title: 3D coherence of light speckles- difference between Airy and gaussian speckles

text: In this article, http://www.dfm.uninsubria.it/laboferri/web_page/articoli_pdf/PRA_2009-Magatti- Three_dimensional_coherence_of_light_speckles-experiment.pdf, the 3D-coherence properties of light speckles are studied at different distances from the source. What I don't understand is the distinction drawn between what the authors call Airy speckles and gaussian speckles. In particular, it's not clear under which experimental conditions Airy or gaussian speckles are produced. How should I choose my setup so that I can know a priori that the speckles will be Airy or gaussian?


title: Dynamics - Beginner question

text: I am trying to understand this question, but I keep coming to a dead end: Two identical rectangular masses rest on top of one another, the top is tied to a wall on the left, while the one on the bottom has a roped attached to it on the right side. What is the max force that can be applied to the rope on the bottom before the block will begin to slide out from underneath the top one. (the coefficient of friction is equal between the block and the ground, and the bottom block with the top block). Ok, so I am pretty sure this question is just asking me to find the net force on the bottom block, as applying a force larger then this will result in an acceleration of the lower block: $$ F_{net}=ma$$ $$ 2F_f=ma$$ $$2F_n\mu=ma$$ $$2mg\times2\mu=ma$$ Ok, I am totally lost as to how solve this question. My intention is to find the net force acting on the lower block. I am pretty sure the the acceleration will be zero, resulting in the right side of the equation equaling zero. But I am not sure how to handle the coefficient of friction.


title: Understanding the operation of Thomas precession

text: How can we physically understand the operation of **Thomas Precession**? This modifies the effective energy of coupling between the spin and the orbital angular momentum of the electron by an extra factor of $\frac{1}{2}$. I know that Thomas precession is related to the fact that two boosts in different directions do not commute and is equivalent to a pure boost plus a pure rotation. But _how does this lead to a precessional motion of the spin_ and how does this effect _physically modifies the dynamics of the electron inside the atom_ such that the _interaction energy picks up an extra factor of $\frac{1}{2}$_?


title: Is this a valid proof that the four-current is conserved?

text: The four-current of a particle moving along a worldine $X^\nu(s)$ is defined as $$j^\mu(x^\nu) = ec \int u^\mu(s)\, \delta^4(x^\nu - X^\nu(s)) \, ds$$ So here's my proof that this is conserved: \begin{align} \partial_\mu j^\mu&= ec\partial_\mu \int u^\mu(s) \,\delta^4(x^\nu - X^\nu(s)) \, ds \\\ &= ec \int u^\mu(s) \,\partial_\mu \delta^4(x^\nu - X^\nu(s)) \, ds \\\ &= ec \int \frac{\partial x^\mu}{\partial s} \frac{\partial s}{\partial x^\mu}\frac{\partial (\delta^4(x^\nu - X^\nu(s)))}{\partial s} \, ds \\\ &= ec \int \frac{\partial}{\partial s} \delta^4(x^\nu - X^\nu(s)) \, ds \\\ &= ec \bigg|_{-\infty}^\infty \delta^4(x^\nu - X^\nu(s)) \\\ &= 0 \end{align} Each step sort of makes sense but working with delta functions makes me feel deeply uneasy and I'm not sure whether this is "valid" by the physicist's standard of rigour or just some hack that appears to give the correct answer.


title: Continuum limit for solid mechanics

text: Is there a rigorous derivation of the limits for continuum properties in solid mechanics? For instance, the stress-strain relationship may be linear for large samples (the slope being the Young's Modulus) but at what limit does that break down? Even if a linear relationship exists for the stress-strain curve between two atoms, or two crystals, the slope likely isn't the same as for the large sample. What spatial scale separates the two regimes and is it similar to fluid dynamics where there is the Boltzmann equation at the non-continuum regime, the Navier-Stokes equations in the continuum regime, and a bridge between the two regimes (Chapman-Enskog equations)?


title: What becomes of the massive clouds expelled from Quasars?

text: What becomes of the massive clouds expelled from Quasar jets? Do they simply disperse into intergalactic diffuse clouds? Or do they on occasion have enough gravity to hold together and or collapse? If they collapse, what becomes of them? Does anybody else get the impression from the following pictures that the clouds hold together? Quasar jet cloud image Google search


title: Why meet at the center of mass?

text: If two objects of different masses are held at a distance $d$ and then I let them go, they will meet at the center of mass of the particle system due to mutual gravitational attraction My question (which may be rather silly) is that why do they meet a the center of mass, why not anywhere else? Is there some sort of mathematical proof behind this?


title: Momentum representation of a state

text: I am trying to figure out the momentum representation of the state which has the properties $$\langle \psi |\hat q |\psi \rangle=-q_0,$$ $$\langle\psi|\hat p|\psi \rangle=p_0, $$$$\Delta q\Delta p=\hbar/2.$$ My approach to this was going to show that this is a Gaussian, because it obeys the minimum uncertainty principle. This would result in a Gaussian centered at $-q_0$. But for momentum, the Gaussian produces an expectation of 0, so is there something that I am missing here? EDIT::: So this is what I have so far. A gaussian centered at $-q_0$ $$A e^{-(q+q_0)^2/2}e^{i p_0x/\hbar}$$ This would have a momentum distribution centered at $p_0$ according to the fourier transform, correct?


title: Definition of "localized state"

text: Is there a definition of a localized state in quantum mechanics? I've seen that some textbooks use this term, but they don't give a definition. Thanks


title: In a moving bus will the ball fall behind me when tossed?

text: Using Newton's law I have to find whether in a moving bus will the ball fall behind me when tossed in air or will come to me only or will fall in front of me. In fact there are two cases; 1- Bus moving with constant speed 2- Bus moving with an acceleration of 1 m/s^2. How do I proceed in each case?


title: Which low energy SUSY models are compatible with LHC and tevatron data so far?

text: The LHC data has eliminated huge swaths of permissible low energy SUSY breaking models. But not all. Which regions of parameter space are still open?


title: Expression for combining several probabilities in a single discriminator

text: I'm doing an analysis of some experiment. Details are not very important, but I've stumbled on an expression that I cannot make sense of. I have several simultaneous measurements of a certain parameter $x_i$. I also have precomputed pdfs, $\rho_i(x_i)$, for that parameter to be "bad". Based on these, I'm computing a confidence interval for each measurement: $$P_i = \int_{|x_i|}^\infty \rho(x)dx$$ Then I'm told that I can combine these into a single discriminator using that expression: $$P = Q \sum_i \frac{(-\ln Q)^{i-1}}{(i-1)!},\quad {\rm where}\quad Q=\prod_iP_i$$ I can understand the product of probabilities, but taking then that cryptic function from it, I'm not familiar with that at all. I'm sure that this is something standard for the processing of experimental data. Can someone clarify what this means?


title: Reflected optical rotation of polarised light

text: Assume polarised light travels through an optically active medium (e.g. water with lots of sugar) with the specific angle $\alpha_s$ of depth $d$ at which end it is reflected (so in total it travels $2d$). Will the light at the end where it initially entered the material be polarised with a relative angle of $\alpha_sd + (-\alpha_s)d = 0$ or with $\alpha_sd + \alpha_sd = 2\alpha_sd$? I believe it is the first, because the rotation is reversed by the medium, but is that actually correct?


title: Quantum tunneling in Field theory with Time dependent potential

text: What should be the limits of integration for euclidean action $S(\phi)$ in 3d and 4d? This action is negatively exponentiated to calculate the decay rate. I suspect that it is variable limit problem.


title: Lorentzforce and righthand rule

text: > Figure 1: ![enter image description > here](http://i.stack.imgur.com/XWSl6.png) > > A spool is hung on a spring. A part of the coil hangs in front of a > homogeneous magnetic field. The coil is incorporated in an electrical > circuit. See Figure 1. > > the coil KLMN has a rectangle shape, has a height of 10.0 cm, a width of 8.0 > cm, and has 200 windings of which two are shown in Figure 1. > > The electrical circuit comprises a power supply, a sliding resistance and > the coil, connected in the manner shown in Figure 1 > > The distance between the bottom of the coil and the borders of the magnetic > field is referred to as y. See Figure 1. If the power source is not yet > connected, y is equal to 5.0 cm. The voltage source will now be connected, > so that the spring will be stretched further and a Lorentz force will be > exerted. The current in the coil can be changed by chaging the resistance. > If there is a larger current, the spring will stretch more. When calculating the LorentzForce : Fl=N*B*I*l Is the length (l) 8cm or 10cm? And why? If the value of the current intensity is at a certain value, the coil length does not appear to increase any further with an increase in the current intensity. Why is this the case when y=10 cm and how to apply the righthandrule in here? I think 10cm, is because the height of the coil. But I don't see why it should be ten, because when there is no current y is already 5. And I don't see how you can use the righthandrule to show that the coil is going upward after y=10cm.


title: Why can electricity flow only in one direction through a diode?

text: A few days ago I was soldering a small thing which contained a diode, a battery and some other useless things. Unfortunately, I soldered the diode reversed and it didn't work. When I reversed it again it started to light. I was trying to find an answer in Google, but the only thing I could find was the fact that electricity in diode can flow only in one direction, but my question is why? How does it work? Is this fact connected with the construction of a diode?


title: why do we get continuous intensity in spectra?

text: we know energy levels of a molecule is discrete. So we should get only sharp peaks for any spectra. But in most of the spectra I see a continuous intensity? why is so? In the following spectra the peaks are understood but why the tail of the peaks don't touch zero intensity line?![Raman spectra of 1,6-dichlorohexane obtained at different excitation lines. Arb: Arbitrary.](http://i.stack.imgur.com/YS7Bp.jpg)


title: can we investigate physics through investigation of pure number?

text: If the consistency between the two is so absolute, why can we not investigate the physical nature of the universe through analysis of pure number? Particularly at the quantum scale?


title: Friction acting on mass on string

text: > _Assume we have some stationary point mass that would slide, if not for some > coefficient of friction, along a light, inextensible string, attached to two > points at different heights. Considering forces on the mass whilst it is > stationary, which direction does the frictional force act?_ My first thought is that it will act along the steeper of the two strings, but the concept of a reaction force (which surely friction must still be acting perpendicular to) seems less clear for a string? ![enter image description here](http://i.stack.imgur.com/293gN.gif) This diagram gives an idea of the kind of set up i'm thinking about (from a random webpage with a sufficiently useful diagram, not specific to this problem), although the string can be of any length and thus don't assume that either string is horizontal.


title: What happens when a black hole dies?

