

text: When we observe a quantum object does it collapse into a point? Or does it collapse into a smaller wave of area that is out of our range of accuracy? My gut tells me the latter.


title: Can TOTEM's T2 detector measure differential cross sections?

text: My current research involves making a prediction for data collected by the TOTEM experiment at the LHC. The experiment is primarily designed to measure the _total_ inelastic and elastic scattering cross sections, and accordingly all the results I've seen from the TOTEM collaboration show $\sigma_\text{inel}(\sqrt{s})$ or $\sigma_\text{el}(\sqrt{s})$. But the calculation I'm doing outputs (essentially) the differential cross section $\frac{\mathrm{d}^3\sigma}{\mathrm{d}^2 \mathbf{p}_\perp \mathrm{d}\eta}$ at a specific value of the pseudorapidity $\eta$ and transverse momentum $p_\perp$. Is TOTEM's T2 detector capable of recording $p_\perp$ and $\eta$ for its detected events, so that it could bin them and thus report the differential cross section? Or is it only capable of measuring total event counts? In the latter case it wouldn't be much use to make a prediction of the differential cross section; I'd have to integrate to get the total cross section.


title: (Canonical) Partition function - what assumption is at work here?

text: The canonical partition function is defined as $$Z=\sum_{s}e^{-\beta E_s}$$ with the sum being over all states of the system. The way I saw this derived was by assuming that for each state, the probability of the system occupying that state is proportional to the Boltzmann factor: $P(E=E_s) = c \cdot e^{-\beta E_s}$. Summing the probabilities to one gives $Z=\frac{1}{c}$. My question is : What principles were used in order to get this probability distribution for the energy?


title: How does angular momentum transfer between a planet and its moon?

text: Could you explain how a moon draws angular momentum from a planet? I know that the gravitational force transfers momentum, but I don't understand the mechanics behind it.


title: What results from particle collision would ensure the existence of the graviton?

text: I understand that particles are smashed together to try to enable us to detect some sort of graviton presence but we can't actually detect a graviton due to the fact that it 'exists' in some extra dimension that we are unable to see, so what are the results we would want to see from colliding 2 particles to be able to be sure of the existence of the graviton?


title: What do bullet trails really look like?

text: In the movie The Matrix, during "bullet time" sequences, bullets are shown trailing evenly-spaced refracting blobs:   * http://2.bp.blogspot.com/_uypMqSrmflo/S18yZJPET_I/AAAAAAAAAGk/CDjoZaOLbwY/s400/131222__matrix_l.jpg   * http://www.mccd.edu/faculty/kanemotok/cpsc01_websites/Fall09/AnaKenney/CGI%20website/images/matrix_l.jpg which presumably represent shockwaves or vapour trails. In reality, if it were possible to (almost) freeze time and move around the scene, what would a real bullet trail look like, if at all visible? Is it significantly different for different types of bullets (ignoring tracer bullets) or atmospheric conditions? These video clips claim to show trails, but they are not at all obvious:   * http://trophyroom.com/video/5FQUc4sg9f/   * http://youtube.com/watch?v=I0Avj0QfBtE


title: Is forward scattering = no scattering?

text: What is forward scattering? If it is equivalent to no scattering, then why not call it "no scattering"?


title: Phase space suppression in loop integrals

text: Often (if not always) when calculating loop integrals in QFT one encounters extra 2 $\pi$'s that serve to suppress higher order corrections more so than the most naive guess would give. This happens to the extent that the fine structure constant is defined as $\frac{e^2}{4\pi}$ in anticipation of the fact that get this extra factor beyond the $e^2$ when we first start doing our perturbation theory after reading the Feynman rules from the Lagrangian. That is, people are so used to these extra factors we anticipate them and go ahead and define our perturbation theory in alpha instead of just $e^2$. Is there a deep reason for this? Will this happen in all diagrams for scalar, vector or Dirac particles? This seems like something worth understanding since it will determine if a theory stays within the perturbative window for larger values of the coupling constant appearing in the Lagrangian than the most naive guess would imply.


title: The mathematical physics of saltatory neuron action potential transmission

text: Some time ago, I tried to understand saltatory electrical potential transmission on a myelinated axon but have not found a clear mathematical model of this mechanism. I understand the cable theory, but that seems only explains the normal continuous transmission within the cable either with or without insulation, instead of the saltatory one between two un-myelinated synapsis of an axon. What is the model that describes the interior and exterior interaction that results in the SALTATORY transmission mechanism? It would be even better if the answer to my question could be accompanied with reference to pertinent papers on the mathematical model on the saltatory transmission. Here is a reference on neuronal transmission modeled as electrical potential transmitted in a cable and as an application of the cable theory. My problem with this theory is that it does not account for the saltatory transmission mode of the action potential in a myelinated neuron.


title: Fermi-Dirac Statistics

text: In Fermi-Dirac statistics the probability of being in a certain energy state is $$f(E) = \left[1 + \exp\left(\frac{E-E_F}{k T}\right)\right]^{-1}$$ In the area that I'm looking at the texts always assume the population's energy is much greater than the Fermi Energy and so approximate this as the Boltzmann Distribution. However, I am interested in the probability and am wondering if there is a way I can express the difference $E - E_F$ as a function of the temperature, the work function or some other common parameters? If anyone has any insight I would appreciate some guidance. Thank you, John


title: What are the physics behind the Coriolis effect?

text: What causes the Coriolis effect?


title: How to add a potential term to the Dirac Equation?

text: I've read that if you have a Hamiltonian for the Dirac Equation, you can add a potential term to it simply by adjusting the momentum operator so that $p^\mu \rightarrow p^\mu-A^\mu$, where $A^\mu$ is the relevant potential. But how do you calculate $A^\mu$? For example, what would $A^\mu$ be for an electron in an electromagnetic field given by the tensor $F^{\alpha\beta}$?


title: How you feel in outer space vs. orbit?

text: I understand that when someone is in low earth orbit, the "pull" of their inertia is equal to the pull of gravity. However, the force of gravity is still acting on them. However, if they are in outer space there the pull in any direction is miniscule, they should feel different that the way they feel in orbit, because you feel forces on your body and in outer space there is no net force while in orbit there is. Am I correct?


title: Bose–Einstein statistics exercise

text: I've a basic Bose–Einstein statistics exercise. I've tried to solve it in two ways, but each way gives a different result. We have $n$ identical bosons without interactions at temperature $T$. There are two states, of energies $\epsilon > 0$ and $0$. The question is: What is the value of $U$?   * The first way is to say that the number of particles of the system is the sum of the average number of bosons for each energy : $$n = \frac{1}{e^{\beta\epsilon-\alpha}-1}+\frac{1}{e^{-\alpha}-1}$$ Given $n$, we can obtain $\alpha = f(n)$ and finally say that $$U = \frac{\epsilon}{e^{\beta\epsilon-f(n)}-1}$$ I've calculated $f$ (it is quite long...), and it doesn't give the same results than the second method.   * The second way is to say that $$Z(\beta) = \sum_{n_0 + n_\epsilon = n} e^{-\beta n_\epsilon \epsilon} = \frac{1 - e^{-\beta\epsilon (n+1)}}{1-e^{-\beta \epsilon}}$$ and then $$U = -\frac{\partial \ln Z}{\partial \beta}$$ In fact, I've seen the solution of the exercise and it is the second answer. But why is the first wrong?


title: Do Gauge Theories (CFTs) Have Phase Transitions as the 't Hooft Coupling is Varied?

text: With an eye toward AdS/CFT, I'm wondering if large $N$ CFTs have a (quantum) phase transition as the 't Hooft coupling is varied. To be more specific -- if I look at correlation functions of low-dimension, single trace operators as a function of $\lambda$, are they analytic, continuous, discontinuous in $\lambda$? How much do we know / any references? I'm already aware of 0811.3001.


title: A resource theory of quantum discord?

text: Local Operations and Classical Communication (LOCC) is the classic paradigm for studying entanglement. These are things that are `cheap' and unable to produce entanglement as a resource for a quantum information processing task. We can also describe equivalence classes of entangled states if elements of each class can be transformed to another in that class under LOCC. We can discuss entanglement distillation of going from M copies of a noisy state to N copies of a more entangled state by LOCC. Finally, if some states are undistillable (i.e. N=0 for all M), given a different state $\sigma$, the original noisy state $\rho$ can be activated (or catalysed if you want $\sigma$ to be unchanged) this to a more entangled state. Recently, a lot of discussion has centred around quantum discord. Quantum discord aims to capture nonclassicality in states, if not necessarily entanglement. Loosely speaking, a quantum state $\rho$ without discord (concordant) is one where there is a basis of product states (e.g. $|\psi_{1}\rangle|\psi_{j}\rangle...|\psi_{n}\rangle$ for $n$ parties) with respect to which $\rho$ is diagonal. Discord (but not entanglement) has been related to mixed state quantum computing as well as quantum state merging. Interestingly, it has been shown that given two (non-equal) concordant states, there exists a protocol that produces a distillable entangled state as shown by M. Piani et al and some similar results in A. Streltsov et al. I am curious at how far this analogy between entanglement and `nonclassicalness' distillation can be drawn, in particular, can we construct a reasonable resource theory of discord? I doubt I am the first to think of this so if anyone has any background on this, I'd really appreciate it. We can restrict to being able to produce concordant states and then operations that preserve classicalness. From a paper by B Eastin we know that unitaries that preserve classicalness amount to a permutation of eigenvalues with a change in product basis; we could go beyond the model of local operations. Has anyone produced any results on distillation of discord? If this is all trivial to some of you, my apologies. I am trying to understand what discord actually means from the useful resource theoretic point-of-view.


