title: Retarded time Lienard Wiechert potential

text: In a potential which needs to be evaluated at the retarded time, is this the time which represents the actual time the "physics" occurred? So $t_{\text{ret}}=t-\frac{r}{c}$, not just because it may be that you are receiving a signal at light speed but because "causality" spreads out at the maximum speed, $c$, is this correct? The Lienard-Wiechert 4-potential for some point charge ($q$): $A^\mu=\frac{q u^\mu}{4\pi \epsilon_0 u^\nu r^\nu}$ where $r^\nu$ represents the 4-vector for the distance from the observer. In the rest frame of the charge $A^i$ for $i=1,2,3$ is clearly zero but from what has been said about the retarded time we have that $A^0=\frac{q}{4\pi\epsilon_0c(t-r/c)}$. Obviously I would like to get $A^0=-\frac{q}{4\pi\epsilon_0 r}$, so where is the misunderstanding of retarded time and instantaneous time? Unless we would like the time since the signal was emitted which is $r/c$? Or if $t$ itself is already $t'-r/c$ and we need to return to the instantaneous time $t$, when the signal was emitted.


title: Nodes and Antinodes for standing wave

text: In the arrangement shown in the figure below, an object of mass m can be hung from a string (linear mass density $\mu$ = 2.00 g/m) that passes over a light (massless) pulley. The string is connected to a vibrator with constant frequency $f$, and the length of the string between point P and the pulley is $L = 2.00 m$. When the mass m of the object is either 16.0 kg or 25.0 kg, standing waves are observed in the string. No standing waves are observed for any other mass between these values. * * * $n$ = number of nodes from 25kg $n + 1$ = number of nodes from 16kg I do not understand why they have only integer 1 difference between the numbers of nodes of 25 and 16kg... Anybody know why they put n + 1 for 16kg? If we do that, the problems becomes easy to solve. Thanks,


title: "Dynamic Albedo of Neutrons" and all these Farnsworth fusors

text: I've checked specs of Russian "Dynamic Albedo of Neutrons" module on Mars MSL rover - it can do 10^8 neutron pulses at 10 pulses per second. It works on D-T reaction in tiny linear accelerator, and power consumption is quite low (14W). If you check specs for all these fusors (Farnsworth, Polywell, e.t.c) - they are happy to have 10^6 neutrons (best ones barely scratch 2*10^8) per second while having huge power consumption and short construction lifetime (<1hour). Does that mean that fusors are just a waste of paper from the science prospective, and real scientists routinely make neutron sources based on nuclear fusion with much better parameters/efficiency? **Update:** Fusor: http://en.wikipedia.org/wiki/Fusor ![enter image description here](http://i.stack.imgur.com/LyXIY.jpg) DAN: http://mars.jpl.nasa.gov/msl/mission/instruments/radiationdetectors/dan/ More info on Russian: http://www.federalspace.ru/main.php?id=58


title: Does tapping at the side of a bottle prevent shaken soda from bubbling over?

text: Anecdotal evidence has it that a bottle of soda that was heavily shaken will not bubble over if tapped at the side multiple times. Yet I wonder: Has the tapping really any effect? Or could it be that the mere time that passes while tapping at the side of the bottle has the carbon dioxide not mixed with the soda anymore?


title: Magnetic induction in solenoid

text: I found an equation in theory about magnetic induction in a solenoid: $B_s=\mu_0 I n$. It should be magnetic induction for infinite length solenoid. I wonder if this is anyhow useful. Where can this be used? ($n = \frac {N}{L} $, where $L$ is length of solenoid and $N$ is number of turns... which doesn't make sense to me, if the length is supposed to be infinite)


title: Calculating Gravitational Attraction Involving Massless Objects

text: Using the formula $F=G\frac{m_1m_2}{d^2}$ where $m_1$and $m_2$ are the masses of two objects, $G$ is the gravitational constant, and $d$ is the distance between the objects, it is possible to calculate the force of the gravitational attraction between the objects in Newtons. However, since light is affected by black holes, the property required to interact gravitationally is energy, not mass. How can I calculate the gravitational attraction between objects using their energies instead of their masses?


title: partition function for Wightman and Haag-Kastler QFT

text: From what I hear, some modern mathematical approach quantum field theory uses the following definition > "A $d$-dimensional $S$-structured quantum field theory $Q$ is a mathematical > object, consisting of its partition function $Z_Q$, its space of states > $H_Q$, and its submanifold operators $V_Q$, satisfying various axioms." > (taken from some lecture notes) In physics textbooks, there is a definition of the partition function $Z$ in terms of an classical field action (much like the wikipedia entry). I would like to know if there is a way to recover the partition function of a Wightman QFT (theory given in terms of fields operator valued distributions) or a Haag-Kastler QFT (given in terms of a local net of operator algebras). I ask this because, from what I understand, those need not to come from a classical field theory derived from an action principle.


Are solitons an example of collective motion?


title: Force exerted on ceiling by a simple mecahnical system

text: I have a simple static mechanical system, but I reach a conclusion that seems to me counter-intuitive: ![enter image description here](http://i.stack.imgur.com/ueJV4.png) There is a pulley fixed to the ceiling and there is a weight fixed to a rope which goes through the pulley and is fixed to a point on the floor. I denote by $T$ the tension in the rope and draw the forces applied to the weight (since the weight is in equilibrium, I should have $T=mg$), and to the pulley. Since the pulley is in equilibrium, the force exerted on it by the ceiling should be $2T$, and thus I arrive at the conclusion that in this setting, the force exerted on the ceiling (by Newton's 3rd law) is equal to $2mg$. Is my reasoning correct?


title: Equilibrium for a rope hanging in a Schwarzschild spacetime

text: **Update:** Trimok and MBN helped me solve most of my confusion. However, there is still an extra term $-(2/r)T$ in the final result. Brown doesn't write this term, and it seems physically wrong. **Update #2:** Possible resolution of the remaining issue. See comment on MBN's answer. Suppose we have a rope hanging statically in a Schwarzschild spacetime. It has constant mass per unit length $\mu$, and we want to find the varying tension $T$. Brown 2012 gives a slightly more general treatment of this, which I'm having trouble understanding. Recapitulating Brown's equations (3)-(5) and specializing them to this situation, I have in Schwarzschild coordinates $(t,r,\theta,\phi)$, with signature $-+++$, the metric $$ ds^2=-f^2 dt^2+f^{-2}dr^2+... \qquad , \text{ where} f=(1-2M/r)^{1/2} $$ and the stress-energy tensor $$ T^\kappa_\nu=(4\pi r^2)^{-1}\operatorname{diag}(-\mu,-T,0,0) \qquad .$$ He says the equation of equilibrium is: $$ \nabla_\kappa T^\kappa_r=0 $$ He then says that if you crank the math, the equation of equilibrium becomes something that in my special case is equivalent to $$ T'+(f'/f)(T-\mu)=0 \qquad ,$$ where the primes are derivatives with respect to $r$. This makes sense because in flat spacetime, $f'=0$, and $T$ is a constant. The Newtonian limit also makes sense, because $f'$ is the gravitational field, and $T-\mu\rightarrow -\mu$. There are at least two things I don't understand here. First, isn't his equation of equilibrium simply a statement of conservation of energy-momentum, which would be valid regardless of whether the rope was in equilibrium? Second, I don't understand how he gets the final differential equation for $T$. Since the upper-lower-index stress-energy tensor is diagonal, the only term in the equation of equilibrium is $\nabla_r T^r_r=0$, which means $\mu$ can't come in. Also, if I write out the covariant derivative in terms of the partial derivative and Christoffel symbols (the relevant one being $\Gamma^r_{rr}=-m/r(r-2m)$), the two Christoffel-symbol terms cancel, so I get $$ \nabla_r T^r_r = \partial _r T^r_r + \Gamma^r_{rr} T^r_r - \Gamma^r_{rr} T^r_r \qquad , $$ which doesn't involve $f$ and is obviously wrong if I set it equal to 0. What am I misunderstanding here? _References_ Brown, "Tensile Strength and the Mining of Black Holes," http://arxiv.org/abs/1207.3342


title: 6 Story Building Swaying, Normal?

text: **Preface:** I'm currently sitting at my desk on a 5th floor in a South Florida office building, as I was earlier this morning when I felt the building sway slightly. It wasn't continuous and the building swayed slightly one way and stopped. The sway was so subtle that others in the office didn't notice it (either walking or moving at their desk). Though, others did notice, corroborating my observations. _Additionally, I've spoken with more of my peers and they've stated that the building has swayed stronger than what I experienced this morning._ **Questions:** I 'm somewhat concerned regarding the overall integrity of the building, its foundation and whether or not this is normal.   * Is it normal for a 6 story building to sway, in Florida on a calm day?   * What could be causing this effect?   * What steps should I take in order to get the issue/issues addressed?   * Should I be concerned?


title: How efficient is the Crookes radiometer?

text: I remember many years ago, I think at 8th grade, seeing the teacher show us a Crookes radiometer. I remember it being very fascinating. Today I read the wiki article on it, after looking up what it was called, but the article wasn't very clear in my opinion. Essentially the molecules that hit the dark sides have more energy and thus exert pressure which causes the device to rotate. And something about Einstein... Among the lacks in the article is that it didn't clarify how effective this device is. It seems to me that relatively little energy input results a rather striking output (at least subjectively). So if I heat such a device with, say, $100 \ W$, how many watts would the output be in comparison? Could such a device be built in large scale and thus used to convert solar energy to movement?


title: Radiation: Inverse square law

text: Gamma radiation follows the inverse square law, I understand this as "double the distance, quarter the intensity" So if you have a gamma source, at the source (distance = 0), the Intensity is $I_0$, and say at distance = 1, the Intensity is $\frac{I_0}{2}$ (You can't work this out just from the fact it follows the inverse square law right? You'd need the constant?) So at distance = 2, while the intensity be a quarter of the original intensity so $\frac{I_0}{4}$ or a quarter of the intensity at the distance(1) that was doubled, so $\frac{I_0}{8}$? I ask because I think this graph, which shows intensity of gamma radiation vs distance according the inverse square law, is wrong? ![enter image description here](http://i.stack.imgur.com/bwkyW.jpg) (also I don't see how it gets from 3x to $\frac{I_0}{8}$ because $3^2=9$)


title: Do I need to study the "Standard Model" before studying String Theory?

text: After this semester, I'll have a background up to a first course in QFT (first 5 or 6 chapters of Peskin and Schroeder). The next step in QFT will be something specific to the Standard Model (Elementary particles, QCD, etc). Is that needed before going to study String Theory or not?


title: Why is energy conserved?

text: From basic principals, how does one prove that energy is conserved? Or a little more specifically - Why does this hold: $$\Delta \mbox{ PotentialEnergy} + \Delta \mbox{ KineticEnergy} = 0 $$ Or, for extra credit, why does this hold: $$\Delta \mbox{ PotentialEnergy } + \Delta \mbox{ KineticEnergy} + \Delta \mbox{ ThermalEnergy } = 0 $$


title: Is energy really conserved?