text: It just vanishes into space leaving nothing behind or does it expel some material? Also, talking more about black holes, as far as i understand the term "temperature", it is defined by the amount of kinetic energy of the particles of a body and from what ive read, there are no particles inside a black hole, i know that you can find the temp of a BH by analyzing its hawking's radiation but i cant really understand how the term 'temperature' can be applied to a black hole. Does it mean that whatever is that it is inside a black hole has kinetic energy? Have physicist already experienced the death of any regular BH (by regular i mean space-sized BHs not micro ones created in labs)? Also, how is the Hawking's Radiation form? Because if it is released by the BH i bet it must have formed outside the event horizon, but how exactly?


title: Faraday law, third Maxwell's equation in Mathematica

text: Three question about this equation: $ \displaystyle\nabla\times\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} $ **1** If I solve this equation with Mathematica, I find the magnetic field $b(x,y,z,t),b:\mathbb{R}^4\rightarrow\mathbb{R}^3$ right? **2** I have put an arbitrary function $e:\mathbb{R}^4\rightarrow\mathbb{R}^3$ as electric field for this experiment, but how can I calculate that function for a real case; what I need to do this? **3** One time that I have both the electric and magnetic field how can I compose the electromagnetic field?               Needs["VectorAnalysis`"]          (*Electric field e : R^4->R^3 *)     e[x_, y_, z_, t_] := {x - 3 y, 4 y + t, y + z + t};          Maxwell = Curl[e[x, y, z, t]] == -D[b[x, y, z, t], t];          DSolve[Maxwell, b[x, y, z, t], {x, y, z, t}]


title: What is the velocity area method for estimating the flow of water?

text: Can anyone explain to me what the Velocity Area method for measuring river or water flow is? My guess is that the product of the cross sectional area and the velocity of water flowing in a pipe is always constant. If the Cross sectional area of the pipe increases at a particular point, then the velocity decreases so that the product $AV$ is a constant. Am I right? If so, how can we extend this to pipes where the water is accelerating & does not have a constant velocity? For example, the system may be under the action of gravity & hence the acceleration of the water is $g$, the acceleration due to gravity?


title: Should I heat my room when I'm not here, energy-efficiently speaking?

text: I was wondering as it's getting cold : is it better for my electricity bill to shut down completely my (electric) heater during day, and to turn it on again when I come home (then it will have to heat the room from something like 5°C to around 20°C), or should I keep the temperature around, say, 15°C while I'm away ? I would naively guess it's better off, but I'm also wondering about inertia of the walls for instance...


title: What is the pressure in a vertical pipe that has moving fluid through it?

text: I have a system like this ![](http://i.stack.imgur.com/PN5Ul.png) When the valve is closed, pressure along any point along the thin tube may be found thru pgh. 1. What about when the valve is opened? 2. Does the top of the jar need to be open? 3. If the top is open?


title: Cosmological and gravitational redshift in physics

text: I am interested in knowing what are the differences between gravitational and cosmological redshifts?


title: What are the current possibilities for large-scale storage of electrical energy?

text: I'm curious as to how large-scale electrical storage is managed now, how much they can store, how well they scale and for how long (and what is their loss rate). ( _I'm not asking about technological solutions now being researched_ )


title: Formulas for ball rolling in a bowl?

text: I'm developing a program where I've a ball/sphere rolling in a bowl from the side at top, till the center at bottom, and I'm trying to get the formulas for:   * The **rotation angle** and the **position of the sphere** , depending on the starting angle measured in radians (0.000 - 6.283), the current seconds and the maximum seconds that the ball will keep rotating and considering its deccelerated. All I knew about the needed physics is almost already forgotten, and I'm getting horrible headaches to get a properly working formula, so hope anyone can help me. Thanks in advance!


title: Quantum circuit decomposition

text: I need to construct a universal quantum circuit decomposition for a three- qubit operation where one qubit is the control bit, and a two-qubit unitary operator acts on the other two depending on the control bit. I can do the decomposition when there's a one-qubit unitary operation with as many control bits as needed; this two-qubit unitary with one control bit I am not clear about. Anyone has any clue what the recipe for this construction is?


title: Intrigued about a polarizer effect

text: This comes from one of my physics course at university (many years ago!), I hope I recall it right. We were studying optics, and were given some polarizer filters to experiment. They were disc- shaped (like a photographic filter). I stacked two polarizers, face to face, and I verified that, when rotating one of them, a particular orientation (and its 180 degrees rotation) blocked the light. That seemed to make perfect sense to me, because I pictured the polarizer as a wire grid (conceptually like this image) ![enter image description here](http://i.stack.imgur.com/snSEH.png) that only let pass light in a given polarization direction: then, adding a second polarizer to the other behind had no effect when the direction of the wires coincided, and result in high light blocking if they were orthogonal. We were all satisfied with this explanation... but then I tried to reverse the front filter (I mean, I turned it face down, always keeping it in front of the other, and keeping its "orientation"). To my surprise, keeping the orientation did not keep the block-pass behaviour. Say, if originally the filters were at the "let it pass" position (paralells wires, in my mental picture), putting the filter face down (which would let the imaginary wires in the same orientation) resulted in light blocking, and I had to turned it 90 degrees to let pass the light. This behaviuor puzzled me a little, as also puzzled my teaching assistant (who didn't come with any explanation). I just assumed that my "grid wire" picture was not very apt, but that was all (not enough scientific spirit on my part, I guess). Can anybody make some sense of this?


title: Can I feel (the results of) electromagnetic induction from overhead rail lines in the human body?

text: A few weeks back, I was standing at a Hornsey rail station (in the UK) which uses overhead lines, and particularly has a number of parallel rail lines, all close to the platform. While I was standing there, I went to hold hands with my girlfriend, and experienced a peculiar sensation in my fingertips, almost as if there were very large ridges on her hands. I've felt this before, touching a metal bodied laptop computer that was seemingly not earthed - dragging my finger across the surface my finger felt as if it was jumping along the surface, despite it being a smooth aluminium surface. Is this a result of the overhead lines inducing a reciprocating current through my arm and/or fingertips? Is there some other feature of the railway that could cause this effect? And were we in any danger of any sort?


title: Why are smaller animals stronger than larger ones, when considered relative to their body weight?

text: I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat. It has been suggested to me that this is due to physics, but I am not even sure what to search for. Could someone explain why indeed it is easy for smaller objects/lifeforms to support several times their own weight, but this is harder as objects/animals become larger?


title: Weinberg's way of deriving Lie algebra related to a Lie group

text: I was reading the second chapter of the first volume of Weinberg's books on QFT. I am quite confused by the way he derives the Lie algebra of a connected Lie group. He starts with a connected Lie group of symmetry transformations and then consider how it would be represented on the Hilbert space. Then, he says that as it is a symmetry transformation and because it is connected to identity continuously (because it is locally path connected), the transformation U(T) corresponding to a particular symmetry transformation T must be unitary by Wigner's theorem. So, he takes a group element close to identity and expand it as a power series and considers their compositions to derive at the commutation relations for the operators representing generators of the Lie algebra. Much of this can be followed on pages 53-55. My doubts and confusions are as follows:   1. Does this give a Lie algebra of the underlying group of Ts or the group induced on Hilbert space of the U(T)s?   2. Is it that we presuppose a Lie algebra with some generators already out there and think of all this stuff on Hilbert space as its representation and then derive the commutation relations of the representation of generators to get the abstract Lie product relations of Lie algebra? If yes, then are the Lie algebras of Ts and U(T)s same or different? Please be precise on this.   3. If all the above is true, then why do we need specifically representations on Hilbert space? Any representation would do, right? And where does all this talk about unitarity of representations used in the following derivation of the Lie algebra? I can't see any application of Wigner's theorem in what follows. I think that all that follows can be done on the representation on any space. But then this brings us to the question as to if a representation of the group exists on that vector space? Is it why we need Wigner's theorem and Hilbert space? Then, why don't we take the representations on 4D space-time itself where a representation exists trivially by definition? If this is true, how do we go in general about finding about what kind of representations can exist for a Lie algebra?   4. In general, provided with a Lie group, how does a mathematician go on to derive its Lie algebra? Should the Lie group be connected? Can Lie algebras be defined for Lie groups which are not connected?


title: Correlation/relationship between human magnetic field and body temperature

text: Is there any relationship between human magnetic field and body temperature? I am looking for studies on this topic.


title: Two forms of Feynman Diagrams?

text: I am confused. I know Feynman Diagrams looks like the ones here. But then I saw these, look in page 17[1247] of this paper. Any idea?


title: Are information conservation and energy conservation related?

text: as evident from the title, are both, conservation of energy and conservation of information two sides of the same coin?? Is there something more to the hypothesis of hawking's radiation other than the fact that information cannot be lost? or can I say energy cannot be lost??


title: Role of time dilation in relativistic beaming

text: ![enter image description here](http://i.stack.imgur.com/oLDKf.png) This picture is an excerpt I chose from a Wiki article under "Relativistic beaming" Source: http://en.wikipedia.org/wiki/Relativistic_beaming For theta=0,what I don't understand is why the equation has the time dilation factor in a way that increases the observed luminosity?. According to my understanding time dilation should reduce the observed luminosity since if we assume that the source of light emits a number of photons=n/second in its frame of reference and it moves at a speed of v to an observer.Assume the gamma factor at speed v is k.To the observer the source emits n number of photons(regardless of frequency)in a time period of k seconds,where k >1.According to this reasoning the effect of time dilation has contributed to reducing the observed luminosity.!


title: Understanding unit vectors

text: Trying to understand how the unit vector ${\mathcal{\hat{r}}}$ defined as $\frac{r' - r}{|r' - r|} $ (where $r'$ is the source point) works in this problem: Work out the electric field, $E$, at point $P$ at height $z$ above a line of length $2L$ and charge $\lambda$ along the x-axis in the $x$-$z$ plane. So I work it out to be: $$ \frac{2}{4\pi \epsilon_0}\int_0^{L} \frac{\lambda dx}{\mathcal{|r' - r|^2}} \hat{r} $$ But the books tells me that there it is actually:$$ \frac{2}{4\pi \epsilon_0}\int_0^{L} \frac{\lambda dx}{\mathcal{|r' - r|^2}} cos(\theta) \hat{z} $$ how is $\hat{r} = \cos(\theta) \hat{z}$? Can someone explain how the unit $z$ vector got in there?