title: Edge theory of FQHE - Unable to produce Green's function from anticommutation relations and equation of motion?

text: I'm studying the edge theory of the fractional quantum Hall effect (FQHE) and I've stumbled on a peculiar contradiction concerning the bosonization procedure which I am unable to resolve. Help! In particular, consider the first few pages of X.G. Wen's paper "Theory of the edge states in fractional quantum Hall effects". Here, Wen defines a fermionic field $\Psi(x,t)$ in (1+1) dimensions in terms of a bosonic field $\phi(x,t)$ as $$ \Psi(x,t) \propto e^{i\frac{1}{\nu}\phi(x,t)} .$$ The number $\nu$ is the filling fraction of the FQHE, which we shall set to $\nu=1/3$ for simplicity. The bosonic field fulfills the somewhat strange commutation relations $$ [\phi(x,y),\phi(y,t)] = i\pi\nu\,\text{sgn}(x-y) $$ which are necessary to make $\Psi(x,t)$ anticommute like a proper fermion $$ \lbrace \Psi(x,t), \Psi^\dagger(y,t) \rbrace = \delta(x-y) .$$ Moreover, the bosonic field satisfies the equation of motion $$ (\partial_t-v\partial_x) \phi(x,t) = 0 .$$ Some operator algebra shows that $\Psi(x,t)$ is also a solution to this equation of motion. However, it appears to me that these two requirements, anticommutation and the equation of motion, already fix the Green's function of the fermion! However, Wen goes on to note that these fermions have the Green's function (equation (2.12) in the paper) $$ G(x,t) = \langle T(\Psi^\dagger(x,t) \Psi(0)) \rangle = \exp[\langle\frac1{\nu^2}\phi(x,t)\phi(0)\rangle] \propto \frac{1}{(x-vt)^{1/\nu}} .$$ I do not understand how this can be. After all, from the anticommutation relations and the equation of motion, we can calculate the Green's function to be $$ G(x,t) \propto \frac{1}{x-vt} .$$ To do this, define Fourier modes $\Psi_k, \Psi^\dagger_k$, obtain the usual anticommutation relations for these, solve the equation of motion, and transform back into real space. The result will be as noted, and the exponent $1/\nu$ will be missing. > Where did the exponent $1/\nu$ go? What's wrong with calculating the Green's > function from the anticommutation relations and the equation of motion? Maybe there's something going on inside the $\delta(x-y)$ part of the anticommutation relations? If so, what exactly? Or maybe something about the ground state? Or something about the bosonization procedure as a whole?


title: why does a larger thermal conductivity provide a smaller temperature gradient?

text: I was thinking about Fourier's Law in heat transfer today and for some reason I am just not understanding the relationships it gives us. Fourier's tells us that if the heat transfer rate is kept constant, then a larger thermal conductivity provides a smaller temperature gradient. I am confused about the physical reason behind this or I am just misunderstanding the definition of thermal conductivity. I thought that since thermal conductivity is ease of heat transfer through a material then a high thermal conductivity would mean heat is easily transferred so one side of the material is at a much temperature than the other and a large temperature gradient is created. In fact, this is actually the opposite


title: Block on a block problem, with friction

text: Consider two blocks, one on top of the other on a frictionless table, with masses $m_1$ and $m_2$ respectively. There is appreciable friction between the blocks, with coefficients $\mu_s$ and $\mu_k$ for static and kinetic respectively. I'm considering the fairly routine problem of determining the maximum horizontal force $F$ (say, to the right) that can be applied to the top block so that the two blocks accelerate together. The problem is not hard to solve symbolically. If the two blocks move together, their accelerations are the same, and the top block doesn't move with respect to the bottom block, so only static friction is in play. In a standard coordinate system (with $x$ oriented to the right), the sum of horizontal forces for the top block is $$F-F_{sf}=m_1a$$ and for the bottom block $$F_{sf}=m_2a$$ where $F_{sf}$ is the force of static friction. Solving for $a$ in these two expressions, and then equating them, gives $$F=\frac{(m_1+m_2)F_{sf}}{m_2}$$ The maximum such force will therefore be achieved when $F_{sf}$ is maxed out at $\mu_s m_1g$, so $$F_{max}=\frac{m_1}{m_2}\mu_s(m_1+m_2)g$$ I understand this solution, but conceptually I don't have a response to the following nagging question: $F_{max}$ is clearly larger than the max static friction force $\mu_sm_1g$ (because $\frac{m_1+m_2}{m_2}>1$), so why doesn't the application of a force of magnitude $F_{max}$ to the top block cause kinetic friction to take over? This line of reasoning would suggest that applying a force $F$ of magnitude greater than $\mu_sm_1g$ would cause the top block to start moving with respect to the bottom block (in which case the blocks no longer accelerate together, as in the above solution). I'm at a loss, conceptually, to say what's wrong here. I suspect it has something to do with being careful about reference frames, but a clear explanation would be much appreciated.


title: Do Killing spinors know global information?

text: The conformal Killing spinor equations on $R\times S^3$ in Minkowski signature are \begin{equation} \nabla_\mu \epsilon=\pm \frac{i}{2}\gamma_\mu\gamma^0\gamma^5\epsilon \end{equation} whose solution is \begin{equation} \epsilon=e^{ix^0\gamma^5/2}\epsilon_0 \end{equation} where $\epsilon_0$ is a constant spinor. Note that we identify $S^3$ with $SU(2)$ and use the left-invariant vector fields of $SU(2)$ as an orthonormal frame. For more detail, please see around Eq.2.20 in the following paper. http://arxiv.org/abs/hep-th/0605163v3 When the superconformal index is interpreted as a partition function on $S^1\times S^3$ in the Euclidean signature, the conformal Killing spinors are modified by the Wick rotation $\epsilon =e^{-x^0\gamma^5/2}\epsilon_0$ as in the following papers. http://arxiv.org/pdf/1104.4482v3 http://arxiv.org/abs/1104.4470 However, $\epsilon =e^{-x^0\gamma^5/2}\epsilon_0$ is not well-defined on the temporal circle $S^1$. This issue also happens on the Killing spinor on $AdS_3$. The metric of $AdS_3$ in the Minkowski signature can be written as \begin{equation} ds^2=-\cosh^2 \rho dt^2 + \sinh^2\rho d\phi^2+d\rho^2~. \end{equation} The Killing spinors on this coordinate takes the form \begin{equation} e^{\frac 12\rho\gamma_3}e^{-\frac {i }2 (\phi+ t)\gamma_1}\epsilon_0 \end{equation} as in http://arxiv.org/abs/hep-th/9310194 . However, when you consider the elliptic genus $Tr(-1)^Fq^{L_{0}-c/24} \overline{q}^{\overline L_{0} -\overline c/24} y^{J}$, the bulk duals which satisfy the E.O.M of the supergravity theory are the Euclidean thermal $AdS_3$ and the family of its $SL(2,Z)$ transformations such as the BTZ BH. Then, again the Killing spinors on the Euclidean thermal $AdS_3$ take the form \begin{equation} e^{\frac 12\rho\gamma_3}e^{(-\frac {i\phi }2 + \frac{t}{2})\gamma_1}\epsilon_0 \end{equation} which are also not well-defnined on the temporal circle. The Killing spinor equations themselves are local, so does the Killing spinor know only local information? If so, how you distinguish the Killing spinors in the NS from in the R boundary condition on 2-torus?


title: diffrent elements from diffrent elements

text: Do we get elements from other elements ? Somebody said they believed the world was visited by aliens and they made slavs of us and then amde us mine gold for them then they leave us to grow into what we no as humanity today. I said it would be easier for them to make the gold if they can make space ships and travel thru space,and i said that we make elements from other elements all the time,i even said the sun takes elements and changes them and bombards us all the time with the new elements. i got the answer back off a self proclaimed astro- no it all scientist geologist brain box ---we need the power of a star to do this and its not so common on earth--i cited the hadron colider as making diffrent elemenst from the 1s it used.i even put plutonium 238 and the break down of it as it goes back to lead. whose right?


title: Pure mathematical exposition vs A "for physicists" approach: Which is better?

text: I'm an undergrad interested in theoretical physics. Don't know enough to decide which specific subfield I may go in, but particle theory (strings, QFT, GR) and mathematically heavy condensed matter seem appealing. I have taken quite a few pure math courses in my undergrad career so far (algebra, analysis, topology etc), but now that I've exhausted most of the undergraduate curriculum, its time to start some advanced topics such as Algebraic Topology and Lie Algebras etc. My concern however is that these courses are offered by the mathematics department and thus the exposition and focus is aimed towards training mathematicians rather than someone who may find them useful in another field. So as expected in such a class, the main goal is to understand and prove the main theorems of these fields. However, as a physicist I'm not sure to what extent I can benefit from such an exposition. It would be highly time consuming to take such an approach to mathematics for someone to which mathematics is not the ends itself, but rather the means. On the other hand, there is the option of using "math for physics" type books such as Nakahara or Frankel to learn these topics, which would be a lot quicker and would have a physicist as a reader in mind, but no doubt, depth of understanding would be lost with such an approach. So my question is: Is it worth it for an aspiring theoretical physicist to learn advanced mathematical topics from graduate math classes (and as a result sacrificing the time spent physics), or would it be a better use of his time to focus on the physics and pick up the math from a text or a class of the "for physicists" type? How would the answer change if applied to a mathematical as opposed to theoretical physicist? Note that over here I'm talking about topics such as Algebraic Topology, Differential Geometry, Representation Theory etc.