text: In high school I was taught energy was conserved. Then I learned that nuclear reactions allow energy to be converted into mass. Then I also heard that apparently energy can spontaneously appear in quantum mechanics. So, are there any other caveats with the conservation of energy?


title: Why can't energy be created or destroyed?

text: My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed. _Why_ is that? Is there a theory or scientific evidence that proves his statement true or false? I apologize for the elementary question, but I sometimes tend to over-think things, and this is one of those times. :)


title: Conservation of Energy in the Universe

text: > **Possible Duplicate:**   >  Is energy really conserved?   >  Why can’t energy be created or destroyed? One of the laws of the universe that dazzles me the most is the law of conservation of energy. I however have a couple of questions regarding this law. Since Einstein's equivalence tells us that matter and energy are different manifestations of the same thing:   * Does this mean that the amount of energy + matter has been the same since the beginning of the universe? If this is true, then another question pops up. Since the universe is expanding at an accelerating rate, and the law of conservation of energy tells us that the amount of energy can't increase:   * does this mean that the universe is getting emptier?  Also, an intertwined question:   * Is there some sort of average 'energy density' in the universe? If so, can we notice the effects of the accelerating expansion of the universe by looking at this 'energy density'? (this might sound like/ be a stupid question).


title: Fluids in thermodynamic equlibrium

text: I am reading about the Euler Equations of Fluid dynamics from Leveque's numerical methods for conservation laws. After introducing the mass, momentum and energy equations, some thermodynamic concepts are discussed, to introduce an equation of state. He says               In the euler equations we assume that the gas is in chemical and thermodynamic     equilibrium and that the internal energy is a known function of pressure and density.      After this , the usual thermodynamics-related EOS discussions are carried out. Now chemical equilibrium I understand (number of moles of the chemical constituents do not change), however I don't understand how the assumption of thermodynamic equilibrium can be imposed. From what baby thermodynamics I know, any thermodynamic analysis is always calculated for quasi-static processes, like 'slowly' pushing a piston in a cylinder of gas. But in fluid _dynamics_ fluids are **flowing** and that too rapidly and from intuition there will not be any thermodynamic equlibrium during fluid flow. Where is my understanding going wrong?


title: How does one get the value of acceleration of gravitation on earth accurately by experiment without electronic device?

text: How does one get the value of acceleration of gravitation on earth accurately to 5 significant digits by experiment without electronic device?


title: Is every material able to exist in every state of matter?

text: I was shocked while reading Kittel's "Introduction to Solid State Physics", that the solid state of noble gases is a well described and makes one of the fundamental achievements of solid state physics. So I wonder: is every possible element and every possible compound able to exist in any of the 3 states of matter?


title: Dark energy vs. gravity

text: If dark energy is everywhere around us, then why don't we get separated? For example why don't I get separated from the pen kept in front of me? Or take a similar example in free space. Is dark energy's power greater than gravitational power?


title: How do draw real path which represents the solution of the Landau-Lifshitz equation

text: ![enter image description here](http://i.stack.imgur.com/Zux0C.png) how to draw two images look like images below with gnuplot.


title: Good algorithm for in-experiment 1-D optimization?

text: I'm running an experiment -- for the question, it doesn't matter which one, but I'm measuring an optical intensity $I$ as a function of two parameters: reflection angle $\theta$ and wavelength $\lambda$. I have motion control in place to move the setup to an angle $\theta_0$, and then I measure $I(\theta_0, \lambda)$ all at once using a spectrometer. I then move to the next angle $\theta_1$ and repeat. Due to the beam travelling through different media at different angles, I have to move the position of the detector $p$ slightly for each angle, which is also automated. I should move the detector so that $M(p) = \sum_\lambda I(\theta_n, \lambda, p)$ is maximized. (M stands for "figure of Merit".) $M(p)$ approximately has the form of a Gaussian plus noise, but I should be able to maximize it without caring what form it has. I have an amateurish algorithm in place to search for the proper position $p$ in order to maximize $M(p)$. The quick-n-dirty algorithm steps $p$ in one direction by a step size $\Delta p$, until the value of $M(p)$ becomes smaller than a previous value. Then it goes back one step and tries a smaller step size $\Delta p$ in the other direction. As you can see, it doesn't account for measurement noise. My thoughts on how to improve it were in the direction of measuring a small number of points spaced $\Delta p$ apart and then fitting a parabola through them. My question is, before I sit down and design a better algorithm, can anyone suggest an already-existing algorithm? I don't think a feedback algorithm (such as PID control) is appropriate, since I'm not trying to maintain a certain setpoint under perturbation of the system -- I just need to optimize to one value for each measurement. For bonus points, can someone point me to some papers on this subject?


title: If blue light has a higher energy than red light, why does it scatter more?

text: As $E=hf=\frac{hc}{\lambda}$, blue light - with a smaller wavelength - should have a higher energy. However, it is the case that blue light scatters the most. Why is it that higher energy rays scatter more?


title: What does it mean that particles are the quanta of fields?

text: I saw the question What are field quanta? but it's a bit advanced for me and probably for some people who will search for this question. I learned QM but not QFT, but I still hear all the time that "particles are the quanta of fields" and I don't really understand what it means. Is there a simple explanation for people who know QM but not QFT?


title: Is the gravity we feel in our everyday life mainly curvature of the time coordinate?

text: The question is pretty clear I think. I only started learning GR but the results I've seen so far and from some lectures I saw from Leonard Susskind on YouTube I understood that for low energy densities the curvature of space-time is mainly in the time coordinate. Is this true, is the gravity we feel here on earth mainly the result of curvature of the time coordinate?


Could anyone explain the following expression: Why can mass not be considered concentrated at CM (center of mass) for rotational motion?


title: One-Plaquette Action and SU(2)'s Irreducible Representations

text: I have a typical single-plaquette partition function for a gauge-field $$ Z=\int [d U_{\text{link}}] \exp[-\sum_{p} S_{p}(U,a)]$$ with $U$ as the product of the the $U$'s assigned to each link around a plaquette. Now the $U$'s are irreducible representations of my group elements, which in my case is SU(2), and lets take the 1/2 representation as an example, then define the character as $\Xi_{r}\equiv \text{Tr}[U]$ which for our case is $\Xi_{1/2}=\text{Tr}[U]$. Now I have to take the product of these representations, however (and here's my question), how do I know which group elements/that-element's-representation to assign to each link? I'm not sure how to compute the Trace without knowing first how to do the product of the link's representations, but I don't even know how to assign the elements to the links. Thanks,


title: Derivation of the supergravity action in 11D

text: The Einstein-Hilbert action of general relativity is uniquely determined by general covariance and the requirement that only second derivatives in the metric appear. Yang-Mills theory can be motivated in a similiar way. In the original paper of Scherk, Julia, Cremer there are some arguments given from which they deduced the form of the action. They are only sketched however. Is there a more complete exposition of the derivation in the literature, or possibly even a uniqueness result as in the case of general relativity or Yang-Mills theory?


title: From knowing just the change in kinetic energy, can we find the friction force and engine power?

text: I understand this topic well enough to get all the task done because they aren't very creative. But for my exam I think I should have this clear. During the acceleration the force from the engine is of course bigger than air resistance and friction. This force, can we find it? And then the entire force the engine applies for the acceleration. Not just the stub you after subtracting for friction and air resistance. $W = F \times s$   $F = m \times a$ We have all the work done by forces at work, and the stretch of road is easy to calculate. If I now do this $\frac{F}{m} =a $ will that output be the correct acceleration? And this force that we found, is that a sum force? Because if that's the force sum on the car I can't find the engines power output which is what I want. And what about the friction force at work, I think we can't find it when we just have the change in kinetic energy. Is that right? Primarily I would like to know if the change in kinetic energy can be tied somehow to the engines output during the acceleration. The book has this nice equation too: $P = F \times v $   But that's just constant speed. Since we know the time maybe this can be used: $P = \frac{W}{T}$   That just seems a little too easy. Edit: I got a B on the exam, which means I'll be at again this fall. Not due to this question. Haha.


title: Is this really a golden ratio spiral?

text: In this blog post, I found this picture: ![enter image description here](http://i.stack.imgur.com/Qi3C2.jpg) Does the water really form golden ratio spiral in such cases? Or is the photo just a provocative example, without physics grounds for claims about "goldness" of the spiral?


title: Is using water in a charcoal smoker less efficient than not using water?

text: I have a charcoal smoker that uses water. My understanding is that the water serves as a buffer and as a way to add moisture to the cooking environment. Some say that using water wastes fuel because you have to heat the water. If I am applying Newton's Conservation Law correctly, it seems to me that the heat put into the water is also released from the water as heat energy which also heats the food. So, I don't see that as wasteful. Is more energy lost in that part of the process than if no water was used? It seems to me that the only waste is the heat left in the water after you are finished cooking. If you start with hot water from the tap, there's even less heat wasted there.


title: Normalization for QFT single particle destruction operator

text: I don't understand a particular statement in the QFT book by Klauber. The particular page I'm having difficulty on is page 67 of chapter 3 (PDF link). The big picture is that the author wishes to investigate what the (operator) solutions to the Klein-Gordon equation, $\phi(x)$ and $\phi^\dagger(x)$, do when acting on the vacuum state $|0\rangle$. As prep for this, he creates a "general single particle state" ("general" meaning non $\mathbf{k}$-eigenstate) by operating on the vacuum with the operator $$C\equiv\sum_\mathbf{k}A_\mathbf{k}a_\mathbf{k}^\dagger,\tag{3-108}$$ $$C|0\rangle=\sum_\mathbf{k}A_\mathbf{k}a_\mathbf{k}^\dagger|0\rangle=A_1|\phi_1\rangle+A_2|\phi_2\rangle+\cdots\equiv|\phi\rangle\tag{3-109}$$ Each $A_\mathbf{k}$ is just a number, the absolute value square of which represents the probability of finding the $\mathbf{k}$ eigenstate for the single particle. The new state $C|0\rangle=|\phi\rangle$ is interpreted as a single particle state in a superposition of $\mathbf{k}$-eigenstates $|\phi_k\rangle$. The subscript $\mathbf{k}$ represents different momenta. For probability/normalization arguments, the numbers $A_\mathbf{k}$ should obey $$\sum_\mathbf{k}\left|A_\mathbf{k}\right|^2=1.\tag{3-110}$$ I feel like I understand the above statements. The author then introduces the "general single particle destruction operator" $$D\equiv\sum_\mathbf{k}a_k,\tag{3-111}$$ and shows that when applied to our general single particle state $|\phi\rangle$ above, the vacuum is (re)produced: $$\begin{eqnarray} D|\phi\rangle&=&\left(\sum_\mathbf{k}a_k\right)A_1|\phi_1\rangle+\left(\sum_\mathbf{k}a_k\right)A_2|\phi_2\rangle+\cdots\\\ &=&A_1\underbrace{a_1|\phi_1\rangle}_{=|0\rangle}+A_1\underbrace{a_2|\phi_1\rangle}_{=0}+A_1\underbrace{a_3|\phi_1\rangle}_{=0}+\cdots+\\\ &\ &+A_2\underbrace{a_1|\phi_2\rangle}_{=0}+A_2\underbrace{a_2|\phi_2\rangle}_{=|0\rangle}+ A_2\underbrace{a_3|\phi_2\rangle}_{=0}+\cdots+\\\ &\ &+\cdots\\\ &=&\underbrace{\left(A_1 + A_2 + \cdots\right)}_\text{can normalize = 1}|0\rangle. \end{eqnarray}\tag{3-112}$$ (Note the subtle but important differences in the underbraces; some are $0$ while others are $|0\rangle$.) **The part I am struggling with** is understanding how the underbrace "can normalize = 1" at the end of $\text{(3-112)}$ can be true given $\text{(3-110)}$. It seems to me that the $A$ terms appearing at the end of $\text{(3-112)}$ are the same ones defined in the construction operator $C$ and normalized so that their absolute values squared sum to $1$. How can their just-plain sum also be of magnitude $1$? I know that one would *like * the underbraced term to sum to zero, but I don't see how that can be. * * * It was suggested I consider the quantity $\langle\phi|D^\dagger D|\phi\rangle$. Here is my attempt to calculate it. $$ \begin{eqnarray} \langle\phi|D^\dagger D|\phi\rangle&=&\langle0|(A_1^\dagger+A_2^\dagger+\cdots)(A_1+A_2+\cdots)|0\rangle=\langle0|\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}|0\rangle\\\ &=&\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}\underbrace{\langle0|0\rangle}_{=1}=\underbrace{\sum_\mathbf{j}\sum_\mathbf{k}A_\mathbf{j}^\dagger A_\mathbf{k}}_\text{Can't simplify}\ne1 \end{eqnarray} $$