title: Collision of 2 neutrons

text: If two neutrons collide in 3D space and we want to determine the final velocities of both nuetrons (3 components for each neutrons), we can use the conservation of momentum equations and the conservation of energy equation. Those equations are 4 while the number of unknowns we have is 6. There are 2 free variables. For classical perfectly spherical objects, the solution is strait forward as explained here. However, neutrons because of their quantum nature can't be treated as classical perfectly spherical objects. We can't define a precise point of collision, so the trick we use for classical objects won't work for neutrons or any quantum level particle, or will it? If it doesn't work, does that mean we can't avoid having free two degrees of freedom in this system?? NOTE: What is meant by a collision is a momentum-exchange event


title: Klein-Gordon propagator integral

text: DISCLAIMER: I edited the question to include the integrals and the picture I'm stuyding Klein-Gordon field in Peskin & Schroeder _An Introduction to Quantum Field Theory_ There is an integral that I don't see, related to the computation of the propagators, in particular for the K-G field. I'm refering to two steps $$\langle 0| [\phi(x),\phi(y)] 0\rangle=\int \frac{d^3p}{(2\pi)^3}\frac{1}{2E_p}(e^{-ip\cdot(x-y)}-e^{ip\cdot(x-y)})$$ Now that integral is written as $$\int \frac{d^3p}{(2\pi)^3} \left\\{ \frac{1}{2E_p}e^{-ip\cdot(x-y)}\bigg|_{p_0=E_p}+\frac{1}{-2E_p}e^{-ip\cdot(x-y)}\bigg|_{p_0=-E_p}\right\\} $$ **Question 1) How can be fixed the vale of $p_0$ to be $\pm E_p$? It is only notation?** Now it says that it can be written as a $p_0$ integral in the following contour ![enter image description here](http://i.stack.imgur.com/1jUt1.jpg) If $x^0>y^0$ then $$\int \frac{d^3p}{(2\pi)^3}\int\frac{dp^0}{2\pi i}\frac{-1}{p^2-m^2}e^{-ip\cdot(x-y)}$$ and the countour is closed below. If $x^0<y^0$ the countour is closed above, giving 0 (this i can see it) **Question 2) Why that particular choice of closed contours?** When I had to do any integral with resiudes theorem, usually they are integrals over $\mathbb{R}$ so that the contour of integration in the complex plane is exactly a semicircle of radius $R$ with $R\to \infty$. Now this is not the case, because the contour does not pass throught the entire real line. **Question 3) The semicircles centered at $\pm E_p$ affect the $p_0$ integration?**


title: How can an asymptotic expansion give an extremely accurate predication, as in QED?

text: What is the meaning of "twenty digits accuracy" of certain QED calculations? If I take too little loops, or too many of them, the result won't be as accurate, so do people stop adding loops when the result of their calculation best agrees with experiment ?! I must be missing something.


title: Ricci tensor for a 3-sphere without Math packets

text: Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it. In the beginning, I got an expressions for Christoffel's symbols: $$ \Gamma^{i}_{ii} = \frac{1}{2}g^{ii}\partial_{i}g_{ii} = 0, $$ $$ \Gamma^{i}_{ji} = \frac{1}{2}g^{ii}\partial_{j}g_{ii}, $$ $$ \Gamma^{k}_{ll} = -\frac{1}{2}g^{kk}\partial_{k}g_{ll}, $$ $$ \Gamma^{k}_{lj} = \Gamma^{k}_{lk}\delta^{k}_{j} + \Gamma^{k}_{jk}\delta^{k}_{l} + \Gamma^{k}_{jj}\delta^{j}_{l} = 0. $$ The Ricci curvature must be $$ R_{lj}=\frac{2}{R^{2}}g_{lj}. $$ But when I use definition of Ricci tensor, $$ R_{lj}^{(3)} = \partial_{k}\Gamma^{k}_{lj} - \partial_{l}\Gamma^{\lambda}_{j \lambda} + \Gamma^{k}_{j l}\Gamma^{\sigma}_{k \sigma} - \Gamma^{k }_{l \sigma}\Gamma^{\sigma}_{jk}, $$ I can't associate an expression (if I didn't make the mistakes) $$ R_{lj}^{(3)} = \partial_{j}\Gamma^{j}_{lj} + \partial_{l}\Gamma^{l}_{jl} + \partial_{k}\Gamma^{k}_{ll}\delta^{l}_{j} - \partial_{l}\Gamma^{k}_{jk} - \Gamma^{k}_{jk}\Gamma^{j}_{lj} + \Gamma^{k}_{lk}\Gamma^{l}_{jl} + \Gamma^{\sigma}_{k \sigma}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{k}_{jk}\Gamma^{k}_{lk} - \Gamma^{l}_{jl}\Gamma^{j}_{lj} - \Gamma^{l}_{kl}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{j}_{ll}\Gamma^{l}_{jj} - \Gamma^{k}_{ll}\Gamma^{l}_{kl} = $$ $$ = \partial_{j}\Gamma^{j}_{lj} + \partial_{l}\Gamma^{l}_{jl} + \partial_{k}\Gamma^{k}_{ll}\delta^{l}_{j} - \partial_{l}\Gamma^{k}_{jk} - \Gamma^{k}_{jk}\Gamma^{j}_{lj} + \Gamma^{k}_{lk}\Gamma^{l}_{jl} + \Gamma^{\sigma}_{k \sigma}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{k}_{jk}\Gamma^{k}_{lk} - \Gamma^{l}_{jl}\Gamma^{j}_{lj} - 2\Gamma^{l}_{kl}\Gamma^{k}_{ll}\delta^{l}_{j} - \Gamma^{j}_{ll}\Gamma^{l}_{jj}, $$ where there is a summation only on $k, \sigma$, with an expression for the metric tensor. Maybe, there are some hints, which can help?


title: Is it possible to have topological degeneracy in 1D ?

text: I mean to have q-fold degenerate ground states on a ring which could not be lifted by local perturbation. If the answer is no, then what is the physical (or mathematical) reason against having such a state in 1D ?


title: Vacuum polarization of dark matter candidates

text: Can a dark matter particle (axions, for instance, but the question applies to any valid candidate for dark matter) induce QED vacuum polarization effects even if being electrically neutral and with no observable magnetic or electric dipolar moment?


title: Coincidence detectors in Bell tests: How close is close enough?

text: When is a coincidence a coincidence? We know that to identify entangled photons, the electronics is set to look for simultaneous clicks at opposite detectors. The size of the window is to some degree arbitrary. What I am wondering is: If you run a Bell-type entanglement experiment where you get, for example, 100 coincidences per second with a 100 picosecond interval, what happens if you double the window to 200 picoseconds? Do you expect, in general, to get 200 coincidences per second? I can think of three possible answers and I wonder which one is most true:   1. If your window is small enough, doubling the window doesn't matter because you are accurately identifying all available entangled particles. You still get 100 coincidences per second (or possible a few more for truly sporadic coincidence counts.)   2. Double the window, double the coincidences. You get 200 per second.   3. Double the window and you get four times the coincidences: 100 entirely in the first half of the window, 100 in the seconde half, and 200 "cross-coincidences". 400 per second in total. I hope this question is meaningful and I wonder what the answer is.


title: Vanishing of photon one-point function in QED

text: I would like to know why the photon one-point function vanishes in QED. I am aware that any $n$-point functions vanishes for odd $n$ because of 'charge- conjugate" argument, this does not apply to $n=1$ case. Here I want to know how to see the quantity $$ \int \frac{d^4x}{(2\pi)^4}tr\left[ \gamma^a(\frac{1}{-k+m})\right] $$ vanishes.


title: Calculating angles for tetrahedral molecular geometry

text: Let's say I have a molecular like CH3F (i.e. fluoromethane), and I'm able to measure the angle (theta) between the C-H bonds. Provided (theta) what is the angle between the C-F bond and the C-H bonds?


title: Rotations in space-time

text: In Landau's Classical Theory of Fields, one finds the statement: > Every rotation in the four-dimensional space can be resolved into six > rotations, in the planes $xy,zy,xz,tx,ty,tz$ (just as every rotation in > ordinary space can be resolved into three rotations in the planes > $xy,zy,xz$). How can I prove this statement? Thanks.


title: Is Space-Time Quantisation necessary or even meaningful?

text: It is believed among people working on Quantum Gravity, that space-time must be quantised at the Planck scale. Although it is very hard to verify such proposition, it is interesting from a philosophical and epistemological point of view. According to this view, no spatial or time interval shorter than the Planck values exist: **For length:** $l_p=\sqrt{\frac{\hbar G}{c^3}}\sim 10^{-35}$m **For time:** $t_p=\sqrt{\frac{\hbar G}{c^5}}\sim 10^{-43}$s. Let us assume there is no structure smaller than $l_p$, and let us imagine some kind of cube of side $l_p$. This cube must be distinct and “distinguishable” from other similar cubes around it. For this to be possible, and for the cubes to be able to display some dynamics, something in the form of continuous space must stand in between these cubes. **Question:** If the above reasoning is correct, then, what is the meaning or need for space-time quantisation?


title: Addition of different physical quantities

text: We all know the "apples and oranges" rule which says that it's meaningless to add or subtract two different quantities like apples and oranges. But the same rule doesn't hold for the multiplication and division. So my question is, why two quantities with different units cannot be added to one another but the multiplication of them is allowed and how this is expressed mathematically?


title: Why trimming the mainsail gives a larger torque to head up

text: I am learning sailing on a 5m catamaran (Nacra 5). I am familiar with basic aerodynamics and the physics of the sail and keel. We learned that when sailing closed hauled, too tight a mainsail tends to bring the boat up to the wind. And that the opposite is true for the jib. For example, one may steer up to the wind to come about, using the mainsail alone, by trimming it tight. My question is why the trimmed mainsail in the above setup gives a larger torque. This also seems opposed to the rule: "The more the mainsail is sheeted out the more the boat tends to come up.", as explained in http://www.sailtheory.com/mandf.html#sailsteering Edit: To my understanding, there are several possible competing effects involved: (1) Effects that tend to INCREASE the mainsail torque to head up: (1A) The direction of the sail force becomes more perpendicular to the boat. This increases the heeling torque. Since heeling motion happens faster than turning, the boat will heel more. This moves the sail force out, which increases the lever arm and the torque to head up. see diagram in: http://www.sailtheory.com/mandf.html#hellingstuur (1B) The sail force moves backwards since the sail is stretched backwards. This increases the lever arm and the torque. (2) Effects that tend to DECREASE the mainsail torque to head up: (2A) The direction of the sail force becomes more perpendicular to the boat. Assuming that the force is perpendicular to the sail, and that the center of rotation is between the mast and the center of force, one sees that pulling the sail in, reduces the torque to head up, as can be seen in the following diagram: ![diagram of the sail force torque](http://i.stack.imgur.com/9I2TX.jpg) As Theta gets smaller, the torque Tau is reduced. This is opposed to what is stated in some of the answers below. (2B) The sail force is reduced since the sail was pulled beyond its optimal angle of attack, thus losing lift and reducing torque. (2C) Due to (2B) heeling torque is also decreased. With a similar reasoning to (1A) this decreases the lever arm and torque. (2D) The sail force moves forward since the aft part of the sail is too flat and stalls. This shortens the lever arm and reduces torque. This is described in http://www.sailtheory.com/mandf.html#sailsteering We know from reality that 1A+1B > 2A+2B+2C+2D. Considering heeling alone, we also know that 1A > 2C. It remains to be explained why in general this is the case.


title: Bose Einstein condensates and gravity

text: Are Bose Einstein condensates affected by gravitational attraction ? In sufficient quantity, do Bose Einstein condensates exert a gravitational force ?