title: Ground state energy of hydrogen molecule ion

text: In this paper, it is mentioned: > Furthermore, since the energy of $H_2^+$in the ground state must be lower > than that of an H atom in the ground state,the negative (attractive) forces > in the $H^+ – H$ interaction must play the dominant role How do we intuitively expect the ground energy of hydrogen molecule ion $H_2^+$ should be less than that of hydrogen atom without doing exact quantum mechanical calculation.


title: Uniform Circular Motion Confusion

text: Having some problems when calculating this problem: A fly lands on an old record player, while a 7'' O.D. 45 RPM record is playing. If the fly lands on the edge of the record, determine the records period of rotation. I do not understand how to calculate (T) tau. w = (2*PI*R)(#cycles/sec) = (2*PI*V) T = 1/V R is 7.0 in (or 0.18 meters) 45 RPM is 4.72 radians/second Therefore to find w: w = (2*PI)(0.18m)(4.72rad/s) = 5.33 meters*radians/second Please notice that I have meters * radians when this should be only radians and seconds. I am just looking to calculate Tau (T)


title: What's the acceleration of pendulum when velocity is zero

text: For a pendulum which has a light string to hang a bob, I know that when the bob swing to the leftmost or rightmost end, the velocity of the bob is zero and the acceleration should be maximized. But if you look at the formula of the angular acceleration $a = v^2/r$, but when $v$ is zero, acceleration will be zero too. So why in the text they said that the acceleration at that point is maximum?


title: How to solve following equation (Yukawa field)?

text: By using Lagrangian for Yukawa interaction, $$ L = -\frac{1}{c}A_{\alpha}j^{\alpha} + \frac{1}{8 \pi c}(\partial_{\alpha}A_{\beta})(\partial^{\alpha}A^{\beta}) + \frac{m^{2}c}{\hbar^{2}}A_{\alpha}A^{\alpha} $$ we can get equation of field dynamics, $$ 4 \pi j_{\beta} = \frac{\mu^{2}c^{2}}{\hbar^{2}}A_{\beta} + \partial^{2}A_{\beta}, \quad j_{\beta} = \frac{v_{\beta}}{c} Q \delta (\mathbf r - \mathbf r_{0}(t)). $$ How to solve it (define $A_{\beta}$)? I tried solve equation by using Fourier transform $$ f(r^{\alpha}) -> f(w^{\alpha}) = \int e^{-ir_{\alpha}w^{\alpha}}d^{4}x, $$ and got $$ w_{\alpha}w^{\alpha}A_{\mu}(w_{\alpha}) + \frac{\mu^{2}c^{2}}{\hbar^{2}}A_{\mu}(w_{\alpha}) = 4 \pi \int \limits_{-\infty}^{\infty} e^{-ictw^{0} + i(\mathbf r_{0}(t) \cdot \mathbf w)}dt $$. I cannot take the integral from the right part, because I don't know how the particle moves (i.e. the dependence $\mathbf r = \mathbf r_{0}(t)$). Can you help me?


title: Human as a heat engine

text: humans are heat engines. heat engines can't operate when the source and the sink are at the same temperatures and also if the temperature of the sink is higher than that of the source. in humans' case, the source is our body itself (technically the mitochondria) and the sink is the surrounding. so how can people keep on working when the temperature >= body temperature. ??


title: Lorenz gauge fixing

text: Is it always possible to define function $\psi$ satisfying the Lorenz gauge equation $$ \partial_{\mu}\partial^{\mu} \psi + \partial_{\mu}A^{\mu} = 0? $$


title: Nonsensical dispersion relations for elastic wave propagation

text: In an earlier question about Einstein notation, a link was provided to a medical paper which used acoustic propagation to noninvasively detect the orientation of muscle fibers. In short, muscle fibers are roughly a transverse isotropic material, with Hooke's law given by $$\sigma=\left( \begin{array}{cccccc} \text{C}_{11} & \text{C}_{11}-2 \text{C}_{66} & \text{C}_{13} & 0 & 0 & 0 \\\ \text{C}_{11}-2 \text{C}_{66} & \text{C}_{11} & \text{C}_{13} & 0 & 0 & 0 \\\ \text{C}_{13} & \text{C}_{13} & \text{C}_{33} & 0 & 0 & 0 \\\ 0 & 0 & 0 & \text{C}_{44} & 0 & 0 \\\ 0 & 0 & 0 & 0 & \text{C}_{44} & 0 \\\ 0 & 0 & 0 & 0 & 0 & \text{C}_{66} \\\ \end{array} \right)\cdot \left( \begin{array}{c} \epsilon _{11} \\\ \epsilon _{22} \\\ \epsilon _{33} \\\ 2 \epsilon _{23} \\\ 2 \epsilon _{31} \\\ 2 \epsilon _{12} \\\ \end{array} \right)$$ where $\epsilon_{jk}=\frac{1}{2}(u_{j,k}+u_{k,j})$ where $\mathbf{u}$ is the material displacement from equilibrium. Solving the equation of motion $\nabla\cdot\sigma=\rho\ddot{\mathbf{u}}$ using plane wave solutions of the form $\mathbf{u}=\mathbf{A}e^{i(\omega t-\mathbf{k}\cdot\mathbf{r})}$ yields the dispersion relation $$\Gamma'\mathbf{A}=\mathbf{0}$$ where $$\Gamma'=\rho\omega^2I-\left( \begin{array}{ccc} \text{C}_{11} k_1^2+\text{C}_{66} k_2^2+\text{C}_{44} k_3^2 & -(\text{C}_{66}-\text{C}_{11}) k_1 k_2 & (\text{C}_{13}+\text{C}_{44}) k_1 k_3 \\\ -(\text{C}_{66}-\text{C}_{11}) k_1 k_2 & \text{C}_{66} k_1^2+\text{C}_{11} k_2^2+\text{C}_{44} k_3^2 & (\text{C}_{13}+\text{C}_{44}) k_2 k_3 \\\ (\text{C}_{13}+\text{C}_{44}) k_1 k_3 & (\text{C}_{13}+\text{C}_{44}) k_2 k_3 & \text{C}_{44} k_1^2+\text{C}_{44} k_2^2+\text{C}_{33} k_3^2 \\\ \end{array} \right)$$ where $I$ is the identity matrix. Since the $k_2$ and $k_3$ directions are identical, WLOG you can assume $k_2=0$ to give $$\Gamma'=\left( \begin{array}{ccc} \rho \omega ^2-\text{C}_{11} k_1^2-\text{C}_{44} k_3^2 & 0 & -(\text{C}_{13}+\text{C}_{44}) k_1 k_3 \\\ 0 & \rho \omega ^2-\text{C}_{66} k_1^2-\text{C}_{44} k_3^2 & 0 \\\ -(\text{C}_{13}+\text{C}_{44}) k_1 k_3 & 0 & \rho \omega ^2-\text{C}_{44} k_1^2-\text{C}_{33} k_3^2 \\\ \end{array} \right).$$ By choosing a particular material and frequency $\omega$, plotting the regions where the dispersion relation holds as a function of wavevector $\mathbf{k}=(k_1,0,k_3)$, one gets beautiful images, such as the following picture of the allowed regions using data values for graphite-epoxy fiber lamina (see Table 3) using `{C33 -> 23.4, C11 -> 2.1, C13 -> 0.943, C66 -> 0.527, C44 -> 1.03, \[Rho] -> 1, \[Omega] -> 4}` and $-6.8\leq k_1,k_3\leq 6.8$: ![enter image description here](http://i.stack.imgur.com/O7GS8.png) The horizontal axis is the $k_3$-axis, and the vertical axis is the $k_1$-axis. The red mode corresponds to a pure shear-wave, and the green and blue modes correspond to quasi-transverse and quasi-longitudinal modes. If you just make up some random unphysical numbers for the $C_{ij}$, you occasionally get something which looks like this, using `{C11 -> 0.192954, C33 -> 0.318008, C44 -> 0.269456, C13 -> 0.782727, C66 -> 0.195521, \[Rho] -> 1, \[Omega] -> 1}` and $-6.8\leq k_1,k_3\leq 6.8$: ![enter image description here](http://i.stack.imgur.com/HT5xn.png) In the above, there is one propagation mode (in blue) which extends out to infinite radius in $\mathbf{k}$-space, which seems to physically makes no sense, since $\omega$ is fixed, so you'd expect the dispersion relation to hold only within a bounded annulus in $\mathbf{k}$-space (infinite spatial frequency makes no sense). So this is probably a simple question, but is there any type of material which can have this sort of dispersion relation, or is there some reason why real materials can never behave in this manner (ie, they need to have bounded spatial wavenumber for a given acoustic frequency)? For reference, here is the code I used to derive the above equations and make the images. The images above are essentially defined by $$c_j(k_1,k_3)=e^{-a|\lambda_j(k_1,0,k_3)|}$$ where $c_j$ is the $j$th RGB color channel, $\lambda_j(k_1,0,k_3)$ is the $j$th eigenvalue of $\Gamma'$ evaulated at $(k_1,0,k_3)$, and $a$ is a constant which affects the visibility of the curves.


title: Seashell occurrance

text: Sometimes, sea shells accumulate on the sea shore, but sometimes they will instead be dragged back out to sea. What are the main physical factors that determine which of these things will happen?


title: If $F=ma$, how do can we experience both gravity and a normal force even though we are not accelerating

text: So as I sit in my chair, I experience a gravitational force pushing me into the chair and I'm also experiencing the normal force of the chair pushing back at me so I don't fall. According to Newton's Laws, $F=ma$ and I understand that gravitational acceleration near Earth is $-9.8\: \mathrm{m/s^2}$ so the normal force is $9.8\: \mathrm{m/s^2}$ times my mass. What I don't understand is that if acceleration is change in velocity and my velocity is not changing (thus acceleration is zero), how is there a force?


title: How (or why) equivalence principle led to Einstein field equations?

text: If equivalence principle was origin of general relativity what was the process that this principle led Einstein to developed his theory of general relativity?