title: Regulating an infinite sum in the $bc$ CFT

text: The EM tensor of the $bc$-CFT is $$ T(z) = \colon \partial b c \colon - \lambda \partial \colon b c \colon $$ After expanding in a mode expansion, we find $$ T(z) = \sum_{m} \frac{1}{z^{m+2}} \sum_n \left( \lambda m - n \right) \colon b_n c_{m-n} \colon \implies L_m = \sum_n \left( \lambda m - n \right) \colon b_n c_{m-n} \colon ~~...(1) $$ Now, one would like to rewrite this expression using creation-annihilation (CA) normal ordering ${}_\circ^\circ~~{}_\circ^\circ$ instead of conformal-normal order $\colon~~\colon$. A simple calculation reveals $$ L_m = \sum_n \left( \lambda m - n \right) {}_\circ^\circ b_n c_{m-n}{}_\circ^\circ + \frac{\lambda \left( 1 - \lambda \right)}{2} \delta_{m,0} $$ The above expression was derived by requiring that $L_m$ satisfy the Virasoro algebra. I am now looking to provide an alternative derivation of the above expression. Here's my procedure. We start with the result of exercise 2.13(a) in Polchinski (which I have already verified) $$ \colon b(z) c(w) \colon - {}_\circ^\circ b(z) c(w) {}_\circ^\circ = \frac{(z/w)^{1 - \lambda} - 1 }{z - w} $$ Now, taking a limit of $w \to z$, we find $$ \colon b(z) c(z) \colon = {}_\circ^\circ b(z) c(z) {}_\circ^\circ + \frac{1-\lambda}{z} $$ from where we can derive the difference between the CA normal ordering and conformal ordering of the mode coefficients $$ \colon b_m c_n \colon = {}_\circ^\circ b_m c_n {}_\circ^\circ + \left( 1 - \lambda \right) \delta_{m+n,0} $$ Now, let us start from the expression $(1)$ and plug in the above. We find \begin{equation} \begin{split} L_m &= \sum_n \left( \lambda m - n \right) \colon b_n c_{m-n} \colon \\\ &= \sum_n \left( \lambda m - n \right) \left[ {}_\circ^\circ b_n c_{m-n} {}_\circ^\circ + \left( 1 - \lambda \right) \delta_{m,0} \right] \\\ &= \sum_n \left( \lambda m - n \right) {}_\circ^\circ b_n c_{m-n} {}_\circ^\circ + \delta_{m,0} \sum_n n \left( \lambda - 1 \right) \\\ \end{split} \end{equation} Comparing with the expression mentioned above, we are required to have > $$ \sum_{n=-\infty}^\infty n \left( \lambda - 1 \right) = \frac{\lambda > \left( 1 - \lambda \right)}{2} $$ I am not able to show this. Anyone has any > idea?


title: Large deflection of cantilever beam

text: How can I find the amount of point force at the end of a cantilever plastic beam that produces e.g. 45deg slop at the end of the beam? Is this the right equation: $$F=\frac{2EI\theta ^2}{L^2 sin(\theta)}$$ I derived this equation by putting $x=\frac{L}{\theta}sin(\theta)$ and $\theta_x=\theta$ in the following eqation from the "Large and small deflections of a cantilever beam" paper: $$x=\sqrt{ \frac{2EI}{F} } \left(\sqrt{sin(\theta)}-\sqrt{sin(\theta)-sin(\theta_x)}\right)$$ where $\theta$ is the maximum slope at beam end and $\theta_x$ is the beam slop at x. Thanks for your help in advance.


title: EPR-type experiments and faster-than-light communication using interference effects as signaling mechanism

text: I understand that faster-than-light communication is impossible when making single measurements, because the outcome of each measurement is random. However, shouldn't measurement on one side collapse the wave function on the other side, such that interference effects would disappear? Making measurements on "bunches" of entangled particles would thus allow FTL communication, by making observed interference effects appear or disappear. How does such an experiment not: 1) Clearly imply that faster-than-light communication is possible? or 2) (if #1 is rejected) Imply that measurement of one half of an entangled pair does not cause the collapse of the other half's wave function. Why doesn't this thought experiment clearly show that if we maintain that FTL communication is ruled out, we must also rule out "universal collapse" in the Copenhagen interpretation? **EDIT: Here is an example of an explicit experiment (though I think experts could come up with something better):** You can entangle a photon with an electron such that the angle of the photon is correlated with the electron's position at each slit of a double slit experiment. If the photon is detected (it's outgoing angle measured), then which-path information is known, and there is no interference. If the photon is not detected, the interference remains. The experiment is designed such that the photon and electron go in roughly opposite directions, apart from the tiny deflection which gives which-path information. You set up a series of photon detectors 100 ly away on one side, and your double slit experiment 100 ly away in the opposite direction. Now you produce the entangled pairs in bunches, say of 100 entangled pairs, each coming every millisecond, with a muon coming between each bunch to serve as a separator. Then the idea is that someone at the photon detector side can send information to someone watching the double-slit experiment, by selectively detecting all of the photons in some bunches, but not in others. If all of the photons are detected for one bunch, then the corresponding electron bunch 200 ly away should show no interference effects. If all of the photons are not detected for one bunch, then the corresponding electron bunch 200 ly away would show the usual double-slit interference effects (say on a phosphorus screen). (Note that this does not require combining information from the photon-detector-side with the electron-double-slit side in order to get the interference effects. The interference effects would visibly show up as the electron blips populate the phosphorus screen, as is usual in a double-slit experiment when which-path information is not measured.) In such a way the person at the photon detectors can send '1's and '0's depending on whether they measure the photons in a given bunch. Suppose they send 'SOS' in Morse code. This requires 9 bunches, and so this will take 900 milliseconds, which is less than 200 years. The point is that such an experiment would only work if you assume that the measurement of the photon really does collapse the wave function nonlocally.


title: Question about the refraction of Fresnel lenses

text: If you line up the suns rays parallel to a Fresnel lens, the light is concentrated, and the focus directly underneath. However, what happens if the sun is off to the side, making the light hit at an angle (ei. 45 degrees)? Will the only difference be the focal point, of will the light be less concentrated? If so, by how much? What about a convex lens? Thanks


title: Particles entangled after the big bang

text: Is that true that the big bang caused the quantum entanglement of all the particles of the universe so every particle is entangled to each other particle of the universe?


title: Solving systems of equations in dynamics

text: I have an exam in two days for first year university physics. Often for dynamics problems, I am required to solve algebraic systems of equations by hand, and this can be very daunting. When I see the solutions, however, the steps that the solver took to seem very clean and almost obvious. Are there some rules of thumb that physicists use to solve small systems of equations, either by elimination or substitution? Here is an example. Find $\frac{m_1}{m_2}$ in terms of only $\theta$ where the known quantities are $\theta$, $m_1$, $m_2$. $$F_T \sin \theta = m_1 g$$ $$F_T \cos \theta = m_1 a$$ $$F_T \sin \theta + F_N \cos \theta = m_2 g$$ $$F_N \sin \theta - F_T \cos \theta = m_2 a$$


title: Why does tension not do work in this pulley system? etc

text: I have a slight difficulty understanding the solution to the following problem: A light inextensible string with a mass $M$ at one end passes over a pulley at a distance $a$ from a vertically fixed rod. At the other end of the string is a ring of mass $m(M>m)$ which slides smoothly on the vertical rod as shown in the figure. The ring is released from rest at the same level as the point from which the pulley hangs. If $b$ is the maximum distance the ring will fall, determine $b$ using the principle of virtual work. The solution is: Let $l$ be the length of string. Therefore, $$x+(a^2+b^2)^{1/2}=l\tag{1}$$ Imagine a vertical displacement $\delta b$ of the ring along the rod. $$\delta x+b(a^2+b^2)^{-1/2}\delta b=0\tag{2}$$ Therefore, $$\delta x=-b(a^2+b^2)^{-1/2}\delta b$$ The constraints over the pulley and rod do no work. By the principle of virtual work, $$Mg\delta x + mg \delta b=0$$ Substituting value of $\delta x$: $$-Mgb(a^2+b^2)^{-1/2}\delta b = -mg\delta b$$ Since $\delta b$ is arbitrary, $$b^2=\frac{m^2}{M^2}(a^2+b^2)$$ $$b=\frac{ma}{(M^2-m^2)^{1/2}}$$ My questions are:   1. How is Equation 2 obtained from Equation 1? (especially with regard to the intermediate steps)   2. When I attempted the problem, I had the tension in the string doing work, because the displacement of the string is in line with the tension in the string. Why is this not being considered? Thanks in advance. ![setup of problem](http://i.stack.imgur.com/AwaiU.jpg)


title: Why do strings split in string theory?

text: In string theory, we are told strings can split and merge if the string coupling is nonzero, even while the worldsheet action remains Nambu-Goto or Polyakov plus a topological term. However, a classical solution, in say the light cone gauge, shows that provided the worldsheet time increases with light cone cone time initially, adding topological terms will not change the solution, and the string will not split. So why and how do strings split?


title: What are the basic postulates of string theory?

text: I read the Popular Science books, _The Brief history of time_ (Hawking), _The fabric of the cosmos_ (Greene), and _The Grand Design_ (Hawking), etc. but did could not manage to understand what string theory is, even at a Popular level. I couldn't understand what exactly it is, even at a Popular Science level. ! Can any one please briefly describe what are its' fundamental postulates?