What types of materials can be electrically charged by rubbing? Is there a certain type of materials in which static electricity can be produced by rubbing together two different materials?


title: Stress energy tensor of a perfect fluid and four-velocity

text: In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\mu\nu}$ with a signature $(-, +, +, +)$ : $\boxed{g_{\mu\nu} = \begin{pmatrix} -c^2 & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{pmatrix}} \qquad\qquad \boxed{ g^{\mu\nu} = \begin{pmatrix} \frac{-1}{c^2} & 0 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1 \end{pmatrix}} $ The tensor of a perfect fluid is: $ \boxed { \begin{aligned} T^{\mu\nu} &= \left(\rho c^2+P\right)u^{\mu}u^{\nu}+Pg^{\mu\nu}\\\ T_{\nu}^{\mu} &= \left(\rho c^2+P\right)u^{\mu}u_{\nu}+P\delta_{\nu}^{\mu}\\\ T_{\mu\nu} &= \left(\rho c^2+P\right)u_{\mu}u_{\nu}+Pg_{\mu\nu} \end{aligned} }$ where $\rho$ is the mass density. We can easily check that $\rho c^2$ and $P$ have the same unit. If we consider only the first component $00$, there is no pressure $P$ involved (I consider a cosmological case where the fluid is at rest in comoving coordinates). The only solution to have that is that $u^0u^0=-g^{00}$, so that the contravariant perfect fluid tensor become : $T^{00} = \left(\rho c^2+P\right)u^{0}u^{0}+Pg^{00}=\left(\rho c^2+P\right)\times{-g^{00}}+Pg^{00}=-\rho c^2 g^{00} = -\rho c^2 \times -\frac{1}{c^2}=\rho$ which is the expected result. So, to have the expected result, one should have $\boxed{\left(u^0\right)^2=+\frac{1}{c^2}}$. Now come the weird part. The metric can be written: $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}=-c^2dt^2+dx^2+dy^2+dz^2=dt^2\left(-c^2+\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2}\right)\approx-c^2dt^2$ which is equivalent to : $\frac{dt^2}{ds^2}\approx-\frac{1}{c^2}$ which means : $\boxed{\left(u^0\right)^2=-\frac{1}{c^2}}$ So we clearly have a problem of sign here between the two expressions of $u^0$. **QUESTION: where is the mistake ?**


title: Is the wave function objective or subjective?

text: Here is a question I am curious about. Is the wave function objective or subjective, or is such a question meaningless? Conventionally, subjectivity is as follows: if a quantity is subjective then it is possible for two different people to legitimately give it different values. For example, in Bayesian probability theory, probabilities are considered subjective, because two agents with access to different data will have different posteriors. So suppose two scientists, A and B, have access to different information about the same quantum system. If A believes it has one wavefunction and B believes it has another, is one of them necessarily "right" and the other "wrong"? If so then the wavefunction is objective, but otherwise it must contain some subjective element.


title: Measuring the vacuum permittivity

text: So I was reading the EF experiment that's used at the MIT to measure the vacuum permittivity and I was thinking about trying it just to see how it works: http://ocw.mit.edu/courses/physics/8-02x-physics-ii-electricity-magnetism- with-an-experimental-focus-spring-2005/labs/experiment_ef.pdf I have some questions about it and I hope you can help me with them:   1. It says "To find the electric force on the foil, assume that the charge density, σ , on the foil, is the same as that of the lower washer". How good that assumption is?   2. It says "Charges on the foil feel only horizontal forces from other charges on the bottom plate, so the vertical force on the foil is due to the electric field of just the top charge sheet". What horizontal forces are they talking about? I thought both the bottom plate and the upper plate were all exerting a vertical force on the foil and that's why I don't understand why the force is due just to the electric field of the upper plate $(V/2d)$ and not from the one of the two plates $(V/d)$.   3. Finally, why washers? I think it really doesn't matter if they're just two discs, does it?


title: Why is it valid to take the partial trace to describe a subsystem?

text: In derivations of decoherence, there eventually comes a point when we are asked to take the partial trace over the environment. Why should this be valid for an entangled system? Why should taking the partial trace give the correct description of the system? Taking the partial trace turns a pure density matrix into a mixed density matrix, but a pure state is always a pure state. In what sense can the reduced mixed density matrix be assigned probabilities?


title: Dirichlet and Neumann Boundary condition: physical example

text: Can anybody tell me some practical/physical example where we use Dirichlet and Neumann Boundary condition. Is it possible to use both conditions together at the same region? If we have a cylindrical symmetric array of lenses to focus ions, which regions it comes the Dirichlet BC and Neumann BC? This types of systems can be simulated using SIMION. I know SIMION solved Laplaces equation. But I am confused with the boundary conditions.


title: what is a kink-kink-meson vertex?

text: These are questions I have after reading the Rajaraman's book "Solitons and instantons". So I think you must have read the book if want to answer. And also know about quantum solitons. Rajaraman derives classically (page 82) an interaction potential energy $V(R)$ of the kink-antikink pair. Then, he says some things that are not obvious for me, but he talk about they as obvious things: "[The potential (3.109) is clearly reminiscent of a one-meson exchange potential extracted from quantum field theory. In fact, in chapter 5 where we quantise this theory, it will be seen to carry a meson of mass $ \sqrt 2 m$, and to yield a kink-kink-meson vertex of order $1/\sqrt\lambda$. If we remember that two such vertices are involved when a meson is exchanged between kinks, and put in sufficient factors of m to meet dimensional requirements, the one- meson-exchange Born amplitude must clearly yield a potential $V(R)$" What is a kink-kink-meson vertex? And how we can show that it is of order $1/\sqrt\lambda$? How are related two such vertices with the one-meson-exchange Born amplitude? and what is this one-meson-exchange Born amplitude he talks about? and finally How is obvious that this amplitude yields a potential $V(R)$?


title: How to define the required aspherical coefficients for a lens?

text: I would like to purchase my own aspherical lens for testing purpose. Then I need to define the required coefficients. But how? My known requirements are f=25mm F/# 2.0 and MTF 0.5 @ 120lp/mm with a wl of 850nM. The distance from the object plane to my lens is 10m. I will test it for an imaging application. Thanks


title: Is there a material with non-monotone density dependence on pressure?

text: Is there a material with such non-linear effects (including chemical reactions, phase transitions etc ...) so that there would be range where pressure increase would yield density decrease? What would be the answer if we consider volume instead of density?


title: Quantum superposition of states: experimental verification

text: How can somebody demonstrate the quantum superposition of states directly by other means than the double slit experiment? And why can't macroscopic objects like a pen be in superpostion of states? Will it ever be possible to have an object like a pen to be in superposition of more than one state?


title: Cyclic Coordinates in Hamiltonian Mechanics

text: I was reading up on Hamiltonian Mechanics and came across the following: > _If a generalized coordinate $q_j$ doesn't explicitly occur in the > Hamiltonian, then $p_j$ is a constant of motion (meaning, a constant, > independent of time for a true dynamical motion). $q_j$ then becomes a > linear function of time. Such a coordinate $q_j$ is called a cyclic > coordinate._ The above quote is taken from p. 4 in Ref. 1. What I don't understand is why $q_j$ is a linear function of time if $p_j$ is constant in time. In other words, why does $p_j$ constant in time imply partial $\frac{\partial H}{\partial p_j}$ is a constant? (In particular, $\frac{\partial H}{\partial p_j}$ could depend on any of the other coordinates or momenta.) Reference:   1. Patrick Van Esch, _Hamilton-Jacobi Theory in Classical Mechanics_ , Lecture notes. The pdf file is available from the author's homepage here.


title: exercise concerning the inclined plane

text: I have an exercise to ask yourself. > A ball of mass m = 5 g starts at rest and travels 50 cm along a ramp > inclined at 45 ° to the horizontal. a) If we imagine frictionless motion of > the ball along the ramp, compute the speed of the ball at the bottom of the > ramp. What is the change in potential energy of the ball? My idea is to use the gravitational acceleration, multiply it by the cosine of $45°$ and find the x-component of the gravitational acceleration, and finally use the formula $ v^2=v_0^2 +2*g*x $ to find the speed at the end of the ramp.   For the second point, my idea is to use this formula $ m*g*h $.   My idea is correct?


title: Compressibility factor

text: Z=pV/RT                          p=pressure of the gas                          V=real volume of the gas                         R=universal gas constant                         T=absolute temperature      My question is that whether p is real pressure of the gas or ideal pressure of the gas...It is mentioned nowhere..If it is real pressure than we cannot apply ideal gas equation and cannot write Z=Vreal/Videal..but in actual compressibility factor is also defined as the ratio of real volume of gas /ideal volume of gas..How it is derived.And please mention whether p is considered as ideal pressure or real pressure....


title: A question on an exercise from Gravitation by Misner, Thorne and Wheeler

text: My question is on problem 4.1 of Gravitation. In a generic case of electric field and magnetic field(i.e not $E=0$ or $B=0$ or $E$ and $B$ perpendicular), define the direction $\hat{n}$ unit vector , $$\hat{n}\tanh (2\alpha)=\frac{2\vec{E}\times\vec{B}}{\vec{E}^{2}+\vec{B}^2}$$ and $\vec{\beta}=\tanh(\alpha)\hat{n}$ is the velocity vector. Show in the frame of the rocket with velocity $\vec{\beta}$, the Poynting vector vanishes. I tried the following but I am stuck at the cancellation. Let the $\bar{\vec{E}}$ and $\bar{\vec{B}}$ be the field in the rocket frame and the field without bars be the field of the rest frame. direction parallel along the velocity of rocket is denoted as subscript $\parallel$ and direction perpendicular to velocity of rocket direction is denoted by $\perp$ as the subscript. By lorentz transformation. \begin{align} \bar{\vec{E_{\parallel}}}&=\vec{E_{\parallel}}, \\\ \bar{\vec{E_{\perp}}}&=\frac{\vec{E_{\perp}}+\vec{\beta}\times\vec{B_\perp}}{\sqrt{1-\beta^{2}}},\\\ \bar{\vec{B_{\parallel}}}&=\vec{B_{\parallel}}, \\\ \bar{\vec{B_{\perp}}}&=\frac{\vec{B_{\perp}}-\vec{\beta}\times\vec{E_\perp}}{\sqrt{1-\beta^{2}}}. \end{align} Note $\beta\times X_{\perp}$=$\beta\times X$ In barred frame, \begin{align} \bar{\vec{E}}\times\bar{\vec{B}} &=(\bar{\vec{E_{\perp}}}+\bar{\vec{E_{\parallel}}})\times(\bar{\vec{B_{\perp}}}+\bar{\vec{B_{\parallel}}})\\\ &=\bar{\vec{E_{\perp}}}\times\bar{\vec{B_{\parallel}}}+\bar{\vec{E_{\parallel}}}\times\bar{\vec{B_{\perp}}}. \end{align} Plug in the lorentz transformation. One gets $$\frac{\vec{E}\times\vec{B}-\vec{E}_{\parallel}\times(\vec{\beta}\times\vec{E})-\vec{B}_{\parallel}\times(\vec{\beta}\times\vec{B})}{\sqrt{1-\beta^{2}}}.$$ Now the last two terms looks generically like $\vec{X_{\parallel}}\times(\vec{\beta}\times\vec{X}$) simplifies to $\vec{X_{\perp}}(\vec{X}\cdot\vec{\beta})$. However, I could not see the cancellation at this stage. Did I do something wrong?