This is a soft question. I'm not sure if it is appropriate for this site. Would someone who is only interested in string theory benefit from working out the problems in Jackson electrodynamics? I have to decide if I should take a graduate course in electrodynamics that uses Jackson. Would that be a waste of time? I think that it would be most suitable to engineers but I'm not sure. Do you think that an undergraduate course would be sufficient? I'm going to specialize in Theoretical high energy physics.


title: Can current and resistance result in establishing voltage?

text: I want to know whether current & resistance can cause a voltage across the circuit, according to Ohm's Law $V=IR$. Is it possible to use a low current to produce a high voltage?! How can this be, as current is directly proportional to voltage?


title: Derivation of superluminality in astronomy

text: Can somebody provide derivation of the relativistic explanation of (apparent) superluminality in astronomy? The derivation on wikipedia seems to be a bit confusing.


title: Levi-Civita symbol in Euclidean space

text: Suppose a component of tensor field is described by $B^k=\varepsilon^{kij} \phi_{ij}$. If we define $B^k$ in an **Euclidean space** then does the rising or lowering of the indices of the Levi-Civita symbol change the sign? I mean does $B^k=\varepsilon^{kij} \phi_{ij}=\varepsilon_{kij} \phi_{ij}$?


title: How to calculate the amout of time for oil to get out from a pneumatic cylinder?

text: I have a system. You can assume it is just a 1" diameter steel pneumatic cylinder (1 foot long). Assume we ignore the bore size issue at this point. One chamber is filled up by oil (6" of volume), this chamber also has a hole to let the oil to go to atmosphere. The other half is connected to a 'x' psi gas pressure source (let x = 20 at this time). My question is how to calculate the amount of time does the system take for all oil to get out from the cylinder? Please give me some direction Thanks


title: When the sun's magnetic field reverses, how can "the north pole (have) already (...) reversed, and (we are) waiting for the south pole to catch-up"?

text: I was reading a popular news account of the sun's upcoming magnetic field reversal at http://www.csmonitor.com/Science/2013/0807/Sun-s-magnetic- reversal-means-big-changes-for-the-solar-system-video, and was confused already in the second paragraph: > Data from NASA-supported observatories indicate that the next flip will > happen in just three to four months – the north pole has already jumped the > gun and reversed, and scientists are now just waiting for the south pole to > catch-up. The completed flip will herald changes throughout the entire solar > system, according to a NASA video. This doesn't make any sense to me. Given that there are no magnetic monopoles, there is no way for the sun to have two "south" magnetic poles. The field lines have to close somehow. (Granted, I don't for the life of me understand what a flux rope is, although the videos are cool.) I conclude that one of the following must be true:   1. Most likely: the reporter misinterpreted / mistranslated-into-colloquial something a scientist said.   2. Reasonably likely: my understanding of magnetism is completely wrong.   3. Unlikely: all of the known physics of magnetism is completely wrong. The reason I think 3. is unlikely is that it would have been the headline, not "Sun's magnetic reversal..." So which is it? More generally, if it's possible to give a somewhat-technical description of "Sun's magnetic reversal", that would be awesome (but I realize that most likely that would require a book chapter, and is outside the scope of this forum). Feel free to gear your answer at the level of an incoming physics PhD student.


title: a question about singularities in gravity and Physics in general

text: I had this doubt bugging my mind for a long time about singularities in Physics. I heard that R.Penrose and S.Hawking have proposed that there could be singularities at Blackholes and at the time of the origin of universe (at the time of Bigbang). The concept of singularity is essentially a mathematical one. (unless you call anything that you cannot explain physically as a singularity). My question is (which i want to get it resolved once for all so that i stop bugging people with it :-))....We know that the gravitational field has a singularity at points where there are point masses, so is the case with point charges and electric field. So what is it they have proved about existence of singularities. I assume that they most probably have started with a universe which doesn't have point masses. If they did'nt start with point masses, then what did they actually started with. I want to know what they have started with in mathematical terms and physical terms, once for all. I am open to read some references as well to resolve this doubt. **EDIT :** I am more puzzled as to why I am not getting an answer to the question I've asked which doesn't involve any analysis but a plain request for information. Did they (R.P and S.H) assume existence of point masses in their model of universe (or objects or whatever it is) or they have assumed smooth matter distributions ? This is a fairly straight forward request for information....I do not understand why I don't get the answer for the question I've asked...rather I get something else.


title: Equivalences for mCi doses and Rads?

text: What would be the magnitude of radiation received for a 50mCi dose of radioiodine I-31 as compared to say RADS? I read that it might be on the order of about 35 RADS. Does this make sense?


title: Why isn't the maximum eigenvalue of $J_z$ squared equal to the maximum eigenvalue of $J^2$?

text: During a standard derivation of the eigenvalues of the angular momentum operators, $J^2$ and $J_z$, where $$J^2|\alpha, \beta\rangle =\hbar^2\alpha|\alpha, \beta\rangle$$ and $$J_z|\alpha, \beta\rangle =\hbar\beta|\alpha, \beta\rangle$$ one can show that $\alpha \geq\beta^2$. Textbooks at this point say that there must exist $\beta_{max}$ for a given $\alpha$. My question is, why cannot we say, based on the last inequality, that $\beta_{max}=\sqrt{\alpha}$?


title: The expansion of space time (EST)..and the one directional flow of time?

text: Time flows one way...cause creates effect. Could the expansion of the universe also cause an expansion of space time in a similar fashion?...and if so, would this phenomenon explain the one way direction of time? I read a article about EST (expanding spacetime theory) that seemed to back this up.


title: MVH amplitudes and the unitarity method

text: In the last 5 years there has been a silent revolution in QFT called the unitarity method and the Maximum Violating Helicity (MVH) Amplitudes that basically consist an alternative way to obtain the same amplitudes that you would obtain by the Lagrangian formalisms and the Feynmann diagrams without breaking the bank in supercluster computing time. Now, there is little known about why/how this method works in the theoretical aspect. Are there interesting things that might result from this in the theoretical domain, or it just happens to be an extremely convenient calculation tool? What are the main insights that the existence of these identities give us? If you feel that a summary of the state of the art is all that can be offered for now, feel free to give that as an answer.


title: What is mean by 'good quantum number' in spectroscopy?

text: In electronic spectroscopy of molecules, why some quantum numbers are considered to be 'good quantum numbers'? For example, $n$ and $l$ are said to be not good quantum numbers while $j$ is considered to be a 'good quantum number'. What is the logic/idea behind these concepts?


title: Conductor resistance calculation method

text: **1.** What are the variables that effect on a conductor resistance (I mean all of them)? First of all I would like to say that I know how to calculate the resistance of a conductor using the method below: $R=\rho\dfrac{\ell}{A}\,$ where $\ell$ is the length of the conductor, measured in metres [m], $A$ is the cross-section area of the conductor measured in square metres [m²], and $\rho$ is the electrical resistivity of the material, measured in ohm-metres $(\Omega·m)$. Now I can calculate the resistance of a wire using the relation between the electrical resistivity of the material, the length of the conductor and the cross-section area of the conductor. But there is a missed variable in the previous method, it's the temperature change $\Delta T$. **2.** So I thought that if I would use the previous method with the temperature change variable calculated, then I should multiply it by temperature change in kelvin (because of the direct proportion between resistance and temperature change)? Is that correct? $R=\rho\dfrac{\ell}{A}\times\Delta T$


title: Conserved currents in higher-spin theories

text: After the proposal of Maldacena (AdS/CFT), there have been numerous attempts to find out gravity duals of various kinds of CFT. Klebanov and Polyakov gave one such correspondence here. The claim is this: The singlet sector of the critical $O(N)$ model with the $(\phi^{a}\phi^{a})^{2}$ interaction is dual to the minimal bosonic theory in $AdS_4$. Of course, we have to take the large $N$ limit. Simplest such $O(N)$ invariant theory (free, with no interactions) is: \begin{equation*} S= \frac{1}{2}\int d^3 x \sum_{a=1}^{N}(\partial_{\mu}\phi^{a})^{2} \end{equation*} How do I derive the conserved currents in this theory? $\phi^{a}$ are $N$-component vector fields and NOT $N \times N$ matrix fields.


title: Simple explanation of relation between speed of sound and r.m.s. speed?