title: Gauge invariance and diffeomorphism invariance in Chern-Simons theory

text: I have studied Chern-Simons (CS) theory somewhat and I am puzzled by the question of how diff. and gauge invariance in CS theory are related, e.g. in $SU(2)$ CS theory. In particular, I would like to know about the relation between large gauge transformations and large diffeos. If you know any good sources, I would be really grateful. Thank you!


title: Why do stars flicker?

text: Why do stars flicker and planets don't? At least this is what I've read online and seen on the night sky. I've heard that it has to do something with the fact that stars emit light and planets reflect it. But I don't get it, isn't this light, just "light"? What happens to the reflected light that it doesn't flicker anymore? I was thinking that it has to do something with Earth's atmosphere, different temperatures or something (if this has any role at all).


title: Help with 1D and 2D density of states

text: I am currently looking at changes in DOS when sampling recipocal space finely. More precisely, I am looking at the expressions $$\rho_\text{1D}(E)\text{d}E = \frac{m}{\pi \hbar} \sum_i \text{H}(E-E_i)\text{d}E, \\\ \rho_\text{2D}(E)\text{d}E = \frac{1}{\pi}\sqrt{\frac{2m}{\hbar^2}} \sum_i \frac{n_i\text{H}(E-E_i)}{\sqrt{E-E_i}}\text{d}E$$ where $E_i=\frac{\hbar^2k^2}{2m}$ is taken from the free-electron model. Here $H$ is the Heaviside step-function, and $n_i$ is a degeneracy factor (set to unity). Let's now say that I can sample $k$-space arbitrarily fine. I want to see how the 1D and 2D cases converge. I can easily see that 2D converges to 3D (i.e. $\sqrt{E}$-behaviour) (see figure below), but what can one expect for the 1D system? Usually, DOS explanations in books and online only focus on the actual expression and not what happens when you sum up all channels. At least it's not something I have seen. Any ideas? ![2D DOS](http://i60.tinypic.com/352md8g.png)


title: Peaks on top of Bremsstrahlung

text: The following is a graph of the intensity of Bremsstrahlung generated by accelerating electrons to hit a target vs. its wavelength. ![](http://i.stack.imgur.com/EbRI9.png) I'm wondering what causes the additional peaks for high energies? I have read in my physics script and on Wikipedia that "these peaks are characteristic of the material used as target", but what exactly is going on at the level of atoms? Does the electron maybe go into some sort of bound state and thus loses - in addition to its kinetic energy - also some potential energy? Why would that only happen for high enough energies?


title: Simulation of an one dimensional driven diffusive system

text: I'm currently writing a simulation in python with scipy and matplotlib to reproduce an one dimensional driven diffusive system described in this paper from M.R. Evans et al. The system consists of positive, negative and hole particles. On the left side of the system the positive (negative) particles are produced on the left (right) side with a possibility a and destroyed at the right (left) side with the possibility b. In some cases the system should show a flip between positive and negative high density states and I'm trying to reproduce this behaviour with my simulation. But all I can see is an increasing current within my simulation and I can't find any problems in my code explaining such behaviour. Does anyone here have some experience simulation such or similar systems? Cheers, Florian


title: Einstein's Cosmic Speed Limit $c$ challenged by Black Holes/Massive Objects

text: To even suggest the great Einstein is incorrect on anything he has written is inconceivably absurd. Being fully aware of this I do indeed dare to not only suggest he is wrong but I cannot believe nobody has questioned his cosmic speed limit to this very day. I do not need any mathematical equations to prove this point. Common sense will suffice. If in fact the cosmic speed limit $c$ is achieved only by light at the estimated 186,000 miles per hour then it must be true that this speed limit is surpassed when light is engulfed by a super massive black hole because of the simple fact that the light cannot escape this giants gravitational pull. Wouldn't this mean that the light must be travelling faster than what it was before meeting this cosmic anomaly?


title: Why can't light escape from a classical black hole?

text: Photons do not have (rest) mass (that's why they can move at speed of "light"). So, my question is how the gravity of classical$^1$ black hole can stop light from escaping? \-- $^1$ We ignore quantum mechanical effects, such as, Hawking radiation.


title: Restrained double pendulum

text: The equations of motion of a double pendulum are well-known. Usually you'd have the them expressed in the rotations $\theta_1(t)$ and $\theta_2(t)$. There are two degrees of freedom. Now consider the case in which the motion at the bottom is constrained in the $x$-direction: ![](http://i.stack.imgur.com/cQRKL.png) There is now only one degree of freedom $\Delta(t)$. Instead of deriving the equation of motion for this system is there a way to use the EOMs of the original (unrestrained) system with a sort of constraint relation to derive the new EOM? EDIT: Lagrangian ![](http://i.stack.imgur.com/nVcU6.png) resulting in the equation of motions ![](http://i.stack.imgur.com/J5blx.png) ![](http://i.stack.imgur.com/1jOup.png) taken from here (note that $\theta_2$ is drawn with the incorrect sign in the figure above). And then the constrained equation is (I think) $$h_1 \sin(\theta_1) = - h_2 \sin(\theta_2) = \Delta(t)$$


title: Symmetric, antisymmetric and mixed symmetry particles

text: Can someone explain to me the concept of symmetric, antisymmetric, and mixed symmetry when talking about the states of identical particles?


title: Are laws of gravity time symmetric?

text: Time symmetry is often explained by the example of orbiting objects... What I can't find an explanation for is the moment when an object enters into orbit around another object. That clearly breaks time symmetry, since once object is in orbit (and you reverse time), it will never leave of it's own. Does this mean laws of gravitational motion are not time symmetric? Or is there some other explanation (e.g. entropy of the system)?


title: Kepler problem: flows generated by constants of motion

text: This is part of an admission exam problem, found at http://www.sissa.it/mp/admission/tests//2008_common.pdf Consider the Hamiltonian of Kepler problem $$H(\boldsymbol{r},\boldsymbol{p})= \frac{|\boldsymbol{p}^2|}{2\mu} +\frac{\alpha}{|\boldsymbol{r}|}, \qquad \mu>0>\alpha,$$ where $\boldsymbol{r}\in M=\mathbb{R}^3\setminus\\{ 0 \\}, \ (\boldsymbol{r},\boldsymbol{p})\in T^*M$ and $|\boldsymbol{r}|=\sqrt{r_1^2+r_2^2+r_3^2}$. The quantities $$\boldsymbol{m}=\boldsymbol{r}\times\boldsymbol{p}, \qquad \boldsymbol{W}=\boldsymbol{p}\times\boldsymbol{m}+ \mu\alpha\frac{\boldsymbol{r}}{|\boldsymbol{r}|}$$ are constants of motion, as is well known. It is stated then that the flows generated by the functions $m_i$ and $W_i,\ i=1,2,3$ are canonical transformations. **I don't understand** is what is meant by this statement: I mean, I know what a canonical transformation is, but I would appreciate some explanation or reference about this precise statement. Thanks in advance for the help, and I hope this is formulated in compliance with the rules of this community.


title: Planar Electric Field in Glass, how to express it?

text: The field must have properties such as `planar wave` and `moving in glass` so $$\bar E=E \left( \hat i t + \hat j \sin(kz-wt) + \hat k \cos(kz-wt+\phi) \right)$$ that is also moving in the x-axis and right-hand-circular-polarized so the phase difference $\phi=\pm \pi/2$ (more on p.557 in Understanding Physics -book and here). I need some natural constants to express the permittivity with the electric field $\bar E$. The wave must move in glass. It has refractive indices so that $n_1 \sin(\theta_1)=n_2 \sin(\theta_2)$, more here. **I. How can I express electric field in Glass?** > $$ \bar D= \epsilon_0 \bar E + \bar P = \epsilon_0 (1+\chi) \bar E = > \epsilon_r \epsilon_0 E$$ > > where the permittivity applies to the electric field. Now you have the > electric field in terms of permittivity and $\bar D$ is the density of the > wave in the material. $\bar E$ is the outside -field while the $\bar P$ is > the inner polarization -thing. **II. How can I get planar wave?** > According to this here, the planar wave can be expressed as > > $$U(\bar r, t) = A_0 e^{i \left( \bar k \cdot \bar r - wt + \phi \right)}$$ > > but now I am confused how I can get the phase difference there. **Perhaps related** >   1. Help me to visualize this wave equation in time, to which direction it > moves? >


title: Is Connes model a composite Higgs in disguise?

text: Most of the 5-dimensional Higgs models can be seen, if I understand correctly, as models where the Higgs is a composite. Now, is this true for Connes models? It is a model of extra dimensions too, in some sense. And when you look at the papers, at some moment a product of two symbols is substituted by a single $\phi$, so it seems that some composition is at work.


title: Calculating the time of dawn

text: Knowing that astronomical twilight (i.e. astronomical dawn) is when the sun is 18 degrees below the horizon, I am calculating the astronomical twilight time this way:               Sunrise of day1 - [ (Sunrise of day1 - Sunset of day0) / 180° ] * 18°      Is that correct?


title: Wilson loops and gauge invariant operators (Part 1)

text: I guess the Hilbert space of the theory is precisely the space of all gauge invariant operators (mod equations of motion..as pointed out in the answers)   * Is it possible that in a gauge theory the Wilson loops are the only observables?  (...I would vaguely think that if a set of Wilson loops one for every cohomology class of the space-time is the complete set of observables then this is what would be "a" way of defining a Topological Field Theory but may be this is also possible for pure gauge theories in some peculiar limit or on some special space-time geometries..)   * When the above is not true then what are all the pure gauge theory observables?..I guess its only the local observables that is missed by the Wilson loops..   * In general is it always true that all gauge invariant observables are precisely all the polynomials in the fields which are invariant under the action of the gauge group? (..and this is a well studied question in algebraic geometry under the name of Geometric Invariant Theory?..)    * If one has matter in the theory then I guess the baryons and the mesons are the only matter observables? I guess there is no gauge group dependence on their existence?  (..though baryons can always be defined for any anti-symmetric combination of the flavour indices I guess mesons can be defined only if equal amount of matter exists in the conjugate representation of the gauge group also..right?..)   * Why are gauge traces of arbitrary products of matter fields neither baryons nor mesons? (...in arbitrary gauge theories is it legitimate to identify these states as ``chiral primaries" in any sense?..)


title: Coulomb potential energy functional derivative

text: I'm having problem understanding how to compute a functional derivative when it's involved more than one integral, such as the coulomb potential energy functional: $$ J[\rho] = \frac 12\int \frac{\rho(r)\rho(r')}{|r - r'|} drdr' $$ According to the functional derivative formula I should do something on these lines: $$ \frac {\delta J}{\delta \rho (r)} = \frac{\partial}{\partial \rho(r)} [ \frac 12 \int \frac{\rho(r) \rho(r')}{|r - r'|}dr' ]$$ In my wrong reasoning I would simply take $\rho(r) $ out of the integral and apply the derivative: $$\frac {\partial \rho(r)}{\partial \rho(r)} \frac 12 \int \frac{\rho(r')}{|r - r'|} dr' = \frac 12 \int \frac{\rho(r')}{|r - r'|} dr'$$ Which is wrong because the correct result should be: $$\int \frac{\rho(r')}{|r - r'|} dr'$$ 1) I'm quite confused by the notation and how to treat a partial derivative by $\rho(x)$ 2) What's the correct way to handle and compute functional derivatives in these cases? I'm actually in a similar situation with a much complex derivative, such as the same thing in the density matrix formalism: $$ J[\gamma_1] = \frac 12 \int \frac{\gamma_1 (x_1', x_1) \gamma_1 (x_2', x_2) \delta(x_1' \- x_1) \delta(x_2' \- x_2) dx_1 dx_1' dx_2 dx_2'}{|x_1 - x_2|}$$ $$\frac {\delta J[\gamma_1]}{ \delta \gamma_1 (x_1'. x_1)}$$


title: Why does thermal resistance go down as temperature goes up?