title: Wave function not normalizable

text: Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\psi(x)$ $$\int_{-\infty}^{\infty}|\psi(x)|^2 dx<\infty \qquad \text{or}\qquad \int_{0}^{\infty}|\psi(x)|^2 dx <\infty$$ What would be the physical implications if one (or both) of those integrals diverges. From the viewpoint that the Copenhagen interpretation is one of the most popular and the wave function is interpretated as a probability distribution in this case; will the wavefunction diverging be valid for any other interpretation of quantum mechanics. Does anyone know any wavefunctions which are not normalizable? What if there was a singularity at 0 that made it diverge at all times. For instance $$\int_{0}^{\infty}|\psi(x)|^2 dx \to\infty\quad \text{but}\quad \int_{a}^{\infty}|\psi(x)|^2 dx <\infty $$ where $a>0.$ Would the second integral count as a valid pdf?


title: Effects of firing shells on the Earth's angular momentum

text: During a certain war, millions of shells were fired by country A towards the west, and even more shells were fired back by country B towards the east. The average momentum of each bullet were the same on average. What was the ultimate effect on the angular momentum of Earth? Here's my try: B fired more bullets than A. Because average momentum of each bullet is the same, there must be a net change in momentum. As such the Earth's angular momentum would have to change. However, the answer was that there was no effect, not even a very slight one. Why is that so?


title: Ideal gas and inelastic collisions

text: Why is it necessary that all inter-molecular collisions in an ideal gas be elastic? My understanding is that a gas behaves ideally so long as the potential energy arising from inter-molecular interactions is negligible compared to the kinetic energy. Collisions are necessary to establish equilibrium, but the nature of the collisions seems to me irrelevant. Further, it seems to me that, following a transfer of heat to the gas, inelastic collisions are necessary in order to properly distribute energy among the internal molecular degrees of freedom (vibration, rotation).


title: What if the Moon was 100x more massive: Binary planets?

text: I was reading a couple of Earth-Moon related questions (Mars just collided with Earth! A question of eccentricity, Could the earth have another moon?) and they got me thinking about planet-moon systems in general. Binary star systems are pretty common. The types of the two stars in the binary can vary pretty widely (main sequence, puslar/neutron star, black hole, white dwarf, giant phases, etc, etc), but some are formed of a pair of roughly equal mass (within a factor of <10). I can't say I've ever heard of a binary planet system, though. Of course a planet with a moon is sort of a binary, but I've never heard of an equal mass binary planet. I think the closest thing in the solar system would be the Pluto-Charon system, with a mass ratio of about 10:1. Is there any reason a binary planet would be unstable? Obviously this is a three-body system, which has some inherent instability, but Earth-Moon-Sun seems pretty stable. Would increasing the mass of the Moon to match that of Earth make the system unstable? How about gas-giants? I think a Jupiter-Jupiter binary close to a star would be short lived because of three-body interactions, but what about further out? Would, for instance, a double-Jupiter or double-Saturn be stable in our solar system? Or is there some tidal effect that would cause the orbit to decay and the binary planet to merge? As an aside, it seems that binary asteroids aren't terribly difficult to find... perhaps we just haven't seen any binary planets yet because they're only stable relatively far away from their star, making them difficult to detect outside our own solar system?


title: Mass of particle w.r.t. dimension

text: I heard in a lecture recently - just as a comment - that a particle which is massive in say $D=4$ can be seen as a massless particle in higher dimensions and vice versa. Our prof didn't give any equations to explain this and I don't really see how this comes about. What's the trick?


title: Determinant of ray transfer matrix

text: Quick question on the wikipedia entry for the _ray transfer matrix_ , which is defined : ![enter image description here](http://i.stack.imgur.com/Jt51r.png) where $x$ is the distance of a point from the optical axis and $\theta$ the angle that the beam profile at that point makes with the optical axis. Now, wikipedia says:![enter image description here](http://i.stack.imgur.com/2eMFv.png) but does NOT provide a reference. Does anyone have any idea on how to get that results? Do you know a website/paper where this is discussed?


title: Under what conditions is a pure fusion nuclear bomb possible?

text: Is it possible to build a pure fusion and powerful nuclear bomb? (because due to the harsh conditions of enriched uranium, I would guess most of today's world countries will want to build pure fusion bombs.) A pure fusion bomb is a nuclear bomb that does not need fission so does not need enriched uranium.


title: Why are two voices singing the same note louder than one?

text: Let's say for example:               Two people sing the same note (frequency) and volume (amplitude) together.      Why is it that the two persons sound louder than they would individually? I would imagine that since the traverse waves generated by the two persons would randomly be louder or softer, based on the phase shift between the two sound waves. But that hypothesis is obviously untrue since in the real world a choir is clearly louder than a soloist.


title: Is it theoretically possible for a person to pass through a solid wall/object?

text: I understand that matter cannot pass through other solid matter because of the electrons that orbit an atom prevents this but I was curious to know if it is theoretically possible to somehow get passed this problem caused by the electron shell and by doing so would this allow th effective passage of matter through matter. My thinking was that on the scale of the atom the amount of space taken up by actual particle mass is small compared to the space between the nucleus and the electron cloud so if the electron problem can be circumvented then all that space could possibly be used to pass through. It would probably take some fanciful piece of tech based on unknown physics but i though i would ask!! Thanks.


title: Is it possible see an invisible light with something similar to a normal lens of an eyeglass?

text: Let's say that I am a spy in a room with a lot of people and I want to communicate with my friend using Morse code via a light signal. Does there exist a type of light (laser or otherwise) that is invisible to the human eye, but that I can see using a modified transparent lens? In other words, can I make an infrared or ultraviolet light visible using something similar to a normal lens of an eyeglass? If yes, is it easy to buy or make it?


title: How would a diffraction pattern change if the atoms were triangular instead of spheres?

text: On a related note, what's a good book/source that would answer questions that go very in depth with these kinds of "what if" questions because I am also asked the same if the atoms were long cylinders instead of spheres.


title: Why are wires in simple circuits approximated as equipotentials?

text: I just answered this question: Voltage in a circuit by asserting that any two points joined only by a wire, but no other circuit elements, are at the same value of the electric potential. It is, of course, true that in the context of electrostatics, the values of the electric potential at any two points inside of a perfect conductor are the same, but the electrostatics argument seems irrelevant for circuits. **Why are we justified in using this same fact for circuits in which we are clearly no longer faced with electrostatics?** **Edit.** As prompted by user twistor59 in the comments below, let me include a bit more detail to clarify the question and hopefully indicate why I don't think the answer is obvious. If we consider any segment of the wire as a small ohmic resistor, then as the resistance of any segment drops to zero, the voltage drop across that segment will be correspondingly small for a given current. In particular, zero resistance implies zero voltage drop. However, if I just have a piece of a perfect conductor sitting somewhere, then it is totally unclear to me why, or that it is even true, that the electric field will vanish in its interior if the situation is non-electrostatic. Perhaps, therefore, the magic lies in the microscopic physics that (in particular) lead to Ohm's Law?


title: How can neutrinos oscillate though the lepton flavors have differing masses?

text: Since the total mass-energy for the neutrino presumably does not change when a neutrino changes lepton flavor, though the mass is different, what compensates for the gain or loss of mass? Does the propagation speed of the neutrino change?


title: Any practical results yet from 'Twistor Uprising'?

text: In Nima's lectures on the 'Twistor Uprising' (for example here), he gushes about about new powerful techniques for calculating amplitudes, such as summing Feynman diagrams using 'BCFW recursion.' He makes it sound like there should be practical results, like new QCD amplitudes for use in Monte Carlo generators, etc. Are there any, or are any on the horizon? Or does he overstate the power of these new techniques? **EDIT:** There is a recent blog post by Matt Strassler seeming to imply that BlackHat (a program that computes QCD amplitudes) uses some of these techniques.


title: How fictitious are fictitious forces?

text: How fictitious are fictitious forces? More specifically, in a rotating reference frame i.e. on the surface of the earth does an object that is 'stationary' and in contract with the ground feel centrifugal and Coriolis forces? Or are these forces purely fictional and used to account for differences in observed behaviour relative to an inertial frame? To give a practical example a turreted armoured vehicle is sitting stationary and horizontally somewhere in the UK. The turret is continually rotating in an anti-clockwise direction. Do the motors that drive the turret's rotation require more power as the turret rotates from east to west and less power as the turret rotates from west to east? i.e. are the turret motors cyclically assisted and hindered by the earths rotation?


title: When is an operator subspace the span of Kraus operators?

text: Let $A$ and $B$ be finite dimensional Hilbert spaces, and let $\mathcal{L}(A \to B)$ be the space of linear operators from $A$ to $B$. Say that a subspace $K \subseteq \mathcal{L}(A \to B)$ is a _span of Kraus operators_ if there are operators $\\{K_i\\}$ such that $\sum_i K_i^\dagger K_i = I$ and $K = \textrm{span}_i\\{K_i\\}$. Equivalently, $K$ is a span of Kraus operators if there is an ancillary Hilbert space $C$ and an isometry $J:A \to B \otimes C$ such that $K = \textrm{span}_{\left| \psi \right> \in C} \\{ (I \otimes \left<\psi\right|) J \\}$. Are there any non-trivial (and hopefully simple) necessary and sufficient conditions for an operator subspace $K$ to be a span of Kraus operators? The only necessary condition I can come up with is that $I \in K^\dagger K := \textrm{span}\\{ x^\dagger y : x \in K, y \in K \\}$, but this is not sufficient.


title: Red Shift and Light Speed

text: I read several questions/answers about red shift on this web site, but there is something I still don't understand. Why is red-shift considered to be the effect of expanding space-time, why it could not be a change in the speed of light. After all, if the speed slowly increases, it could affect the wavelength of the light. And perhaps something else we call "constant" could be changing and affect the wavelength of light. I don't understand why space-time would change but no other physics constants or even laws.


title: At what angle do billiard balls scatter if they collide off center?