text: In an ideal gas, the speed of sound $v_s$ is related to the r.m.s. molecular speed $v_m$ by $$\frac{v_s}{v_m}=\sqrt{\frac{\gamma}{3}} \qquad ,$$ where $\gamma=C_p/C_v=7/5$ for a diatomic gas. I understand how to prove this relation from first principles. However, it seems mysterious to me that it pops out like this in the end. I suppose it's inevitable for dimensional reasons that there will be some relation of this form, since if we want to describe a sample of an ideal gas, a sufficient set of unitful parameters is $m$ and $kT$, and there is a unique way of combining these to give units of velocity. (In a solid or liquid, we have other parameters such as the Young's modulus.) However, this type of dimensional argument doesn't prove that the ratio of the speeds is of order unity. Is there any straightforward physical plausibility argument for the fact that this ratio of speeds is constant, and for the fact that the ratio is of order unity? I guess it's implausible to have $v_m \gg v_s$, since it seems like then a sound wave would sort of get scrambled and lose its identity because of molecular motion, and this scrambling would take place in much less than a period.


title: Strain and stress tensor

text: I have problem by definition of strain and stress. From Gockenbach's book that our reference for FEM, we have $$\epsilon=\frac{\nabla u+ \nabla u^T}{2},$$ that $u$ is vector displacement, and $\nabla u$ is the Jacobian of $u$. So we have $\epsilon$ is symmetric and also $\sigma$, that is $$2\mu \epsilon+\lambda tr(\epsilon)I$$ My problem is that I see everywhere this statement: if $\epsilon$ is symmetric or if $\sigma$ is symmetric we have... why? I can not see the case that they not be symmetric,


title: Find E and B from vector potential

text: I have a vector potential given by: $\mathbf{A}(x,t) = \mathbf{e}_{y}\frac{1}{2} e^{-(x-ct)^{2}/{4a^{2}}}$ Now, the question is "Determine the **E** and **B** under the condition that the scalar potential vanishes $V = 0$." But I'm not quite sure what it means when $V=0$ ? As far as I can see, the **B** -field is given by: $\mathbf{B}=\nabla \times \mathbf{A}$ And then I have that: $\mathbf{E} = -\nabla V$ So is it just this straightforward ? That I find the **B** -field from **A** , and since $V = 0$, the **E** -field is zero, or am I doing it wrong ? Thanks in advance :)


title: Quantum field theory meson scattering calculation (scalar yukawa theory)

text: Please see this question for a clear background of the notation I use. My issue is that I want to use Wick's theorem to calculate the amplitude of meson $\psi(p_1)\psi(p_2)\rightarrow\psi(p_1')\psi(p_2')$ scattering. I can quickly get to the point where I need to evaluate: $\langle p_1',p_2'|:\psi_1^\dagger\psi_1\psi_2^\dagger\psi_2:|p_1,p_2\rangle$ In my notes, the next step is $\langle p_1',p_2'|:\psi_1^\dagger\psi_1\psi_2^\dagger\psi_2:|p_1,p_2\rangle=\langle p_1',p_2'|\psi_1^\dagger\psi_2^\dagger|0\rangle\langle0|\psi_1\psi_2|p_1,p_2\rangle$ where he seems to have changed orders of the fields and inserted $|0\rangle\langle0|$, $|0\rangle$ being the vacuum of the _free_ theory. Can anyone explain what this particular step is? From thereafter the derivation is pretty clear to me


title: Conserved charges given conserved current via Noether's theorem

text: Let $j^{\mu}_{a}$ be the conserved current associated with an infinitesimal symmetry transformation, cf. Noether's theorem. The conserved charge associated with $j^{\mu}_{a}$ is $$Q_a = \int d^{d-1}x j^{0}_a,\,\,\,\,\,\,\,(1)$$ where $j^0_a$ is the time component of $j^{\mu}_a$ and $d^{d-1}x$ is the spatial integration measure. The time derivative is $$\dot{Q_a} = \int d^{d-1}x \partial_{0} j^{0}_a = -\int d^{d-1}x \partial_{i}j^i\,\,\,\,\,\,(2) $$ which vanishes by the generalized divergence theorem. I was wondering if someone could explain how equations (1) and (2) are arrived at. I am wondering if in eqn (1), the author skipped a step and by integrating over the timelike component of $j$ reduced the integral to an integral purely over space and by setting $\mu = 0$ as a result of already integrating over the time component. I guess that $\dot{Q_a} = \frac{d}{dt}Q_a$ and since $j$ is a function of coordinates, when that derivative is brought into the integral, it turns to a partial and the subscript $0$ denotes a time derivative? Many thanks for any clarity on the above.


title: Electric Potential due to conducting sphere and conducting shell

text: 1. The problem statement, all variables and given/known data A solid conducting sphere having a charge Q is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of – 3Q, the new potential difference between the same two surfaces is???? Answer is V.Let R be the radius of the outer surface of shell and r of inner surface and let the radius of sphere be X. Consider epsilon not as 'ε'   1. Relevant equations potential at a point =Q/4∏ε.   1. The attempt at a solution  If Q charge is on the surface of sphere then it will induce -Q charge on the inner surface of the shell and Q on the outer surface of shell.Potential at the surface of sphere due to charge inside sphere is Q/4∏εX but I am unable to calculate NET potential at surface of sphere(potential due to charge outside and inside sphere). I know that potential on the outer surface of shell is Q/4∏εR.


title: If QFT is a sum over 1-D topologies and String Theory over 2-D topologies, what is the corresponding theory for N-D topologies?

text: My understanding is that perturbative QFT can essentially be described as a weighted sum over 1-D topologies (ie Feynman graphs), and String theory is essentially the generalization to a sum over 2-D topologies. Why do we stop here? Is there a name for the theory defined as the sum over 3-D, or N-D topologies?


title: Will pseudo force effect buoyant force in a liquid?

text: Imagine a body in a tub which is in a lift. When the lift is stationary, the body is floating. If the lift accelerates upwards with a constant acceleration, what will happen to the buoyancy? Will its value change or not? I do not think it shouldn't. But my friend says it will. But why?


title: Microscopic model of RLC circuit equation by analogy to continuous medium mechanics

text: According to the analogy of mechanics and electricity, the 1-D system of damped oscillation is similar to the RLC circuit. The equation of damped oscillation is $$ f=m\frac{dv}{dt}+\gamma v+kx$$ where $f$ is the external force other than the linear restoring force $-kx$, $\gamma$ is the damped coefficient. Meanwhile, the equation of RLC circuit is $$ EMF=L\frac{dI}{dt}+RI+\frac{Q}{C} $$ If we introduce the analogy $$ f\sim EMF,\quad v\sim I,\quad m\sim L,\quad \gamma\sim R,\quad k\sim 1/C $$ the two systems have the identical form. Now I want to generalize this analogy to 3-D microscopic form. It is easy to generalize the mechanical oscillating equation to the equation of continuous medium. However, I don't know how to generalize the equation of RLC circuit to 3-D microscopic form. The term $RI$ may alter to $\vec{j}/\sigma$, but how about the other two terms $LdI/dt$ and $Q/C$? What are the microscopic forms of the inductance $L$ and the capacitance $C$?


title: Phase transition through Molecular Dynamics Studies

text: I am doing non-equilibrium molecular dynamics to see the phase transition of single crystal metals in extreme condition (shock compression). I am using NVE to conserve total energy, every thing going fine and I am seeing a kink in P-T curve at the transition point and getting discontinuity in density-T profile, but I am interested to know how can I explain such phenomenon by radial distribution function (RDF). Most of the authors prefer to explain with the help of RDF variation, can you give me a idea or a good reference ,where I get the relation between this two? Any discussion will be a great help.


title: A man running on the treadmill

text: Imagine a man is running on a treadmill. His inertia with respect to floor will be zero because he is not moving with respect to floor. If both he and the tread mill suddenly stops he will not fall, according to Newton's first law. Imagine an ant on the conveyor belt. He doesn't worry about our reasoning. He just believes in 1st law, but can't explain it using the 1st law. (Note: the ant will measure two momentum of the man. same but opposite in direction) Now imagine we are a 2D being, living in a closed world like the surface of the Earth. Is Newton's first law not applicable here? Then what will be the definition of inertia in this world? I think I should mention my current level of understanding. I read general relativity only in popular books.


title: Deriving diffusion coefficients from velocity field?

text: If I know the velocity, $\mathbf{v}(\mathbf{r},t)$, everywhere, is it possible to determine the diffusion coefficient, $D(\mathbf{r},t)$ everywhere as well? Would it be possible from using Fick's Law $$j = -D\cdot\nabla L$$ Or can it be found from the mean-squared displacement with $$\frac{\partial \langle r^2\rangle }{\partial t} = 4D$$ ($6D$ in 3D) What would be the best way to go about this?


title: Does a high energy photon experience deceleration or direction-change when it impinge into water or something due to the change of refraction index?

text: Wave experiences refraction when it propagates into another medium which has different refraction index. Lights surely does experience refraction at the border of mediums which have different refraction index because of its wave- like property. But I am wonder if high energy photons, whose energy exceeds MeV or GeV, also experiences refraction.