text: Here is the thermal resistance data for three speaker coils disengaged from the speaker cone. Any ideas? I would think it would be a horizontal line. ![http://i.stack.imgur.com/pjKyE.png](http://i.stack.imgur.com/kV0Sd.png) http://en.wikipedia.org/wiki/Thermal_resistance Absolute thermal resistance of the part, K/W edit: I also want to point out that the same trend occurred with the loudspeaker functioning as a normal speaker (before I tore out the voice coils to test separately). The reason for this test was that I thought the drop in thermal resistance was due to unwanted cooling effects (actually desirable, but not at the moment) from diaphragm movement, but the drop in thermal resistance vs temperature is still there. edit: I am going to run the same experiment using a 1/4W resistor today and report my findings. For x_C = 40, 60, 80:   1. Heat oven to x_C   2. Measure electrical resistance   3. put .2W of power into resistor   4. Let the resistance measurement settle (thermal time constants)   5. Measure increased electrical resistance   6. Calculate resulting change in temperature of resistor   7. Calculate resulting thermal resistance (dT/dR)


title: Finding force exerted in an Inelastic Collision

text: I did a lab today in Physics in which we launched ball from a spring loaded cannon directly into a pendulum that captured the ball, held it, and swung upwards with it (representing a totally inelastic collision). One question in particular has confused me: > If the collision between the projectile and pendulum had lasted 1 > millisecond, what would the average force have been which the projectile > exerted on the pendulum for the long-range case? My attempt at a solution is as follows: From all the searching I've been doing online, I've found the equation $F = {{p_f}-{p_i}\over {t}}$. I know $p_f$, $p_i$, and I'm given t. Is my understanding right? Can I go right ahead and crunch these numbers, or do I have an incorrect equation?


title: Why does cornstarch and water mixed together become solid when put under stress?

text: Here is one of the many videos on youtube showing how a cornstarch and water mixture can become very hard when put under stress. I haven't found a good explanation of why it behaves the way it does. Is there any paper that has been published on this that explains the weird properties of the liquid? If not, could someone here try to explain what is going on?


title: Partition Function for the Jaynes-Cummings Hamiltonian

text: I was wondering if anyone here has calculated before the partition function for the Jaynes-Cummings Hamiltonian: $H_{JC}=\omega_0 (a^\dagger a + \sigma^+ \sigma^-)+g (a \sigma^+ +a^\dagger \sigma^-)$ (Here I have assumed resonance and $a,a^\dagger$ are bosonic operators and $\sigma^-,\sigma^+$ are ladder operators for a spin one half particle) For the Hamiltonian above the energy of the ground state is zero and corresponds to 0 excitations in the harmonic oscillator and the spin being down. The excited eigenenergies come in pairs and are given by: $\omega_{n\pm}=n \omega_0 \pm \sqrt{n}g$. I am interested in knowing the partition function: $\mathcal{Z}=\text{tr}( \exp(-\beta H_{JC} ))=1+2\sum_{n=1}^\infty\exp(-\beta \omega_0 n )\cosh(\beta g \sqrt{n}) $ I tried Mathematica to get an analytic expression for the above sum and it did not work. Any thoughts on whether the summation can be expressed in terms of some special function or how to calculate it numerically in an efficient and reliable way?


title: Rigid body rolling quesion

text: ![enter image description here](http://i.stack.imgur.com/o8Kjj.jpg) Hey, im having a bit of trouble with the problem in the added photo. So, there is the cylinder which is attached by a massless rope to a massless pulley, to a box (assume it is a pointed object). Now, Im supposed to find the acceleration (linear) of the cylinder assuming its going down. So, I know I can write torques and Newton's laws equations but i could not figure out why my professor said that the acceleration of the box (upwards) is twice the size of the linear acceleration of the cylinder. Can anybody help figure this issue please.


title: Diagonalizing/eigenvalues of the infinite dimensional matrix of N harmonic oscillators on a ring

text: I have trying to show that the continuum limit of N quantum harmonic oscillators gives rise the the klein-gordon field. However, instead of a usual finite string, I want to do it on a ring. Hence, my Lagrangian is $$L=\frac{m}{2}(\dot{q_1}^2+\dot{q_2}^2+.... \dot{q_n}^2)-\frac{m \omega^2}{2}[(q_1-q_2)^2+(q_2^2-q_3^2)+....(q_n-q_1)^2]$$ So that the matrix for V is $V=\begin{pmatrix} 2 & -1 &0 &. &. &. &-1\\\\-1&2&-1\\\0 &-1 &2 &-1 \\\\.\\\\.\\\\.\\\\-1 &&.&.&.&-1 & 2 \end{pmatrix}$ So that $L=\frac{m}{2}[\dot{x}^2-\omega^2 x^{T}V{x}]$. All the quantities here are matrices. **How do I find the eigenvalues of this matrix?** I tried to find a recursion relation between the characteristic polynomial of $N$ and $N-1$ dimensional matrix, but I failed. Is this the correct method? What other method is there? After finding the eigenvalues, the lagrangian can be written separated into its normal modes, and the propagator or kernel can be found out easily using that of the free particle. The limit of this as $N\to \infty$ should be the klein gordon field. But I am stuck on this. Any help will be appreciated.


title: Calculation the reflection coefficient of a mutlilayer material

text: For our project we have to study an infrared filter. This filter is composed of glass and several layers (nanolayers of titanium oxide, silver and cupper deposited on one side of the glass).   Now we have to determine the reflection coefficient of the coating (i.e. the nanolayers alone) given that of the glass and that of the filter as a whole. Now our supervisor claims that this is calculated by the formula: $R_{filter} = R_{coating}R_{glass}$. This formula seems a bit odd to us though, f.i. because it doesn't hold in general (one can imagine a material composed of two materials, where one has 100% reflectance and the other 50%, so this material should have 100% reflectance, while according to the formula it has 50%). I wonder if somebody maybe knows if this formula holds, and maybe can explain which assumptions one could make under which one can derive the formula.


title: Some questions about the spectral function

text: _If you think that this question is likely to get closed then please do not answer and only say that in the comments since this system doesn't let me delete the question once it has answers._ Everytime I am faced with an analysis of the spectral function it looks like a "new" unintuitive set of jugglery with expectation values and I am unable to see a general picture of what this construction means. I am not sure that I can frame a coherent single question and hence I shall try to put down a set of questions that I have about the idea of spectral functions in QFT.   * I guess in a Dirac field theory one defines the spectral function as follows,  $\rho_{ab}(x-y) = \frac{1}{2} <0|\\{\psi_a(x),\bar{\psi}_b(y)\\}|0>$ Also I see this other definition in the momentum space as, $S_{Fab}(p) = \int _0 ^\infty d\mu^2 \frac{\rho_{ab}(\mu^2)}{p^2 - \mu ^2 + i\epsilon}$ Are these two the same things conceptually? I tried but couldn't prove an equivalence. (I define the Feynman propagator as $S_{Fab} = <0|T\big(\psi_a(x)\bar{\psi}_b(y)\big)|0>$)   * Much of the algebraic complication I see is in being able to handle the quirky "_" sign in the time-ordering of the fermionic fields which is not there in the definition of the Feynman propagator of the Klein-Gordon field (..which apparently is seen by all theories!..) and to see how the expectation values that one gets like $<0|\psi_a(0)|n><n|\bar{\psi}_b(0)|0>$ and $<0|\bar{\psi}_b(0)|n><n|\psi_a(0)|0>$ and how these are in anyway related to the Dirac operator $(i\gamma^\mu p_\mu +m)_{ab}$ that will come-up in the far more easily doable calculation of the spectral function for the free Dirac theory.    * Is it true that for any QFT given its Feynman propagator $S_F(p)$ there will have to exist a positive definite function $\rho(p^2)$ such that a relation is satisfied like, $S_F(p) = \int _0^\infty d\mu ^2 \frac{\rho(\mu^2)} {p^2 - \mu^2 +i\epsilon}$ So no matter how complicatedly interacting a theory for whatever spin it is, its Feynman propagator will always "see" the Feynman propagator for the Klein- Gordon field at some level? (..all the interaction and spin intricacy being seen by the spectral function weighting it?..)   * One seems to say that it is always possible to split the above integral into two parts heuristically as,  $$\begin{eqnarray}S_F(p) &=& \sum (\text{free propagators for the bound states})\\\ &&\+ \int_\text{states} \big( (\text{Feynman propagator of the Klein-Gordon field of a certain mass})\\\ && ~~~~~~~~~~~~~~\times(\text{a spectral function at that mass})\big)\end{eqnarray}$$ Is this splitting guaranteed irrespective of whether one makes the usual assumption of "adiabatic continuity" as in the LSZ formalism or in scattering theory that there is a bijection between the asymptotic states and the states of the interacting theory - which naively would have seemed to ruled out all bound states? To put it another way - does the spectral function see the bound states irrespective of or despite the assumption of adiabatic continuity?


title: Asking about centrepetal acceleration

text: please look at the fig first ![enter image description here](http://i.stack.imgur.com/KXMi6.jpg)    1) How can you claim that the triangle ABC is same as the triangle PQR? 2) How can you claim that the angle between V1 and V2 is same as the angle between AC and AB? I think i have forgotten some basic concept from geometry.


title: Compound pendulum clarification?