text: The angle defined by joining a line from the centers of the balls must be important. But do they follow this angle when viewed in the rest frame of one of the balls or in the CM frame? The spheres are hard and friction is ignored. I've tried googling a lot. All of the examples include the final angle as given. Please if give an argument for the answer as well. Thanks in advance.


title: UV-IR cancellation of the open string cylinder diagram and the field theory limit

text: In string theory, the ultraviolet divergences of open string loop diagrams are reintepreted as closed string infrared divergences, by seeing that an annulus with a small loop is also a long tube. In theories with these divergences, like the bosonic string, can one see the equivalence of these divergences when one takes the limit of vanishing string length $\alpha' \to 0$? In other words, can one make a one-to-one mapping between quantum field theory loop divergences and tree-level gravitational infrared divergences and see, in some regulated limit, that they are indeed the same? Or have I misunderstood something?


title: In what way is the Riemann curvature tensor related to 'radius of curvature'?

text: In Misner, Thorne & Wheeler, they say, in their delightful 'word equations' that $$\left(\frac{\mathrm{radius\,\, of \,\,curvature}}{\mathrm{of\,\, spacetime}}\right) = \left(\frac{\mathrm{typical\,\, component\,\, of\,\, Riemann\,\, tensor}}{\mathrm{as\,\, measured\,\, in \,\,a \,\,local \,\,Lorentz\,\, frame}}\right)^{-\frac{1}{2}}.$$ My question is: does this definition of radius of curvature (and others like it - where tensor are described in words) depend on the valence of the Riemann curvature tensor?


title: What exactly is the difference between advection and convection?

text: After reading Wikipedia articles on advection and convection, I still cannot determine whether there is a consensus on a difference between these two terms. Sometimes, the term convection seems to include advection and diffusion, sometimes not. After all, equations exhibit two different terms for advection and diffusion, and we don't call the sum of them a convective term. When we talk about a convective term, it is indeed the advective term! It seems:   * transport of pollutants in a river by bulk water flow downstream relates to **advection**   * boiling water for pasta relates to **convection** Maybe, a conserved property such as energy can be advected, but we only use convection for transport of "ensembles of molecules". So, is there a difference? Could we figure it out with examples?


title: Was the 2013 meteor over Russia stronger than an atomic bomb?

text: Reports of the Russian meteor event (2013) say that it released more energy than 20 atomic bombs of the size dropped on Hiroshima, Japan: > Scientists estimated the meteor unleashed a force 20 times more powerful > than the Hiroshima bomb, although the space rock exploded at a much higher > altitude. Amy Mainzer, a scientist at NASA's Jet Propulsion Laboratory, said > the atmosphere acted as a shield.   >  The shock wave may have shattered windows, but "the atmosphere absorbed the > vast majority of that energy," she said. > > http://abclocal.go.com/kfsn/story?section=news/national_world&id=8994311 Really? Wouldn't that have done more damage than was seen? How does the damage depend on how quickly the energy is released?


title: Classical limit of a quantum system

text: If we have a one dimensional system where the potential $$V~=~\begin{cases}\infty & |x|\geq d, \\\ a\delta(x) &|x|<d, \end{cases}$$ where $a,d >0$ are positive constants, what then is the corresponding classical case -- the approximate classical case when the quantum number is large/energy is high?


title: What would be the effects of an inflating universe?

text: If our universe is inflating, which means it is getting bigger and bigger in size, and every thing it consists of is also enlarging, in a special way such that the mass of the universe remains constant and only the volume changes. In that case, how will we observe the changes in the physical layout of the universe? In particular, are these guesses correct?   1. We will observe that the speed of light is decreasing, as light usually travels $3*10^8$m, in a second, but now since the universe is inflating, it will travel something lesser then that for instance $1.5*10^8$   2. As far as observation is considered, I dont think we will see any changes, since the retina of our eyes would then also have enlarged,   3. As far as physics of the universe is considered I think the planets will keep to their orbits as their mass has not changed a bit and they maintain same distance with other celestial bodies as they used to,


title: Can electron exist as a standing wave inspite of successive superposition?

text: With the development of quantum mechanics, it was found that the orbiting electrons around a nucleus could not be fully described as particles, but needed to be explained by the wave-particle duality. In this sense, electrons do not orbit the nucleus in the sense of a planet orbiting the sun, but instead exist as standing waves. The lowest possible energy an electron can take is therefore analogous to the fundamental frequency of a wave on a string. Higher energy states are then similar to harmonics of the fundamental frequency. I got a question here, if electrons exist as standing waves (matter waves), can they get superposed by non-mechanical wave such as electromagnetic wave? My book says that, the wavelength of an electron with mass $9.11\cdot10^{-31}Kg$ and moving with the velocity of $10^6m/s$ has the wavelength of about $7.28\cdot10^{-10}m$. The wavelength associated with the moving electron is said to be of the same order of magnitude as of $X$-rays, which can be easily measured. If assume that electron gets superposed with electromagnetic radiation, I thought by superposition, electron would loss its existence as standing wave (I didn't get any source to support this view, if it is wrong, please explain). Actually atoms get incident by electromagnetic radiation at every instant, if we assume electrons to get superposed by electromagnetic wave, electron can't exist as stationary wave, but according to schrodinger model, electron is said to exist as stationary wave. I don't know whether I have gone wrong any where or _is it that electron can't exist as stationary wave?_ if any is the case please explain. ![standing wave](http://i.stack.imgur.com/bzoQv.gif)   [ **standing wave** ]


title: Torque, lever and mass

text: The Force used in a catapult is exerted near its axis. If we double the length of the arm of the catapult, but still use the same Force at the same point as before near the same axis, does the distance in the Torque equation change?Edit (In an experiment we have found that the longer arm/lever makes the ball fly a longer distance, but how do we compute this into the equation?) Also, how can one combine the mass of the ball used by the catapult with the calculation of the torque?


title: Vortex Air Intake

text: Why is a vortex air intake in an engine cause the fuel to burn more efficiently within a vehicle. Why is this more efficient.


title: Difference between electrostatic and magnetic field

text: ![enter image description here](http://i.stack.imgur.com/EmB8t.jpg) In this, F (the force set by the field) is perpendicular to the speed of the particle. In which case does F change the absolute value of V? Why or why not?


title: Kepler's law of periods question, application

text: I'm a year 12 student studying in Sydney Australia. I would like to ask about a physics question which I'm currently troubled on. I have made several attempts of answering the question but no luck. Includes: Trying to sub the information into the Kepler's 3rd law, which is $r^3 / t^2 = GM/ 4\pi^2$. Could someone please hint, or explain HOW this question can be answered as it would help me greatly. > Geostationary satellites orbit with a radius of 42260km. Use this > information to find the period of a satellite which orbits with a radius of > 15000km.


title: A bomb can be made to explode or implode?

text: A bomb can be made to implode or explode. Can one be made to explode while creating a vortex at the same time?


title: Where does the Maxwell-Boltzmann distribution come from?

text: I understand that Maxwell-Boltzmann distributions arise for distributions of weakly interacting particles at equilibrium. But I'd like to know if there's a deeper reason behind why they are specifically Maxwellian. I apologize if the question is poorly formed, it just popped into my head and I thought I'd ask about it.


title: What kind of damage is expected to happen to the earth in case of being hit by a direct gamma ray burst?

text: I understand how the ozone layer would be quickly depleted and the UV radiation from the sun would reach the ground...etc I understand all this, but what confuses me is where all this huge amount of energy would go if our atmosphere stopped it ? As far as I know, gamma rays will be stopped by the earth's atmosphere and the rays will not touch the ground, also, a typical GRB has about 10^43 to 10^45 joules of energy for the gamma photons, so where would this huge amount of energy go if our atmosphere absorbed it other than ozone depletion ? Would it heat the atmosphere to thousands of degrees ? Or am I just wrong about something here ?


title: Am I interpreting Gauss' Divergence Theorem correctly

text: I was reading Introduction to Electrodynamics by Griffiths and I wanted to check if I understood Gauss' Divergence Theorem correctly. The theorem states: $$\int \int \int_V \vec{\nabla} \cdot \vec{C} \hspace{0.1cm} dV = \oint\oint_S \vec{C} \cdot d\vec{a}$$ where the LHS is integrating over a volume and the RHS is integrating over a _closed_ surface. Here is my analogy. Suppose you have some pipes, like this: ![enter image description here](http://i.stack.imgur.com/DcZYq.png) Imagine that all pipes are outputting water at an equal rate. Now, I say that the LHS of the theorem says you can **calculate the flux** of water by adding up all the **divergences$^{\dagger}$** across the **volume** that the water creates. In contrast the RHS says that you can calculate the total flux by adding the **individual fluxes** at the **boundaries** (indicated in red). **Is a correct way of thinking about the theorem?** $^{\dagger}$ Where the divergence is a measure of the spread of a vector (of a vector function) at a given point.


title: Do lawn mowers still cut the grasses if they aren't attached to the ground?

text: A rotary lawn mower cuts the grasses by rotating it's sharp blade on it but if the grasses aren't attached to the ground will they still cut them? What forces act on this system? I know the answer is yes but I can't imagine how it works.


title: Question: if a nuclear reactor of a submarine explodes will it contaminate the surrounding waters?

text: After watching a Naval War movie I began to ask myself this question and I would like to know.


title: Is there really time reversibility in physics?

text: First, how i got to the question. I was randomly looking at this page: http://en.wikipedia.org/wiki/Langton%27s_ant describing some sort of cellular automaton. Here is the path of this automaton after more then 10000 steps: ![Langton's ant trajectory](http://upload.wikimedia.org/wikipedia/commons/d/d3/LangtonsAnt.png) You can clearly see that after some time ant gets stuck in the loop (actually its a highway) and will be going on it forever. Here comes the problem with many faces: If we turn back time and start ant from one of the loop(highway) points we will never get to the original messy part, so obviously this computation is not time reversible since there is no way out of the loop! :) (But there should be ! :D) Another thing is that information is also not time reversible - since if at some point in computation operation DELETE is used to erase all the previous data there is no way to get that data back. Ok. So far so good. Lets take this question one step further. How is that so? They always told us laws of physics are time reversible and so on. My intuitive answer is following. We can say that in the universe there are so called LAWS (physics and so on) and OPERATIONS. What is the difference? LAWS do not change the objects on which they are operating and hence probably are time reversible. OPERATIONS on the other hand can change objects, their number in the system and hence are not time reversible (if I delete some files there is no algorithm how to restore them again) I understand its very close to the idea and interpretation of information. So when they show us movie backwards and we think time got reversed its actually false! Yes you can reverse the motion of the people and objects, but you can not reverse speach for example since there is no logical connection between phrases being said. Here we can see how the content of the movie (life) is actually twofold - time revesible part (geometry in the poorest sense) and informational part (speech, computations and so on). From all above it seems that the universe overall is NOT time reversible, and there is no way to go back in time! :) Or is there?