title: Argument for symmetry of potential

text: Consider the following electrostatic charge configuration of a spherically symmetric, perfect conductor with total charge $Q = 2q$, where $q > 0$. A point charge $q$ is placed at the position shown. ![Charge configuration](http://i.stack.imgur.com/3Gzvx.png) We are asked to calculate the electric potential $V(r)$ a distance $r$ from the centre for $r > c$ and $b < r < c$ with the convention $V(\infty) = 0$. This question seems to presuppose that the potential $V$ is even symmetric. It is clear that this is the case $r = c$, since a conductor is always an equipotential surface. However, I don't see why there could't be asymmetries for $r > c$, considering the fact that the charge distribution is not symmetric. So what is the argument for this symmetry? If anyone answers using Maxwell's equations, could you use the integral versions?


title: How does series circuit differ from parallel circuit?

text: Why does same current $I$ flow in the series circuit but it isn't the case with resistors in parallels?


title: Algebraic/Axiomatic QFT vs Topological QFT

text: Can anybody please tell me a good source investigating the relation between Algebraic/Axiomatic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT)? Or is there none?


title: Dirac trace theorem

text: I am unable to prove exactly one trace identity that appears in the appendix of Peskin and Schroeder's QFT book. Can someone help me? The theorem [Appendix A.4 eqn (A.28)] says that the order of $\gamma$ matrices inside of a trace can be reversed: $$\text{tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\cdots)=\text{tr}(\cdots\gamma^\sigma\gamma^\rho\gamma^\nu\gamma^\mu)$$ Can someone help me with this trivial thing...?


title: London into Australia in 90 minutes

text: Me and my friend are having a debate on whether it would be possible for a human to travel at 15,000 miles an hour from London to Australia in the matter of 90 minutes. Would a human be able to survive travel in such at fast speeds knowing he will have to overcome immense amount of g's. Basically is it possible for a human? or will he suffer death in the process?


title: Superconductor general concept questions

text: I was thinking about building an electric motor using superconductors and I have some general concept questions in regards to how the behavior might be different from ordinary wires.   1. The Meissner effect, expulsion of magnetic fields from the superconductor. If you construct a solenoid out of superconducting wire does this Meissner effect significantly change the field pattern compared to a regular wire solenoid?   2. If a solenoid is created out of superconducting wire is the current limited to the magnitude of the magnetic field it creates because if the magnetic field strength is too large it will return the superconductor to a non superconducting state?   3. If you have a superconducting wire with a current flowing through it and you expose it to a magnetic field does it still experience a force (obeying the same rules as a regular wire with current flow) or because of the Meissner effect the magnetic field never reaches the current carriers and so no force is experienced? Or is there only a fraction of the expected force experienced because the magnetic field does not penetrate through all the current carriers?   4. If you have two or more strips of superconducting tape stacked on top of each other with current flowing in the same direction through them and then you have a magnetic field strength probe and you probe the space above the stack, would the field be just the sum from each strip or would the Meissner effect in the strips closer to the top surface cause magnetic shielding of the lower strips and so the real magnetic field strength near the surface of the top strip would be less than the sum of the magnetic fields from each current carrying strip?    5. Is current flow in a superconductor only a surface effect? (i thought it was) Why did I see one company selling 3D wires? is that really isolated 2D stacks?   6. Is the only way to make a loop of superconductor run a continuous current is by induction? Thanks for reading and thanks in advance for any responses.


title: A tensor summation question

text: With the definition of the tensor \begin{equation} J_{ij} = I_{ij} - \tfrac{1}{3}\delta_{ij}I^{k}_{k}. \end{equation} I have seen the quantity \begin{equation} J_{ij}J_{ij} \end{equation} written as \begin{equation} I_{ij}I_{ij} - \tfrac{1}{3}I_{ii}I_{jj}. \end{equation} How is this possible?


title: Calculating the Suns vector and distance from the Earth at one time

text: I am developing a C++ program where I'd like to be able to see the Suns direction vector and distance from Earth at a given time, since 2000 Jan 0.0 TDT. The reference frame I want to calculate it is from the Earth, with the up vector running along the axis of the poles. What I have tried so far has been based off http://www.stjarnhimlen.se/comp/tutorial.html#5, but its not entirely the output I'm after; and I feel I can probably achieve this much easier. Any ideas? Sorry if this is off-topic.


title: What does Planck/WMAP/COBE actually measure when studying the CMB?

text: I have been reading a lot about Cosmological experiments such as Planck or WMAP that aim to investigate the CMB's power spectrum. I am unclear though as to what theses sky surveys actually measure, is it the distance to regions of temperature fluctuations?


title: Electron Incident On A Finite Potential Barrier

text: This is problem 2.8.3 from Miller's _Quantum Mechanics For Scientists And Engineers_. I'm getting stuck when I try to figure out the wave equation on the right-hand side of the barrier. The original problem is: > Graph the (relative) probability density as a function of distance for an > electron wave of energy 1.5 eV incident from the left on a barrier of height > 1 eV. Continue your graph far enough in distance on both sides of the > barrier to show the characteristic behavior of this probability density. In the text, Miller solves a similar problem where the energy of the incident particle is less than the energy of the barrier. However, in this case, since the particle's energy is greater than the energy of the barrier, I'm using the same form of the wave function on either side of the barrier. Namely, $$ \psi_{left}(x) = C \; exp(ik_{left}x)+D \; exp(-ik_{left}x) $$ $$ \psi_{right}(x) = G \; exp(ik_{right}x) $$ where the potential is zero for $x \lt 0$ and 1 eV for $x \ge 0.$ Substituting the above into Schrodinger's equation produces $k_{left}=\sqrt{\frac{2m(E-V_{left})}{\hbar^2}}$ and $k_{right}=\sqrt{\frac{2m(E-V_{right})}{\hbar^2}}.$ Using the boundary conditions that $ \psi_{left}(0)=\psi_{right}(0) $ and $\psi_{left}^\prime(0)=\psi_{right}^\prime(0),$ we obtain, $$ C+D=G $$ $$ ik_{left}C-ik_{left}D=ik_{right}G. $$ From this, we can solve for $D$ and $G$ to obtain, $$ D=\frac{k_{left}-k_{right}}{k_{left}+k_{right}}C $$ $$ G=\frac{2k_{left}}{k_{left}+k_{right}}C. $$ Substituting back into the wave functions gives, $$ \psi_{left}(x) = C \; exp(ik_{left}x)+\frac{k_{left}-k_{right}}{k_{left}+k_{right}}C \; exp(-ik_{left}x) $$ $$ \psi_{right}(x) = \frac{2k_{left}}{k_{left}+k_{right}}C \; exp(ik_{right}x). $$ A this point, I feel like I've done something wrong, since $|\psi_{left}|^2$ is a sinusoidal standing wave, but $|\psi_{right}|^2$ is constant. Have I made a mistake? Will an incident electron with energy greater than that of an infinitely long potential barrier have a probability distribution that is sinusoidal before the barrier and constant after the barrier? Also, although the original problem uses the word "relative", I'm still troubled by the fact that even in a potential well situation (which is different from the statement of the problem) the above solution would lead to a contradiction. Namely, a finite and constant probability distribution outside of the well would not tend to zero at plus infinity. Can someone either confirm this reasoning, or perhaps point out where I made a mistake? Please note that I've tagged this question as homework, although it does not pertain to any actual homework assignment. I'm just reading on my own.


title: Name for celestial "Prime Meridian"?

text: Is there name for the line that goes from celestial pole to pole at RA 0 degrees 0 minutes 0 seconds? On Earth we would call it the Prime Meridian. Is it called the "Celestial Meridian"?


title: Heat of adsorption from fugacity data

text: I have a set of data relating the fugacity ($\approx$ pressure) to the loading for a given set of temperatures. There are three temperature sets each having five fugacity vs. loading points. The heat of adsorption is given by: $$\Delta H=\frac{\mathrm{d}\bigl[\ln\frac{P}{Po}\bigr]}{\mathrm{d}\bigl[\frac{1}{T}\bigr]} $$ for fixed loading. How would I transform this relationship given that I can calculate $H$? Note, that any practical way to get $H$ from my data would be appreciated - I tried fitting loading to $P$ with a linear function but that failed miserably. (In retrospect this is obvious if you work out the above equation.)


title: Mass defect- From where mass is being lost?

text: As a school student, I have wondered while studying mass defect the following mysterious problem **My assumption**   1. Just like a car's mass is constituted by each part of it(i.e total mass of car will be equal to mass of body+engine etc..), I assume that the mass of nucleus is the sum total of the mass of the particles the nucleus contains, i.e `mass of nucleus =mass of protons +mass of neutrons`   2. I assume all the protons, neutrons are _exactly_ the same, they have (each kind) _exactly_ same mass   3. I assume the charge of proton is distributed uniformly. **Details** A quote from Wikipedia: http://en.wikipedia.org/wiki/Binding_energy > It is observed experimentally that the mass of the nucleus is smaller than > the number of nucleons each counted with a mass of 1 a.m.u.. This difference > is called mass excess. **The mysterious problem I encounter** There is no change in the number of neutrons or protons in the nucleus of the atom. And my idea was as the mass of nucleus is the total mass of protons and neutrons, if the mass of nucleus is to change what I can think about the possibilities are   1. Either the number of neutrons or number of protons should change   2. In order to fit the 2nd assumption, some part of the mass of every neutron and every proton will go (probably as energy) so that all the protons are same and all the neutrons too are same. The 1st possible explanation is obviously wrong as we don't see any change in the number. The 2nd possible explanation is just 'like a reason for the sake of giving a reason' logically difficult to convince myself. If that were true there should have been many types of protons and different types of neutrons (both classified according to their mass), which again is un-intuitive. **The question**   * What's the possible explanation to the question **from where the mass is being lost?**   * Either one of my assumption is wrong (but isn't my assumption just logical?) or there should be some other reason how the mass can vary (may be we should better define what is the mass of the nucleus). In that case, another question , If my assumption 1 is false _What constitute the mass of a nucleus?_   * Isn't there something like a standard proton or a standard neutron (By standard I mean a fixed mass fixed charge etc)   * By my third assumption, If the proton loses some mass shouldn't it lose some of it's charge. And suppose we forget `E=mc^2` can't we say the difference in the potential energy of electron-proton (as it changes when the charge changes) is an explanation of how the nucleus gets binding energy.