text: I read in a book the following about compound pendulum and small displacements:   1. There are two points only for which the time period is minimum.    2. there are maximum 4 points for which the time period is same.  Why is this? Can someone please explain? I am familiar with maximum time period being when $k=l$.   In general, time period is $$T=2\pi \sqrt\frac{k^2+l^2}{lg}$$ for small angle approximation. $k$=Radius of gyration about the centre of gravity, $l$=distance of point of suspension from Centre of Gravity, $g$=gravity


title: Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

text: **Background:** Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, the so-called Born-reciprocal relativity is based on the "phase-space"-like metric $$ds^2=dx^2-c^2dt^2+Adp^2-BdE^2$$ and its full space-time+phase-space extension: $$ds^2=dX^2+dP^2=dx^\mu dx_\mu+\dfrac{1}{\lambda^2}dp^\nu dp_\nu$$ where $$P=\dot{X}$$ Note: particle-wave duality is something like $ x^\mu=\dfrac{h}{p_\mu}$. In Born's reciprocal relativity you have the invariance group which is the _intersection_ of SO (4 +4) and the ordinary symplectic group Sp (4), related to the invariance under the symplectic transformations leaving the Poincaré- Cartan two-form invariant. The intersection of SO(8) and Sp(4) gives you, essentially, the unitary group U (4), or some "cousin" closely related to the metaplectic group. We can try to guess an extension of Born's reciprocal relativity based on higher accelerations as an interesting academical exercise (at least it is for me). In order to do it, you have to find a symmetry which leaves spacetime+phasespace invariant, the force-momentum-space-time extended Born space-time+phase-space interval $ds^2=dx^2+dp^2+df^2$ with $p=\dot{x}$, $ f=\dot{p}$ in this set up. Note that is is the most simple extension, but I am also interested in the problem to enlarge it to extra derivatives, like Tug, Yank,...and n-order derivatives of position. Let me continue. This last metric looks invariant under an orthogonal group SO (4+4+4) = SO (12) group (you can forget about signatures at this moment). One also needs to have an invariant triple wedge product three-form $$\omega_3=d X\wedge dP \wedge d F$$ something tha seems to be connected with a Nambu structure and where $P=\dot{X}$ and $F=\dot{P}$ and with invariance under the (ternary) 3-ary "symplectic" transformations leaving the above 3-form invariant. **My Question(s):** I am trying to discover some (likely nontrivial) Born- reciprocal like generalized transformations for the case of "higher-order" Born-reciprocal like relativities (I am interested in that topic for more than one reason I can not tell you here). I do know what the phase-space Born- reciprocal invariance group transformations ARE (you can see them,e.g., in this nice thesis BornRelthesis) in the case of reciprocal relativity (as I told you above). So, my question, which comes from the original author of the extended Born-phase space relativity, **Carlos Castro Perelman in** this paper, and references therein, is a natural question in the context of higher- order Finsler-like extensions of Special Relativity, and it eventually would include the important issue of curved (generalized) relativistic phase-space- time. After the above preliminary stuff, the issue is: > What is the intersection of the group SO (12) with the _ternary_ group which > leaves invariant the triple-wedge product > > $$\omega_3=d X\wedge dP \wedge d F$$ More generally, I am in fact interested in the next problem. So the extra or bonus question is: what is the (n-ary?) group structure leaving invariant the (n+1)-form $$ \omega_{n+1}=dx\wedge dp\wedge d\dot{p}\wedge\cdots \wedge dp^{(n-1)}$$ where there we include up to (n-1) derivatives of momentum in the exterior product or equivalently $$ \omega_{n+1}=dx\wedge d\dot{x}\wedge d\ddot{x}\wedge\cdots \wedge dx^{(n)}$$ contains up to the n-th derivative of the position. In this case the higher- order metric would be: $$ds^2=dX^2+dP^2+dF^2+\ldots+dP^{(n-1)}=dX^2+d\dot{X}^2+d\ddot{X}^2+\ldots+dX^{(n)2}$$ This metric is invariant under SO(4(n+1)) symmetry (if we work in 4D spacetime), but what is the symmetry group or invariance of the above (n+1)-form and whose intersection with the SO(4(n+1)) group gives us the higher-order generalization of the U(4)/metaplectic invariance group of Born's reciprocal relativity in phase-space? This knowledge should allow me (us) to find the analogue of the (nontrivial) Lorentz transformations which mix the $X,\dot{X}=P,\ddot{X}=\dot{P}=F,\ldots$ coordinates in this enlarged Born relativity theory. **Remark:** In the case we include no derivatives in the "generalized phase space" of position (or we don't include any momentum coordinate in the metric) we get the usual SR/GR metric. When n=1, we get phase space relativity. When n=2, we would obtain the first of a higher-order space-time-momentum-force generalized Born relativity. I am interested in that because one of my main research topics are generalized/enlarged/enhacend/extended theories of relativity. I firmly believe we have not exhausted the power of the relativity principle in every possible direction. I do know what the transformation are in the case where one only has X and P. I need help to find and work out myself the nontrivial transformations mixing X,P and higher order derivatives...The higher-order extension of Lorentz-Born symmetry/transformation group of special/reciprocal relativity.


title: Attenuation mass-thickness for sunset light

text: We are able to look directly at the sun near sunset and sunrise, which clearly demonstrates the fact that our atmosphere attenuates visible light. Let's imagine it follows the typical attenuation profile. $$ I = I_0 \, e^{-(\mu/\rho)\rho \ell} $$ Where $(\mu/\rho)$ is the mass-attenuation coefficient in units of $(m^2/kg)$, and $\rho l$ is the mass-thickness (or the area-density, I think it has a few names) in units of $(kg/m^2)$. The effect of the "soft" sunlight is then predicted as a result of the fact that the mass-thickness of the atmosphere between our eyes and the sun diverges fairly fast as the angle to the sun falls to zero degrees above the horizon. Let's say that a person is standing on (perfectly spherical) Earth, with their eyes at a known elevation, looking at the sun which lies at a known angle above the horizon. What is the expression for the mass-thickness of air in that line of sight? The reason I find this non-trivial is that I can't figure out if the density profile of the atmosphere should matter or not. You could reduce it to simple geometry and get an answer, but is there a coherent argument for that being correct? With a clear expression, I'm actually kind of curious if you could measure the mass-attenuation coefficient with just a digital picture. The sun's intensity starts out constant over the circle, and you know the angle between the top and bottom of the sun exactly. So if you could extract intensity data over the vertical diameter maybe you could then do a least- squares function fit to extract out that attenuation coefficient, and even do it for each of the 3 colors. I don't plan on doing that, but it would be a cool science project.


title: In which direction is the acceleration directed in a non uniform circular motion?

text: Acceleration is directed towards the center of the circle in a uniform circular motion. Is it same for the non-uniform circular motion?


title: path integrals: how/why can the phase be identified with the action?

text: In _Peskin & Schroeder_, chapter 9 introduces the functional methods. The idea, to recall, is simply to sum over all the possible paths: > $U(x_a,x_b;T) = \sum_{\text{all paths}} e^{i . \text{phase}} = \int Dx(t) > e^{i . \text{phase}} $ Then, it happens that the phase could be **identified** with the action. But I have not understood how this could be done ... Could anyone clarify that point? Thanks in advance.


title: Charged black holes in equilibrium

text: Consider a pair of (possibly rotating) charged black holes with masses m1, m2 and like charges q1, q2. It seems that under certain conditions gravitational attraction should exactly cancel electrostatic repulsion and a stationary spacetime will result. > What are these conditions? The point charges analogy suggests the equation k q1 q2 = G m1 m2 However, it is by no means obvious this equation is the correct condition except in the large distance limit. Also: > Is it possible to write down this solution of Einstein-Maxwell theory in > closed form?


title: Ex 0.2.1 in Sachs and Wu's textbook

text: In the next attachements are: 1\. Exercise 0.2.5 which I want help with.   1. Proposition 0.2.1 and its proof. Now, basically a few things are changed in the theorem, I don't think I can use here the definition of s(t) in the proof of prop0.2.1 cause its s(t)=0, I don't think I can use this trick here. Other thoughts that I had, obviously if I plug m=0 into prop0.2.1 I get that I should have: $$\frac{d\gamma^1}{du}=\pm \frac{d\gamma^2}{du}$$, and $$\frac{d\gamma^2}{du}=a$$. My question is how do I satisfy condition b in the theorem, I guess this x should be $$\pm Id +constant$$ ![1](http://i.stack.imgur.com/uNJZS.png) ![2](http://i.stack.imgur.com/Gftvm.png) ![3](http://i.stack.imgur.com/Eo0PX.png)


title: Bessel vs. modified Bessel in radial equation of hydrogen

text: I am trying to understand the difference between Bessel functions and modified Bessel functions (simply googling is yielding complicated, non-intuitive answers). I was under the impression that one allowed for a complex parameter while the other did not - is this true? My question stems from trying to understand the radial part of the Hydrogen eigenproblem (with $u = rR(r)$): $$ \frac{d^2u}{dr^2} = \left[ \frac{l(l+1)}{r^2} - k\right] u(r) $$ which is solved by a linear combination of Spherical Bessel functions and Neumann functions: $$ u(r) = Ar j_l(kr) + Brn_l(kr) $$ Is this solution valid for both real and imaginary $k$? For reference, this linear combination is from Griffiths' _Introduction to Quantum Mechanics_ , Equation 4.45.


title: Membrane that allows liquid to pass only when forced under pressure

text: Is it possible to have a membrane that will not let a liquid through it at normal pressures due to gravity, but pass that liquid when substantially pressurised? For instance, a few inches of water (say 0.1psi) would be blocked, but 100psi would pass through.


title: Which similar properties must objects have to sustain quantum entanglement?

text: > Quantum entanglement occurs when particles such as photons, electrons, > molecules as large as buckyballs, and even small diamonds interact > physically and then become separated; the type of interaction is such that > each resulting member of a pair is properly described by the same quantum > mechanical description (state), which is indefinite in terms of important > factors such as position,momentum, spin, polarization, etc. Source: > Wikipedia All objects can get quantumly entangled, but to get a more sustained quantum entanglement with a long dechoherence time, the objects has to have equal properties. Do all properties have to be equal or could some properties be different? For instance: Form seem to be important for quantum entanglement, but is size important? Could a small diamond be quantum entangled with a larger diamond with the same shape?


title: Angular momentum of the electric field of a point-like electric charge and the magnetic field of a monopole

text: I am currently reading "Magnetic Monopoles" of Ya. Shnir. My problem is I can not retrieve a result the author provides in the first chapter of the first part. In this chapter, he studies the non-relativistic scattering of an electric charge on a magnetic one. The author writes [p.5, near eq. (1.13)]: > ... the appearance of an additional term in the definition of the angular > momentum $(1.11)$ originates from a non-trivial field contribution. Indeed, > since a static monopole is placed at the origin, its magnetic field is given > by $(1.1)$. Then the classical angular momentum of the electric field of a > point-like electric charge, whose position is defined by its radius vector > $\mathbf{r}$, and the magnetic field of a monopole is a volume integral > involving the Poynting vector > > \begin{align} \tilde{\mathbf{L}}_{eg} &= \dfrac{1}{4\pi}\int \mathbf{r'} > \times \left [ \mathbf{E} \times \mathbf{B}\right] d^3r'\tag{L.1}\\\& = - > \dfrac{g}{4\pi} \int d^3r' \left( \mathbf{\nabla}'\cdot \mathbf{E}\right) > \hat{\bf r}' \tag{L.2}\\\ &= -eg\hat{\bf r} \tag{L.3} \end{align} > > where we perform the integration by parts, take into account that the fields > vanish asymptotically and invoke the Maxwell equation > > \begin{equation}\left(\mathbf{\nabla}' . \mathbf{E} \right) = 4 \pi e > \delta^{(3)}\left( \mathbf{r} - \mathbf{r}'\right)\end{equation} ... The magnetic field is $\mathbf{B} = \dfrac{g}{r^3} \mathbf{r} \tag{1.1}$ The generalised angular momentum is $\mathbf{L} = \mathbf{r} \times m\mathbf{v} - eg \hat{\bf r} \tag{1.11}$ The author gives how he got $(L.2)$ from $(L.1)$ but I do not know how to do? Have you any idea?


title: When is it useful to distinguish between vectors and pseudovectors in experimental & theoretical physics?