title: Do photons in interferometer violate the law of conservation of mass?

text: I mean Mach-Zehnder Interferometer, where light split into two shortest paths. Is light after splitting wave or particle? Is it a particle? How could one photon change to two? If the wave, does the sum of the energies of both waves divided by the square of the speed of light equal to the weight of the original photon? ![Mach-Zender interferometer](https://upload.wikimedia.org/wikipedia/commons/d/d8/Mach_Zehnder_interferometer.svg) Thank you.


title: Understanding the Bose-Fermi dichotomy

text: I'm an amateur on a quest to understand QM. In various places (such as early in chapter III.4 of the Feynman Lectures) I have seen an argument that looks like it's trying to convince me that any class of indistinguishable particles must _necessarily_ behave either like bosons or like fermions -- _i.e._ , that this is not only an empirical fact, but a deductive consequence of the basic structure of QM. I have trouble understanding these arguments. They seem to pull an assumption out of nowhere, that interchanging two of the identical particles must necessarily correspond to multiplying the wavefunction by a constant. Once this is given, I can see that this constant must be either $1$ or $-1$, but where does that assumption come from? If I write down a wavefunction _without_ knowing in advance that the particles are identical, there are oodles of ways to write one that _doesn't_ react to an interchange merely by a constant factor. Is there something that _a priori_ excludes all those states from consideration? I've been able to reconstruct an argument that works for a world containing only $2$ instances of the indistinguishable particles, to wit: > What it formally means for the particles to be indistinguishable must be > that the operator $P$ which interchanges the two particle _commutes with > every observable_ , in particular with the Hamiltonian. > > If we have any random state of the system, we can write it as a sum of > eigenstates of $P$, and because $P^2=1$, the eigenvalues must be $\pm 1$. > Now whenever time passes (we apply $H$) or we make an observation (and apply > some other observable operator), the operator we use commutes with $P$, and > therefore we can diagonalize $P$ and the operator _simultaneously_ and see > that the $+1$ eigenstates never contribute any amplitude to the $-1$ > eigenstates and vice versa. > > Thus if _at some point_ in history we have made an experiment to determine > out whether our two particles are bosons or fermions, right after the > experiment the state of the world is going to be a pure $+1$ or $-1$ > eigenstate -- and because of the above reasoning it is going to _stay_ in > that eigenspace for all eternity henceforth. > > So in this case it's of no use to consider mixed states. The problem with this argument is that it doesn't seem to scale beyond two particles. With $n$ particles there are a lot of different permutations of them one could consider, and these permutations don't commute internally, and so I cannot be sure that they will _all_ diagonalize simultaneously with the operator I'm observing. This is probably not just a problem with my proof skills, as dimension- counting shows. Let's consider a discrete system where each of $3$ particles can be in one of $3$ states. The state space has dimension $27$ over $\mathbb C$ -- but the "fermion" subspace (consisting of those states that go to their negative under an interchange of _any_ pair of particles) has dimension $1$, and the "boson" subspace (those states that stay unchanged under any transposition) has dimension $10$ if I count it correctly. There are $16$ dimensions left unaccounted for! Is there a principled reason why all of those states are necessarily unphysical or uninteresting? In Zee's _Quantum Field Theory in a Nutshell_ (which I bought by mistake but don't have anywhere near the prerequisites to understand) there's some mumbling about "anyons", which as far as I can decipher it seems to claim that a strict Bose-Fermi dichotomy is only necessary if one has a _continuous_ set of base states of dimension at least $3$ -- by some topological considerations that mostly went over my head. Am I misunderstanding it when I think Feynman and others claim that the dichotomy is a derived result?


title: maximum positive electric charge of solid body

text: What are the limiting factors on the positive charge of a solid body? If I assume a 'perfect insulator' environment that would not exchange charge with my solid body, I would guess that I can remove electrons from the body until it's internal chemical bond structure made by them breaks down and the body disintegrates. Are there any other other limits that would be hit first? How to calculate the charge at which the body breaks away?


title: black hole event horizon

text: Given gravitational time dilation, under what conditions will a test particle cross an event horizon before the black hole evaporates? Assume zero background radiation.


title: How does the holographic principle imply nonlocality?

text: For example in the discussions here and here there are comments by Ron Maimon: > Your complaint about locality would be more serious if holography didn't > show the way--- the CFT in AdS/CFT produces local AdS physics, even though > the description is completely and ridiculously nonlocal and > Once you realize that gravity is defined far away on a holographic screen, > the idea of hidden variables becomes more plausible, because the physics of > gravity is nonlocal in a way that suggests it might fix quantum mechanics How is gravity nonlocal? I thought GR was explicitly Lorentz Invariant? Or are these statements more philosophical (something I would not expect from Ron), ie, just a statement that the boundary is "far away" and isomorphic to the interior... * * * **EDIT:** Ron gave an answer that is very difficult for me to parse. Can someone who is a bit more pedagogically inclined interpret what he says? I asked him to clarify various points in the comments, with little luck. I'm not even sure how he is defining 'locality': > The nonlocality of gravity doesn't mean that Lorentz invariance is broken, > Lorentz invariance and locality are separate concepts. It just means that to > define the state of the universe at a certain point, you need to know what > is going on everywhere, the state space isn't decomposing into a basis of > local operators. I do not see how this does not violate Lorentz invariance. If your state at time t depends on parts of the universe outside your light cone, this is clearly a-causal. > "Locality" is a bit of an overloaded term, and for this discussion I will > assume that it means there are bosonic operators at every point which > commute at spacelike separation (Bosonic fields and bilinears in Fermi > fields). This means that that the orthogonal basis states at one time are > all possible values of the bosonic field states on a spacelike hypersurface, > and over Fermi Grassman variables if you want to have fermions. I do not understand this definition, and frankly it seems unnecessarily complicated and non-transparent. Is this a different definition of 'locality' compared to what is used, for example, in Bell's famous paper?


title: Bond Angles - H2O vs CO2

text: H2O has a 109.5 degree bond angle, but CO2 has exactly 180 degrees. Is there a qualitative reason for this? It's hard to believe CO2 is exactly 180 degrees unless there were some symmetry, but the same symmetry argument should apply to H2O then. So, is it really exactly 180 degrees?


title: Fields of Steady Currents Using Electrostatics

text: Suppose you have a uniform ring charge rotating at constant angular velocity so that you also have a uniform ring of steady current, and thus you can use the Biot-Savart Law to compute the magnetic field. But I also remember from college that to good approximation you can use electrostatics to compute the electric field due to the charge if the current is steady, in spite of the fact that charges are moving. I never understood how this works. Can someone offer insight?


title: How is it possible that quantum phenomenons (e.g. superposition) are possible when all quantum particles are being constantly observed?

text: I don't understand how quantum mechanics (and therefore also quantum computers) can work given that while we work with quantum states, particles that this quantum state consist of cannot be **observed** , which is the most fundamental requirement. If I am not mistaken, by "observed" we mean interaction with any other particle (photon, gluon, electron or whatever else). So my very important questions:   1. Aren't the particles this quantum state consists of interacting with each other? Why doesn't that cause the state to collapse?   2. Aren't all particles in the universe interacting with Higgs field and gravitons etc? Why doesn't that cause every quantum state to collapse? I feel there is something very fundamental in quantum mechanics that I am not aware of, hence I would be very pleased to have these questions answered.


title: Number of Images from pair of plain mirrors?

text: Basically not exactly a homework, but I came across this problem. I have two plain mirrors angled at 60 degree angle. An object is placed in between symmetrically. What will be the number of total images obtained? What I mean is that there will be images of images, how can we count them (If at all), and how are they related to angle between the mirrors? Any help will be appreciated. ![enter image description here](http://i.stack.imgur.com/REcne.jpg)


title: Rear Projection 3d figures?

text: From the wikipedia page on a now prominent 3d virtual idol "From 'Things often confused with holograms' on the Wikipedia page linked above: 'In 2010, there was a series of concerts ... which included Hatsune Miku ... performing on stage as a "holographic" character. This effect was actually achieved through a special method of rear projection against a semi-transparent screen." How exactly was she created/made since it supposedly is not a hologram?


title: How do we measure $i[\hat\phi(x),\hat\phi(y)]$ in QFT?

text: What operational procedure is required to measure $i[\hat\phi(x),\hat\phi(y)]$ in an interacting (or non-interacting) QFT? [assume smearing by test- functions, or give an answer in Fourier space, for $i\bigl[\widetilde{\hat\phi}(k),\widetilde{\hat\phi}(k')\bigr]$, to your taste; give electromagnetic field indices to $\hat\phi(x)$ and transpose the question to QED if the scalar field case seems unreal in an essential way]. In a non-interacting QFT, $i[\hat\phi(x),\hat\phi(y)]$ is a c-number, but in an interacting QFT it may presumably be an observable independent of $\hat\phi(x)$ and $\hat\phi(y)$. How do we measure it, however? If we cannot measure it, is it effectively chosen by convention? An answer that describes the (operational) difference between $A=\hat\phi(x_1)...\hat\phi(x_i)\hat\phi(x_{i+1})...\hat\phi(x_n)$ and $B=\hat\phi(x_1)...\hat\phi(x_{i+1})\hat\phi(x_i)...\hat\phi(x_n)$ might be equally good (though $A^\dagger\not=A$ and $B^\dagger\not=B$). As always, a reference would be good. EDIT: Following Luboš' answer, and more particularly his comments, for which again thanks, $i[\hat\phi(x),\hat\phi(y)]$ is zero if $x-y$ is space-like, both in non-interacting and interacting quantum fields. If a quantum field $\hat\phi(x)$ is the basis for an empirically adequate quantum theory, measurement operators at time-like separation from a given space-like hyperplane at time $t$ can in principal be expressed as a nontrivial function of the field and field momentum at time $t$, $\hat\phi(y)=\int[\hat\phi(t,\mathbf{x}')f_t(\mathbf{x}')+\hat\pi(t,\mathbf{x}')g_t(\mathbf{x}')]\mathrm{d}^3\mathbf{x}'$, for two functions $f_t$ and $g_t$, which for a given theory are determined by the Lagrangian. Given many implicit assumptions, measuring the commutator $i[\hat\phi(t,\mathbf{x}),\hat\phi(y)]=-g_t(\mathbf{x})$ (because $[\hat\phi(t,\mathbf{x}),\hat\pi(t,\mathbf{y})]=i\delta^3(\mathbf{x}-\mathbf{y})$) is equivalent to measuring $g_t(\mathbf{x})$. If we also measure $i[\hat\pi(t,\mathbf{x}),\hat\phi(y)]$, we can determine (or at least significantly constrain) the Lagrangian. _If we do not know the Lagrangian_ , therefore, it would seem to be desirable to have an operational way to measure $i[\hat\phi(x),\hat\phi(y)]$ in order to _measure_ the Lagrangian (instead of the usual approach of guessing the Lagrangian and seeing how close the consequent predictions are to the measurements; the haphazardness of this process is something of a crisis in HEP). I would welcome comments on this or an answer that takes this EDIT into account. Alternatively, an operational approach to _measuring_ the Lagrangian would implicitly be a measurement, in principle, of $i[\hat\phi(x),\hat\phi(y)]$; is there such a thing?