title: Does the current light pollution set a fundamental limit for the range of Earth-based telescopes?

text: As far as I know, all deep-sky pictures are captured with the Hubble Space Telescope. If there would be no atmospheric distortion, could we make deeper pictures in the optical spectrum with terrestrial telescopes, say, due to bigger mirrors? Or does the terrestrial light pollution limit the range due to the signal-to- noise ratio?


title: Why did increasing the Ackermann geometry in my race car make it faster in corners?

text: Ackermann geometry is used to account for the different radius arcs that the front tires follow when the steering wheel is turned from center. It's often expressed as a percentage: e.g. 25% Ackermann, 100% Ackermann. I had observed the following physical phenomena: increasing the Ackermann geometry in my race car made it faster. Why did this work?


title: How can a black hole emit X-rays?

text: Considering that a black hole's gravity prevents light from escaping, how can a black hole emit X-rays? Visible light and X-rays are both electromagnetic radiation, shouldn't the black hole's gravity prevent X-rays from escaping?


title: Imaginary time in QFT

text: I'm reading chapter 4 of **Introduction to Quantum Field Theory** by _Peskin & Schroeder_. In the $\phi^4$ theory, the authors state that the ground state of the interaction theory $|\Omega\rangle$ can be written as ($\hbar=1$ ) $$|\Omega\rangle=\lim_{T\to\infty(1-i\epsilon)}\left( e^{-iE_{0}T} \langle \Omega | 0 \rangle \right)^{-1}e^{-iHT}|0\rangle$$ where $E_{0}=\langle\Omega|H|\Omega\rangle$,$|0\rangle$ is the free theory vacuum and $E_n, |n\rangle$ are the eigenvalues and eigenstates of the hamiltonian $H$ .I'm trying to understand that letting the time be "slightly ($\epsilon$ small in some sense) imaginary" is the way to get rid of $n\neq0$ terms in the following series, via a decaying real exponential: $$e^{-iHT}|0\rangle=e^{-iE_{0}T}|\Omega\rangle\langle\Omega|0\rangle\ + \sum_{n\neq0}e^{-iE_{n}T}|n\rangle\langle n|0\rangle $$ $$ \sum_{n\neq0}e^{-iE_{n}T(1-i\epsilon)}|n\rangle\langle n|0\rangle= \sum_{n\neq0}e^{-iE_{n}T}|n\rangle\langle n|0\rangle+ \sum_{n\neq0}e^{-\epsilon E_{n} T}|n\rangle\langle n|0\rangle$$ **Question 1)** I don't really see the oscillating term can be neglected as $T \to \infty$ **Question 2)** How can be physically justified the following substitution $T\to T-i\epsilon$ ? Thanks for your time


title: What happen when two galaxies collide?

text: As the title suggests, I know collision of galaxies happens. Questions are   1. Why do they collide (obviously because of gravity, but in what conditions)?   2. Aftermath of the collision?   3. If there is a super massive blackhole in the center of the galaxies, what happen to them?


title: Is there any evidence for the claim that the moon was once part of the Earth?

text: There is a hypothesis that says a part of the Earth was split away and became the Moon. Is there any scientific evidence for this claim?


title: What kind of telecommunication systems do astronauts use in space?

text: What kind of telecommunication systems do the astronauts use in space? I ask about both earth-to-space and space-to-space.


title: Why do chimneys have these spiral "wings"?

text: While walking around I noticed something very peculiar. Many chimneys had spiral "wings", while others didn't. I came up with two possibilities:   1. The wind circles around the chimney upwards which pushes whatever gases being released even higher into the sky.   2. The wind circles around the chimney downwards which prevent the chimney from going left or right and rather "push" it downwards to make it more steady. I feel that both of those possibilities are silly. So, **why do _some_ chimneys have those spiral "wings"? And why other _taller_ chimneys don't?** ![spiral_chimney](http://i.stack.imgur.com/BrjLl.jpg)


title: Two interacting particles on sphere drift to sphere poles

text: Suppose we have two particles which can move on sphere of radius $r$, and they attract to each other so that their potential energy is $g(d)=ad$ where $d$ is distance between them. I've found Lagrangian, it looks like this (in spherical coordinates): $$L=\frac{r^2}2\left(m_1\left(\dot\theta_1^2+\dot\varphi_1^2\sin^2\theta_1\right)+m_2\left(\dot\theta_2^2+\dot\varphi_2^2\sin^2\theta_2\right)\right)-g\left(l_{arc}\left(\theta_1,\varphi_1,\theta_2,\varphi_2\right)\right),$$ where $l_{arc}$ is arc length between two points on sphere: $$l_{arc}=2r\arcsin\left(\frac1{\sqrt2}\sqrt{1-\cos\theta_1\cos\theta_2-\cos\left(\varphi_1-\varphi_2\right)\sin\theta_1\sin\theta_2}\right)$$ So, equations of motion for particle $i$ look like: $$\frac{m_i r^2}2 \frac{\text{d}}{\text{d}t}\begin{pmatrix}2\dot\theta_i\\\ 2\dot\varphi_i\sin^2\theta_i\end{pmatrix}=-\begin{pmatrix}\frac{\partial g(l_{arc})}{\partial \theta_i}\\\ \frac{\partial g(l_{arc})}{\partial \varphi_i}\end{pmatrix}+\begin{pmatrix}\sin\left(2\theta_i\right)\dot\varphi_i^2\\\ 0\end{pmatrix}$$ I solve a Cauchy problem, so here're initial conditions: $$\theta_1(0)=\frac\pi2+10^{-4}\\\ \theta_2(0)=1.05\cdot\frac\pi2\\\ \varphi_1(0)=-1.5\\\ \phi_2(0)=-1.45\\\ \dot\theta_1(0)=0.003\\\ \dot\theta_2(0)=0.003\\\ \dot\varphi_1(0)=-0.01\\\ \dot\phi_2(0)=-0.01$$ Now as I solve this problem, it appears that the system drifts to the pole of the sphere whenever the particles interact (here $r=100,\; m_1=m_2=1,\; a=1$): ![enter image description here](http://i.stack.imgur.com/K1RxF.png) I've tried multiple methods of solving this problem in Mathematica using `NDSolve` with big range of steps, but still the trend is the same, it seems to not depend on method of solution. So, I seem to have made a mistake somewhere in formulating the problem. If instead of attracting the particles repulse, they again appear attracted by poles: ![enter image description here](http://i.stack.imgur.com/JUmfY.png) Is my derivation correct - i.e. finding Lagrangian and deriving equations of motion? What could be the reason for such strange error?


title: Developement of modern condensed matter physics

text: Do you know any resources describing historical aspects of developments of modern condensed matter physics (many body physics etc)? Thanks.


title: Symmetric transverse traceless tensors of rank $s$ and $(s,0,0,..,0)$ representations of $SO(n)$

text: Can someone help see this connection as to why a spin $s$ (an Integer) particle is to be thought of as a symmetric transverse traceless tensor of rank $s$ and that they lie in the $(s,0,0,..,0)$ highest weight representation of $SO(n)$? * * * Is it true that if the particle is on a space of the form $G/H$ only then is there a consistent notion of "spin" using the representations of $H$?


title: Optical Spectra and the Diffraction Grating

text: I'm doing a lab where we are identifying different gases by their diffraction spectra. This what my lab book says: > Verify and calibrate your spectrometer, by measuring Hg and Ne. Measure 5 or > more separate lines for each gas ($j=1,2...5$). For each line, make 4 > separate measurements of $\theta$ and estimate $\delta\theta$. Calculate > $\lambda$, _presuming_ a diffraction separation of $d=1016$ nm. Now obtain > $\overline{\lambda_{j}} \pm \overline{\delta\lambda_{j}}$. Plot > $\lambda_{j}$ vs $\lambda_{j}^{theory}$, and obtain your best estimate of > the true value of $d$. I'm extremely confused please help! how do I plot $\lambda_{j}$ vs $\lambda_{j}^{theory}$, and how does this help determine $d$?