text: My understanding of pseudovectors vs vectors is pretty basic. Both transform in the same way under a rotation, but differently upon reflection. I might even be able to summarize that using an equation, but that's about it. Similarly, I can follow arguments that pseudovectors behave differently in "mirrors" than vectors. But my response to this is always: Okay, so what? When would I ever "do physics" in a mirror? The usefulness eludes me. I'd like to gain a better understanding of the importance of this difference.   * When is it useful for an _experimental physicist_ to distinguish between the two?   * When is it useful for a _theoretical physicist_ to distinguish between the two? I believe symmetry is important to at least one of these, but would appreciate a practical rather than abstract argument of when one has to be careful about the distinction.


title: Black holes: Behaviour of light near an event horizon

text: Electromagnetic waves travel mostly in vacuum medium, in outer space, but sometimes in gaseous media, such as in gaseous atmospheres involved in nebulae. If electromagnetic-waves come anywhere in the vicinity of a black- hole, the whole of it enters the black-hole. Can we suppose, that the event- horizon of the black-hole acts in a way like an interface, separating two optical media with different refractive indices **:** vacuum (the medium from which EM waves are "incident"), and the unknown medium beyond. If, this supposition holds true, then, we would know one thing at least regarding this unknown medium, that, EM waves travel in this medium with a velocity > c, as this medium is rarer, than vacuum itself. (Question of an amateur)


title: Are the inner planets on planar orbits because there was more dust in the inner solar system (early on in planetary accretion)?

text: Question inspired by a question thread here. So when there's lots of dust in a galaxy, the galaxy tends to collapse into a spiral galaxy (to maintain angular momentum and to minimize gravitational potential energy). Is this the same thing that happens in the inner regions of the solar system? The outer regions have less dust so the orbits of minor planets "out there" tends to be more elliptical. And could this perhaps mean that the orbits of planets tend to be more coplanar around stars of higher metallicity?


title: How much time can I power my laptop by eating one dessert?

text: Random question that popped into my mind after a 4-hours power outage. Let us assume that I am eating an extra dessert (250 kcal) and that I am using a bike and a generator to power my laptop (it consumes 10W while idle). How much time can I use the laptop? I computed this as follows: $$ t = \frac{E}{P} = \frac{250 kcal}{10W}=\frac{250 * 4,18KJ}{10J/s} = 1.045 * 10^5s=30h $$ The above calculation assumes 100% efficiency. But even if we assume 10% efficiency, I would still be able to power my laptop for $3h$ and all this by eating just one extra dessert. Questions: Are my calculations correct? Are my assumptions reasonable? Why don't we see a proliferation of bike-laptop devices?


title: What is the most efficient machine for translating gravitational potential energy of one mass into kinetic energy of a different mass?

text: As the question states, what is our current best machine for translating falling gravitational potential energy, such as a large weight, into launching a smaller projectile vertically? A lever? A trebuchet? What sorts of efficiency levels have been achieved? What I'm looking for is the best way to translate: $$ g * h * m_{\textrm{weight}} * \textrm{efficiency} = \frac{1}{2} * m_{\textrm{projectile}} * v^2. $$ # Edit Here's an example: I want to launch a 10kg mass at 100m/s using only gravitational energy (no chemical rockets, no rail guns, etc.). What's our most efficient machine to do this, and what percentage of the energy of the falling weight will actually be transferred to the projectile? This would get me an idea of the size of the weight and how much it has to fall to accomplish this. The only machine I know of that does this on a large scale and for decent size masses is a trebuchet, so this might be a good starting point unless there are others I'm unaware of.


title: Magnetic flux of a loop due to another loop

text: The scenario is the following, I am given 2 loops with the same radius, r, a distance of d, and same current of I. In the left loop the current goes counter clockwise, in the right loop the current is clockwise. The two loops centers lie on the same axis which are perpendicular to the plane of the loops. I am asked to find the magnetic flux of the left loop due to the current on the right loop. I know that the magnetic flux of a loop is $\phi=B\pi r^2$, where $B=\dfrac{ \mu_0I}{2R}$ So how exactly do I find the Total magnetic flux on the loop due to the magnetic flux on the other? Am I going to add or subtract? I get confused at this part


title: Why does your reflection stay the same size when you move further away from the mirror?

text: This was an experiment I saw in my son's workbook. It said to mark out the top of your forehead and the bottom of your chin on a mirror using a whiteboard marker. Then slowly move backwards, and investigate what happens to the size of the reflection subjective to the two marks made. It actually got me quite flabbergasted. I always thought the reflection would get smaller as you moved away from the mirror. Why is this?


title: Max speed as a function of engine power

text: My question started out as finding the maximum speed of a go kart, taking into account only the drag forces as force opposing the motor. I've done some investigation to find: $$ F_{drag}=\; \frac{\rho v^{2}\ C_{x} A_{f}}{2} $$ Where $C$ = drag coefficient, and $\rho$ is the density of air. Further, with $ F_{motor}=\dfrac{P_{motor}}{v} $ and the condition that at maximum speed, the acceleration will be 0, net force will be 0 as well, so $F_{motor}=F_{drag}$. Combining equations, $F_{drag}=\dfrac{P_{motor}}{v}$, so $\dfrac{P_{motor}}{v}=\; \dfrac{\rho v^{2} C A_{f}}{2}$ Simplifying gives $P_{motor}=\; \dfrac{\rho\cdot v^{3}C\cdot A_{f}}{2}$ Solving for $v$ yields: $$v=\; \sqrt[3]{\dfrac{2P_{motor}}{\rho C A_{f}}}$$ Converting to units of miles per hour for $v$, and horsepower for $P_{motor}$: $$v=\; 2.2 \sqrt[3]{\dfrac{HP_{motor} \cdot 745\cdot 2}{\rho\mbox{C}A_{f}}}$$ From this pdf, I've found values of $\rho = 1.2 \frac{kg}{m^{3}}$, $C=.8$, and $A_{f}=.57$. So! Questions:   1. Are my equations correct? I'm particularly concerned about the third root portion, given that drag is a quadratic. Doesn't connect in my mind, so perhaps my substitutions are faulty?   2. Are the values I have for those constants for ideal values, (i.e. perfectly enginneered kart, very unlike what I'd be able to construct myself). If so, by what general percent will my values be different?   3. Finally, is it unrealistic to call all forces other than drag insignificant? If so, by what general percent will my values for speed be too high? For reference, this calculation has a 5 hp engine at a max speed of 52 mph, 15 hp at 75 mph, and 35 hp at 100 mph.


title: Why does the moon face earth with the same side?

text: I know that the rotation period of the moon equals its revolution period. It's just so astonishing that these 2 values have such a small difference. I mean, what is the probability of these 2 values to be practically the same? I don't believe this to be a coincidence. It's just too much for a coincidence. What could have caused this?


title: Has anyone on Earth ever seen the dark side of the moon and if so where are the pictures?

text: If the Moon rotates then we should see the dark side right? But as far as I know the Moon only shows one side to Earth, how can this be if it is rotating?


title: Why lunar day lasts exactly one month?

text: Therefore we don't see dark side of the moon. Is the core of the Moon closer to the Earth than the center of the moon? Or what is the reason? ![Lunar motion](http://i.stack.imgur.com/xZ3f6.jpg)


title: What happens when atoms speed up to extreme velocity?

text: The title might be misleading, but my question is in regard to what happens when we reach temperatures close to absolute zero (Kelvin). I've found different quotes as to what happens on the low end of the scale: > At about 10 micro degrees Kelvin, Rubidium atoms move at about 0.11 mph > (0.18 km/hr) — slower than a three-toed sloth, says physicist Luis Orozco of > the University of Maryland. But what happens if we turn it around, and (hypothetically) increase the speed of the atoms to speeds close to c?


title: Four questions on color

text: First, I apologize for being a mathematician and having no scientific background in physics. The following questions on color came up in a discussion at lunch and I would be very happy to get some answers or corrections of my understanding.   * The first question is, what a color the human eye can see is in the first place. According to this wikipedia article, the human eye has three cone cells $L$, $M$ and $S$ responsible for color vision which can be stimulated each in a one dimensional way, i.e. ''more'' or ''less''. Combining the stimulation of all three, one gets a $3$-dimensional space called the LMS color space. Is a color the human eye can see just a point (in a specific range) in this LMS-space? A negative answer could be that time plays a role (switching between e.g. two points of the LMS-space with a certain frequency defines a color). The model is made in the way that only one point in the space at a time represents a state of the eye at that time and not many. For the following questions, I suppose that the answer to the above question is positive, e.g. a color the human eye can see is defined by a specific point in the LMS-space. By experiments, one could possibly determine the body in the LMS-space, the human eye can see (this is what I meant by the ''specific'' point). Of course, this may differ from person to person but perhaps one can average over some and get an average body $AB$ in the LMS-space. Points inside are now exactly the colors the human eye can see.   * Now there is this CIE color space which is vizualized by this $2$-dimensional picture looking like a sole of a shoe. As far as I understand, there is a bijection (continuous?) from the LMS-space above to this CIE-space where two axes describe the so called **chromaticity** and one axis describe the so called **luminance**. What is the figure shoe sole now exactly? Is it a certain hyperplane in the $3$-dimensional CIE color space to which the luminance-axis in perpendicular? (Perhaps one can say something like ''it is the color at full luminance''. Note, that black isn't in here.)   * The RGB color space has the drawback of having a gamut not filling the whole visible range. Why doesn't one use the frequencies of the $L$, $M$ and $S$ cone cells as the primary colors for technical devices like screens? Couldn't one describe the whole visible range then (i.e. wouldn't the gamut be the whole shoe sole then)?   * My last question concerns this old painters belief, that every color can be obtained by mixing red (R), yellow (Y) and blue (B) or the belief that every color can be obtained by mixing cyan (C), magenta (M) and yellow (Y). Both models, the RYB and the CMY model, are subtractive. When one mixes these colors, doesn't one ignore the luminance? I.e. don't the painters want to say that they can obtain each **chromaticity** by mixing these colors? (Of yourse, if they want to say this, it is also not really true since the gamut of these three colors determine a certain region in the shoe sole plane from the previous questions which is not the complete sole.)


title: How to get funding for research on something that can revolutionize the quantum world!

text: I think this can revolutionize the quantum world! Any ideas on how to impress physicists to get a full fledged funding for research?


title: How does a system interact with environment in quantum mechanics? When would this interaction occur? What is it?

text: As title says, how does a system interact with environment? I realize that this interaction can lead to interference terms and non-diagonal terms in density matrix being reduced (quantum decoherence). But what exactly is this interaction of system and environment? What would be the example of this interaction and when does this interaction occur?


title: A vehicle moves from rest to a certain distance. Would the distance time graph always be curved?

text: A vehicle moves from rest to a certain distance. Would the distance time graph always be curved ? Why cannot it be a straight line ? For example, a car moving from rest to 100m, with a constant velocity of 10m/s . So, if i draw a distance time graph, the line will start from s=0 and end at s=100 ( with t=0 to t=10 ). Would this graph always be curved ? Why can't it be a straight line ?