title: Mass Effect on Slingshot Motion?

text: For my physics class (I'm a high school student), we created slingshots. Our task is to predict the distance a projectile, launched from a slingshot using surgical tubing, would land. We aren't given the mass of the projectile until the day of testing, so we would have to use a graph to find out the distance. My partner and I conducted a series of tests comparing mass to distance from a given pullback length ($55$ cm). The angle the projectile is launching from is $54^\circ$. K is not linear, but for our set pullback length our K value is $100$ (N/m). Our results show a cubic, or possibly quadratic graph. Why is this? Why would the relationship between mass launched and distance follow this shape? Here is a graph of our data: ![enter image description here](http://i.stack.imgur.com/IA0kW.png)


title: Which symmetric pure qudit states can be reached within local operations?

text: There are two pure symmetric states $|\psi\rangle$ and $|\phi\rangle$ of $n$ qudits. Is there any known set of invariants $\\{I_i:i\in\\{1,\ldots,k\\}\\}$ which is equal for both states iff $|\phi\rangle=U^{\otimes n}|\psi\rangle$ for a $U\in\text{SU}(d)$? There is a theorem by Hilbert saying that for a compact group acting on a linear space there is a finite number of polynomial invariants characterizing orbits. However, (as far as I know) it does not provide and explicit construction of the invariants. The problem is easy when $n=2$ (Schmidt decomposition) and $d=2$ (Majorana representation). Partial solutions, and solutions with changed assumptions (e.g. linear operators instead of unitary operations) are welcome as well.


title: Asymptoticity of Pertubative Expansion of QFT

text: It seems to be lore that the perturbative expansion of quantum field theories is generally asymptotic. I have seen two arguments. i)There is the Dyson instability argument as in QED, that is showing the partition function is nonanalytic around the expansion point, by analyzing the ground state or instantons or somesuch. This is a wonderful argument but it requires some non-trivial knowledge about the behavior of your QFT which may not be available. ii) There is some attempt at a generic argument which merely counts the number of Feynmann diagrams at each order, says this grows like $n!$ where is $n$ is the order of the expansion. and so our series looks like $\sum n!\lambda^n$, which is asymptotic. This is of course wholly unsatisfactory since it ignores interference among the terms (even granting the presumption that all the diagrams are of the same order, which feels right). It is true the series is still asymptotic if we take the diagrams to have random phase, but this ignores the possibility of a more sinister conspiracy among the diagrams. And we know that diagrams love to conspire against us. So is there any more wholesome treatment of the properties of the perturbative expansion of QFT? I came to thinking about this while considering the properties of various $1/N$ expansions so anything known in particular about these would be nice.


title: Gravitational redshift derivation

text: When we derive gravitational potential it is dependant only on (rest) mass $m$, but i have seen a derivation of gravitational redshift equation placing relativistic mass $\widetilde{m}$ instead of rest mass $m$ like this: $$ \begin{split} W_1 &= W_2\\\ h\nu_1 + 0 &= h \nu_2 + \left(-\frac{GM \widetilde{m}}{r}\right)\\\ h\nu_1 &= h \nu_2 -\frac{GM \widetilde{m}}{r}\\\ h\nu_1 &= h \nu_2 -\frac{GM h \nu_{2}}{c^2r}\\\ \nu_1 &= \nu_2 \left(1 -\frac{GM}{c^2r} \right)\\\ \end{split} $$ I think this is wrong! What led me to this conclusion? In my other topic people warned me that i can't use $\frac{1}{2}mv^2$ for a photon. So i think that i have no right to swap $m$ with $\widetilde{m}$ than state that photon has mass and use $\widetilde{m}$ in equation: $$ W_p = -\frac{GMm}{r} $$ Photon has mass $m=0$ period! And should be unaffected by gravitational field. I know experiments show us different story but still... * * * **QUESTION 1:** I would need some explaination on this fragile topic. **QUESTION 2:** Could anyone tell me allso when i have the right to use relativistic mass $\widetilde{m}$?


title: Does Kaluza-Klein theory successfully unify GR and EM? Why can't it be extended to the Standard Model gauge group?

text: As a quick disclaimer, I thought this might be a better place to ask than Physics.SE. I already searched there with "kaluza" and "klein" keywords to find an answer, but without luck. As background, I've been reading Walter Isaacson's biography of Einstein, and I reached the part where he briefly mentions the work of Kaluza and Klein. Also, I did my undergraduate degree in theoretical physics, but that was quite a few years ago now... The way I understand the original work of Kaluza and Klein is that you can construct a theory that looks like a 5-D version of GR that reproduces the Field Equations and the Maxwell Equations. An important bit is that the 4th spatial dimension is the circle group $U(1)$, which we now know as the gauge group of EM. I guess the first part of the question should be whether I've understood that correctly. Then, if you build a Kaluza-Klein type theory, but use the SM gauge group $U(1)\times SU(2)\times SU(3)$ instead of $U(1)$, what do you get? Is it gravity and the standard model together? If not, then what? The Wiki article on Kaluza-Klein theory says that this logic "flounders on a number of issues". The only issue is explicitly states is that fermions have to be included by hand. But even if this (Kaluza-Klein)+(SM gauge) theory only describes interactions, isn't that okay, or at least a big help?


title: Quantum mechanics in macroscopic systems

text: I don't understand the superposition principle in quantum mechanics or the collapse of wave-function (I think it's impossible for me to understand it) My question is: Is it possible to demonstrate the quantum mechanical behaviour (Superposition and wavefunction collapse, etc.) in some macroscopic systems under certain conditions so it may be better to understand it? Is it possible to make the wavefunctions of atoms in some material more or less coherent?


title: Does electric potential have a temperature?

text: When I took my first thermo class a tucked away chapter introduced Exergy in terms of electrical energy, meaning that the amount of electrical energy you could get from something is functionally its exergy. Wikipedia doesn't go so far, but this has bothered me for years. > Forms of energy such as macroscopic kinetic energy, **electrical energy** , > and chemical Gibbs free energy **are 100% recoverable** as work, and > therefore have an exergy equal to their energy. However, forms of energy > such as radiation and thermal energy can not be converted completely to > work, and have exergy content less than their energy content. Let's say that we have 1 Volt of potential. The smallest unit of energy we could isolate from this would be a single electron accelerated over the potential. Numerically and naively I could write: $$ 2/3 k T = V e$$ Where $e$ is the electron charge. Doing this I would get something like $17,000 K$ for the temperature. This is high enough to approximate to infinity for conversion processes where your sink is room temperature. I can imagine accelerating something with a greater charge than electron over that potential. This method would find a higher temperature, which presents a counter-argument. So what is it, **is there a theoretical limit to the energy you can get from electricity, giving it a functional temperature?** That would imply an unavoidable loss from DC-DC voltage converters. Another thing that bothers me is saying that radiation can't be converted completely to work. Thermal radiation, yes, but most nuclear radiation is at $keV$ or $MeV$ energies, which would give a dramatically higher temperature than electrical energy (if I've understood correctly).


title: Computability of physics

text: As far as I understand a Universal Quantum Simulator can simulate any quantum system and thus anything that exists in the universe. Also, a quantum computer can implement such a quantum simulator. Further, from what I've read, a quantum computer does not have the ability to compute something that is not computable by a classical Turing Machine, although it can perform certain computations much more efficiently. However, I recently saw this: Classical problem becomes undecidable in a quantum setting. The actual paper: Quantum measurement occurrence is undecidable I don't understand this, unfortunately, so I wanted to ask, does this mean that physics is not computable by a classical computer (Turing Machine)? What about by a quantum computer? Or is this article saying something different altogether?


title: Gravitational force from spherical shell

text: Say we have a point mass $\mu$ located at $(0,0,R)$ and a spherical shell ( _not_ considering it's volume!) of radius $R$ located in the origin. So we have a particle standing right on top of the sphere and we want to determine the total gravitational force exerted on the particle by the spherical shell. Since the force is directed along the z-axis, we only have to consider the z-component of the force. Moreover, suppose the shell has homogeneous density $\sigma$. We can then write $dM = \sigma dA = \sigma R^2 \sin \theta d\theta d\phi$. So we have $$ dF_z = \frac {(z-R)G\mu dM}{\sqrt{x^2 + y^2 + (z-R)^2}^{\ 3}} = \sigma \mu G R^2 \frac { (z-R)\sin \theta d\theta d\phi }{ \sqrt{x^2 + y^2 + (z-R)^2}^{\ 3} } $$ We then integrate over the entire sphere: $$ F_z = \iint \sigma \mu G R^2 \frac { (z-R)\sin \theta d\theta d\phi }{ \sqrt{x^2 + y^2 + (z-R)^2}^{\ 3} } $$ This turns out to equal $-2\pi G \sigma \mu$ and $\sigma = \frac M{4\pi R^2}$, where $M$ is the total mass of the spherical shell, which yields $-\frac {GM\mu}{2R^2}$. What I do not understand why there is a factor 2 in the denominator, should it really be there? Just for the protocol, we get the classical formula if the particle is outside the shell and zero if it is inside the shell.


title: How to work out the work done to put a satellite into orbit?

text: I'm having troubles doing questions that involve how to put a satellite into orbit. So we have a satellite in orbit 20500 km above earth, (and earth radius = 6380 km), the satellites mass is 1000 kg, and let's use the earths mass as $ 6.0 \times 10^{24} $. So to find GPE, is it correct to say you use $ -G( \frac{m_1m_2}r) $, and thus $ = -6.67 \times 10^{-11}( \frac{6.0 \times 10^{24} \times 1000}{6380000 + 20500000}) = -1.489 \times 10^{10} $. Similarly, GPE of the satellite on earth before launch = $ = -6.67 \times 10^{-11}( \frac{6.0 \times 10^{24} \times 1000}{6380000}) = -6.273 \times 10^{10} $. I'm pretty happy with this up to here, but I don't understand the next step. Intuitively, should it not be Work done = Final GPE - Initial GPE, therefore = $-1.489 \times 10^{10} - (-6.273 \times 10^{10}) = 4.784 \times 10^{10} $. However the answers say it's the other way around- $-6.273 \times 10^{10} - (-1.489 \times 10^{10} ) = -4.784 \times 10^{10} $. I just can't wrap my head around this- why is it initial GPE - final GPE, and shouldnt it be a +ve answer since you're doing work upon the object? Thanks in advance