title: Time diffeomorphisms breaking in inflation

text: I am currently working on the topic of inflation. It seems that at the stage of inflation, the universe can be described as a de Sitter space. In such a space, all spacetime diffeomorphisms are preserved. (That is something I don't really understand but I keep reading that so I admit it for now.) Now, I read that, in order to give the FLRW universe that we know today, the time diffeomorphisms are broken and therefore there is a Goldstone boson associated with this symmetry breaking. I also read somewhere else that it is not the inflaton field that is concerned by this broken symmetry but rather the fluctuation field $\delta \phi$ defined as $\phi(x,t) = \overline{\phi_0} (t) + \delta \phi (x,t)$.   * What is the Goldstone boson ? It is not the inflaton field...   * Does it makes sense to say that only time diffs are broken, since space and time coordinates are not so evidently separated ? I know that the background field $\overline{\phi_0}$ gives a clock but I'm still confused.   * Why is FLRW not invariant under time diffs ? Because of the evolution of $a(t)$ ?   * I thought that in order to apply Goldstone's theorem, the broken symmetry had to be continuous. When we speak about broken time diffs, is it a continuous symmetry ? (Maybe because it's only shift in time ?) Sorry if it is not clear, as you can see I'm confused about this topic. Thank to anyone who can provide some answer, even partial. EDIT : Found in 0905.3746, section 2. > In inflation there is a physical clock that controls when inflation ends. > This means that time translations are spontaneously broken, and that > therefore there is a Goldstone boson associated with this symmetry breaking. > As usual, the Lagrangian of this Goldstone boson is highly constrained by > the symmetries of the problem, in this case the fact that the spacetime is > approximately de Sitter space, in the sense that $|\dot{H}| / H^2 \ll 1$. > The Goldstone boson, that we can call $\pi$, can be thought of as being > equivalent, in standard models of inflation driven by a scalar field, to the > perturbations in the scalar field $\delta \phi$. I believe it makes the things a bit more clear, at least for my first question.


title: How to check $\renewcommand{\vec}[1]{\mathbf{#1}} \vec{v'}\cdot\vec{V}$ and $\vec{v}'^2$ are time derivatives of some other functions?

text: From Landau, Lifshitz Mechanics p.127 $\renewcommand{\vec}[1]{\mathbf{#1}}L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U $ He states that "$\vec{V}^2(t)$ can be written as the total derivative with respect to $t$ of some other function.". I understood that but the point that I could not understand is why we can't write the terms $\vec{v'}\cdot\vec{V}$ and $\vec{v}'^2$ as the total derivative with respect to $t$ of some other function ? $\vec{v}'$ is the velocity of particle measured in non-inertial frame. $\vec{V}$ is translational velocity of non-inertial frame with respect to inertial frame.


title: Do particles gain mass only at energy levels found during the big bang?

text: I am trying to make sure my understanding is correct. At energies and temperatures found during the big bang (or at CERN recently), the Higgs mechanism comes into effect. When it does, there is a break in electroweak symmetry (because the field is unstable?) and a condensate is formed. This condensate is what allows bosons to gain mass. So at the moment of the big bang, all the bosons (and fermions I suppose) were massless (due to symmetry), and then at some point after the big bang when the temperature of the universe was just so, the Higgs mechanism took hold and gave the particles mass. Is this correct?


title: Why does a mirage disappear as you approach it?

text: Related: why do mirages only appear on hot day? The title says it all: why does a mirage disappear as you approach it? The related question does not answer this.


title: What is the experimental reason to believe in a Unified Theory?

text: Is there any experimental findings on behalf of the belief that all forces can be unified to a single force? Or the idea has it's inspiration from kind of thought experiment or philosophical belief.


title: Why do we need different ensembles in statistical mechanics?

text: Why do we study these different ensembles, microcanonical, canonical, grand canonical ensemble ? Are they used for studying different physical system or scenarios?(e.g. in some system you can only treat it as mirocanonical, and in other cases you can only apply canonical ensemble) Do they have the same result at thermodynamic limit?


title: Velocity from velocity potential

text: I have this homework question and I get a different answer to the solutions. In Cylindrical polar coordinates $(r,\theta,z)$, the velocity potential of a flow is given by: $$\phi = -\frac{Ua^2r}{b^2-a^2}(1+\frac{b^2}{r^2})cos\theta$$ Find the velocity. I get the velocity as: $$v = (-\frac{Ua^2}{b^2-a^2}(1+\frac{b^2}{r^2})cos\theta + \frac{2Ua^2b^2}{(b^2-a^2)r^2}cos\theta)e_r + (\frac{Ua^2}{b^2-a^2}(1+\frac{b^2}{r^2})sin\theta) e_{\theta}$$ The answer misses out the second term in the $r$ direction, but I can't see where I've gone wrong. Any help appreciated.


title: Transformation rule of a partial derivative

text: We know the following transformation rule: $$ \partial'_b = \frac{\partial}{\partial x'^b} = \frac{\partial x^c}{\partial x'^b} \, \frac{\partial}{\partial x^c} = \frac{\partial x^c}{\partial x'^b} \, \partial_c. $$ Here $\partial_c$ is covariant tensor of rank $1$. But what is the corresponding transformation rule for $$ \partial^m :=g^{mn}\partial_n?$$ How can we derive the transformation rule?


title: References on wave solutions in continuum mechanics

text: I am interested in literature on known wave solutions in continnum mechanics, precisely the following mechanical equation: $$\rho\partial_t^2u_i = C_{ijkl}\nabla_j\nabla_ku_{l}$$ My interest is spread around its study for cubic lattices, which doesn't seem straightforward to solve analytically to me. But I couldn't find a reference book or articles illustrating a peculiar solution of it.


title: What are the steps and energetic requirements for getting from the HST to the ISS?

text: There have been several general critiques of the realism some of the physics depicted in the movie "Gravity" notably regarding the the orbital mechanics of the transition from the Hubble Space Telescope to the International Space Station. I understand that these center around the energetic requirements of getting from the orbits occupied by HST to the ISS, but am foggy on the details. In particular, I'm curious what steps would be involved, and how much energy would be required at each, and whether it would be possible, with any practicable technology, to carry an source that could provide sufficient energy. My (vague) intuition, from what I can gather, is that one would first need to (1) decelerate from the HST orbit and begin to fall towards the ISS orbital altitude (I'm never clear on whether one also acquires a higher orbital velocity as one falls) (2) accelerate to the orbital velocity for the ISS orbital once one arrives at the ISS orbital altitude, and then maneuver to (3) catch up with the ISS and (4) change direction to match its different orbital inclination. What are the actual steps involved in getting from the HST to the ISS? How much energy would be required at each step? How big a fuel tank or other energy source would be required to provide this energy? Could a human possibly carry such a source, or survive its use?


title: Derivation of neutrino oscillation phase factor

text: As we know, the neutrino $\nu_{\alpha}$ with flavor $\alpha=e,\mu,\tau$ is a linear combination of mass eigenstates: $$ |\nu_{\alpha}\rangle=\sum_iU_{\alpha i}|\nu_i\rangle,\quad i=1,2,3 $$ where the propagation of the mass eigenstates $|\nu_i\rangle$ can be described by plane wave solutions: $$ |\nu_{i}(t)\rangle = e^{ -i ( E_{i} t - \vec{p}_{i} \cdot \vec{x}) } |\nu_{i}(0)\rangle. $$ Suppose that $t=T$ and $|\vec{x}|=L$. Then the phase factor can be given by $$ \begin{split} E_iT-p_iL &= E_i\frac{L}{\sqrt{1-m_i^2/E_i^2}}-\sqrt{E_i^2-m_i^2}L \\\ &=E_iL\left(1+\frac{m_i^2}{2E_i^2}\right)-E_iL\left(1-\frac{m_i^2}{2E_i^2}\right) +\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right) \\\ &=\frac{m_i^2L}{E_i}+\mathcal{O}\left(\frac{m_i^4}{E_i^3}\right) \end{split} $$ However, this result is not correct. The artilce Neutrino oscillation on Wikipedia says that $$ |\nu_{i}(L)\rangle = e^{ -i m_{i}^2 L/2E }|\nu_{i}(0)\rangle $$ which differs in a factor of $1/2$. I can't figure out what is wrong with my derivation above (Maybe the formula for time interval $T$ is not correct, but I am not sure). Any help is greatly appreciated.


title: Vorticity - Paradox

text: ![Fluid Element Deformation](http://i.stack.imgur.com/Py4Tq.png) I am having trouble grasping the concept of vorticity. Above is a picture of fluid element under deformation (rotational). Say we ignore the vertical motion, which is $\frac {\partial {v}}{\partial x} = 0$. Which can be considered the case in the boundary layer close to a flat plate. Assume flow is incompressible, and steady. Now the text says that above motion is rotating the fluid element. The problem I am facing is that after some time $\Delta \theta_1$ would grow larger, larger,... What happens after sufficiently long time. This would converge itself to straight line (approx.). So all rotations will cease after some time. But we know that rotations will keep happening. My question is if the above picture is the way it is done, then after long time all fluid element will be a straight line, then how come rotation will keep happening in the fluid. **Note:** You may say that going to that asymptote line will take time $t \to \infty$. But then rotation is slowing down. But take this experiment. Say we have a laminar flow along a flat plate, and consider the area close to flat plate (boundary layer). Now if you place a vorticity meter in the boundary layer, you will be able to see that the meter is rotating with uniform (almost) angular speed. So somehow the fluid element is retaining it's own angular speed. Where the hell am I wrong...? This is killing me!


title: How long would a lever have to be to move the planet Earth?

text: > Give me a lever long enough and a fulcrum on which to place it, and I shall > move the world.   > -Archimedes How long would that lever have to be? That is to say, how long a lever would be needed on Earth, to lift a sphere with the mass of Earth if a human of average size were to sit on the other side of the lever?