title: localization of wave or particles

text: > Nonlinear field theories contain a large number of localized solutions. I have found this text in a article. What I don't understand is "what is localized?". Is it refer defining position of a particle or a wave? Can someone give me an elaboration with example?


title: Moduli potential in Type IIB String Theory

text: In the book _String Theory and M-Theory_ by K. Becker, M. Becker and J.H. Schwarz:   1. Why is the potential for moduli given by eq (10.168): $$\tag{10.168 }V(T,K) ~=~ \frac1{4\mathcal{V}^3} \Big( \int_{CY_4} F \wedge \star F - \frac16 \chi T_{M2} \Big)?$$   Maybe the answer of this question is trivial, but I cannot see how $T$ and $K$ enter this picture and get $V(T,K)$ as their potential.   2. How can one arrives at eq (10.181): $$\tag{10.181} V~=~e^{\mathcal{K}} \Big( G^{a \bar{b}} \mathcal{D}_a W \mathcal{D}_\bar{b} \bar{W} - 3|W|^2 \Big)?$$


title: Random orientation percolation (Grimmett model) from the viewpoint of statistical mechanics

text: This is a rather soft question, but I would like to know how physicists would approach a problem which seems to be hard from the mathematical prospective. The Grimmett percolation model is defined as follows. Consider the $\mathbb{Z}^2$ lattice and randomly assign an orientation for every bond of the lattice in the following way. Let $p$ be the percolation parameter and for every horizontal bond orient it rightward with probability $p$ and leftwards otherwise. The same rule applies for vertical bonds: a bond is oriented upwards with probility $p$ and downwards otherwise. Thus we get an oriented graph supported on the whole lattice $\mathbb{Z}^2$. Grimmett conjectures that for all $p \neq 1/2$ there is an infinite directed path from the origin to infinity. By coupling with classical bond percolation it is not hard to show that at $1/2$ the system does not percolate - so presumably $1/2$ is a critical point. On the other hand, the conjecture is quite old and seems to be very far from resolution mainly because all the known methods used for other percolation models fail to work. I am wondering if this model has any meaning in statistical mechanics and if so how a physicist would approach Grimmett's conjecture?


title: Vacuum Expectation Value and the Minima of the Potential

text: Often times in quantum field theory, you will hear people using the term "vacuum expectation value" when referring to the minimum of the potential $V(\phi )$ in the Lagrangian (I'm pretty sure every source I've seen that explains the Higgs mechanism uses this terminology). However, a priori, it would seem that the term "vacuum expectation value" (of a field $\phi$) should refer to $\langle 0|\phi |0\rangle$, where $|0\rangle$ is the physical vacuum of the theory (whatever that means; see my other question). What is the proof that these two coincide?


title: If you flew into a black hole

text: Would you outlive everyone? I'm coming from the point of view that time would be experienced more slowly (although not from your point of view) the denser the gravity gets.


title: What is the interpretation of the Chern-Simons electromagnetic spin density?

text: Hans de Vries (who happens to be a no-longer-active physics.SE user) has an online book (referenced below) in which ch. 6 is a presentation of an object he calls the Chern-Simons current, electromagnetic spin density, or Chern- Simons electromagnetic spin: $ C^a = \frac{\epsilon_0}{2} \epsilon^{abcd} A_b F_{cd} = \epsilon_0 \epsilon^{abcd} A_b \partial_c A_d $ . He has a long and detailed presentation of this thing, including graphs and examples. Unfortunately I'm not having much luck extracting from this what he claims is the interpretation of it, whether his interpretation is standard, and whether it has a classical interpretation. He references Mandel and Wolf (which I don't have access to), but what they apparently present is a different expression, $\epsilon_0 \textbf{E}\times\textbf{A}$, and refer to it as the intrinsic angular momentum of the electromagnetic field. De Vries says that $C^a$ is the natural way of making this tensorial. It seems hard to check whether what he's saying is standard, since he says, "The derivations (which I had to do myself since somehow one can't find these anywhere) and many details can be found in my paper..." (linking to a paper that duplicates the material in the book). The expression is manifestly classical, so I don't really see how it's to be interpreted as the intrinsic or spin contribution to the field's angular momentum. The classical/quantum interpretation is also obscured because factors of $\hbar$ start appearing at de Vries' eq. 6.6, but these equations are supposed to be justified somewhere later on. It seems odd to me that this is written as a product of the four-potential and a derivative of the four-potential. This makes it not manifestly gauge- invariant. If I was going to write down a density of angular momentum for the electromagnetic field, I would start from the stress-energy tensor, which is a product of $F$ with $F$, and therefore independent of gauge. It's not at all obvious to me what one would even mean by a spin density for the electromagnetic field. I guess for a classical fluid of electromagnetic radiation in equilibrium (e.g., the kind of environment we had during the early universe), I would define a comoving frame and look at the amount of angular momentum $L^{ab}=r^ap^b$ in a small volume element. But that clearly isn't going to work for the electromagnetic field in general, since you can't define a comoving frame for, e.g., an electromagnetic plane wave. It does make sense that the expression is manifestly translation-invariant, since if there is some sensible way to split the angular momentum into spin and orbital parts, only the orbital part should depend on one's choice of axis. De Vries, Understanding Relativistic Quantum Field Theory, http://www.physics- quest.org/ , ch. 6


title: A dielectric table is being inserted between a plate capacitor and $\triangle U<0$ how to deduce the table is attracted to the plates?

text: I am practicing for an exam in my Physics $2$ course. One of a previews exam questions described a plate capacitor and asked to calculate the initial energy $U_{0}$, then a dielectric table was inserted between the plates and I was asked to calculate the energy, $U_{1}$, at this state. I got that there was less energy in the final state than there was at the initial state, i.e $$ \triangle U=U_{1}-U_{0}<0 $$ The question asked if the table is attracted to the plates or repealed from it, and the answer claimed that since there was less energy at the final state then it means that the table is attracted to the plates. I lack intuition on this and I don't understand how the conclusion was made. I tried to think it with terms of work done by the electrical field: If I assume that the table is attracted to the plates then I think that the work is negative (since I don't have to do actual work because there is an attraction), and since there was negative work done $W<0$ I have $$U_{1}=U_{0}+W<U_{0}$$ The other case leads to $U_{1}>U_{0}$ and so I get that the first is the one that occurs. But I am not to sure about this argument since $$W=\int F\cdot dl$$ and since there is an attraction the direction of the path is the same as the direction of the force and so $W>0$. Can someone please help me understand why there is an attraction ?


title: A question about an identity in deriving Born-Infeld action

text: I have a question in David Tong's Example Sheet 4 Problem 5b, how to verify the last equation (*) on p.2? (There is a solution for example sheet 3, but seems to be no solution for example sheet 4.) > Problem 5b: > > Show that the equations of motion arising from the Born-Infeld action are > equivalent to the beta function condition for the open string,   >  $$\beta_\sigma\left(F\right)=\left( \frac{1} {1 -F^2} \right)^{\mu > \rho}\partial_\mu F_{\rho\sigma }=0 $$ **Note:** To do this, it will prove > very useful if you can ﬁrst show the following results:   >  $$∂_μ\left[\operatorname{tr} \ln(1 − F^2)\right] = −4 ∂_\rho > F_{μ\sigma}\left(\frac{F}{1-F^2}\right)^{\sigma \rho } $$ > > which requires use of the Bianchi identity for $F_{\mu \nu }$ and > > $$ \tag{*} \begin{align} \partial _\mu \left( \frac{ F}{1-F^2} > \right)^{\mu\nu} &= \left( \frac{ F}{1-F^2} \right)^{\mu\rho} \partial_\mu > F_{\rho\sigma} \left( \frac{ F}{1-F^2} \right)^{\sigma \nu} \\\ &\qquad+ > \left( \frac{1} {1 -F^2} \right)^{\mu \rho} \partial_\mu > F_{\rho\sigma}\left( \frac{ 1 }{1-F^2} \right)^{\sigma \nu} \end{align}$$ In addition, as given in question 5a > $$F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} $$ My attempt to prove the problem: LHS $$\partial_{\mu} \left( \frac{ F}{1-F^2} \right)^{\mu\nu} = \partial_{\mu} \left[ F^{\mu}_{\alpha} \left( \frac{1}{1-F^2} \right)^{\alpha \nu} \right] = \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + F^{\mu}_{\alpha} \partial_{\mu} \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} \tag{1} $$ Using the formula in matrix cookbook for the derivative of inverse matrix, Eq. (53) Eq. (1) becomes $$\left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 F^{\mu}_{\alpha} \left[ \frac{1}{1 -F^2} \left( \partial_{\mu} F \right) F \frac{1}{1-F^2} \right]^{\alpha \nu} $$   $$= \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} \tag{2} $$ The second term in Eq.(2) cancels the second term in the RHS in the problem sheet equation. We then need to show $$ \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} - \left( \frac{1}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{1}{1-F^2}\right)^{\sigma \nu} =0 \tag{3} $$ then I didn't find a way to show Eq. (3) hold. I tried to combine the second and third term, and rearrange them, but didn't got a simple expression.


title: Can renormalization group evolution be used to capture emergence?

text: I am posing this question with condensed matter systems in mind. **Is it, in principle, possible to obtain emergence using the renormalization group (RG)?** I read in X.-G. Wen's book (Quantum Field Theory of Many-Body systems) that " _we cannot use the renormalization group approach to obtain the emergence of qualitatively new phenomena_ " (Sec. 3.5.4 on page 110). It seems to me that the statement should have been something like RG, **in practice** , cannot be used to capture emergence. Or, more precisely, local perturbative RG schemes are not suited for capturing emergence. Here is what confuses me: if emergence is nothing but the statement that the low-energy degrees of freedom may not be the same as the high-energy, microscopic degrees of freedom in a theory, then **in principle** RG should be able to capture emergence, right?


title: Refraction and scattered light for NLCs

text: For helping with judging NLC candidates (are they NLC or not) I have a set of formulas to calculate the minimum altitude (in km) of the candidate given an observed altitude (in degrees) of the candidate (using known stars, like Auriga, as reference points) and the date/time (giving the Sun's altitude). A typical observed altitude is 10 degrees, largely unaffected by refraction and a typical result is 70 km (ruling out any other type of cloud). The formulas are limited to only work in the horizontal direction (azimuth) of the Sun (that is under the horizon), but this is not a problem in practice. Reasonable assumptions are used, except one: no atmospheric effects like refraction and scattering are accounted for.   1. Are NLCs only visible to the naked eye if directly exposed to sunlight (the Sun directly visible by the NLC)? Or is scattered light enough to make them visible?   2. What about refraction? Light coming to the NLC goes through near vacuum when it is close to the NLC so the refraction must be less than for an observer on the ground, but what is the exact figure (when the Sun is at the horizon as seen by the NLC)?


title: A question about the constraints in BRST-Fock theories

text: In _BRST Symmetry in the Classical and Quantum Theories of Gauge Systems_ , Henneaux says the Fock representation is not applicable to an odd number of constraints. Then he goes on to say that the Kugo-Ojima quartet requires the constraints to be in pairs. For BRST theories, when are they not in pairs?