title: Physical applications of matrices and determinants

text: Other than notation devices, I don't see any direct application of matrices/determinants in physics. For example, they are just a different way to write a partial derivative and determinants find if they can be explicitly solved if written down as simultaneous equations. Calculus, for instance, can be directly applied to physical problems, but I don't know of any other application of matrices other than representing equations in a different notation. And in most of the cases like vector products, you just realise that a huge term can just be written down as a determinant, so it is essentially a notational tool. They are used in tensor calculus, but for similar reasons. Can someone please guide me on more applications with good sources?


title: Azimuthal quantum number,l and magnetic quantum number,m are from angular momentum?

text: Azimuthal quantum number $l$, and magnetic quantum number $m$ are defined when we do derivation for $L^2f=l(l+1){\hbar}^2f$ and $L_zf=l{\hbar}f$. This is my own conclusion after studying introduction to quantum mechanics by Griffiths. Correct? But I don't know why does $L_zf=l{\hbar}f$ become $L_zf=m{\hbar}f$? I wonder why when we operator $L_+$ on an eigenstate, the only change is that $m$ increases by one not for $l$, e.g.$$L_+ Y_{2}^{1}= AY_{2}^{2},$$ why not $l$ change? Thanks


title: The commutator of scalar field

text: I have a real scalar field which is given by the propagator as: $$[\phi(x),\phi(y) ] =\int \frac{d^3 p}{(2\pi)^3} \frac{1}{2E_0} (\exp(-ip\cdot (x-y)) -\exp (ip\cdot (x-y)))$$ And I am asked to show that this equals $$\frac{1}{4\pi r} \partial_r J_0(m\sqrt{t^2-r^2})$$ Where $J_0$ is bessel function of order 0, $t=x^0-y^0$, and $r=|x^i-y^i|$. I am clueless as to how to show this. I mean the first integral is an integral with integrand $2\imath \sin(p\cdot(x-y))$. Anyone care to enlighten me, or me a thorough reference for this calculation?


title: Concepts in Gabriel Kron's later papers

text: Gabriel Kron was an important research electrical engineer known for applying differential geometry and algebraic topology to the study of electrical system. Towards the end of his career he published a number of unusual, even by his standards, papers on concepts with names like _polyhedral networks_ , _self organizing automata_ , _wave automata_ , _multidimensional space filters_ and _crystal computer_ , which I think are more or less synonymous. I have obtained a few of these papers and did not understand them at all. If they were not written be Kron, I would be suspicious of them. I have not been able to find any significant secondary literature on these ideas. The few citations I have tracked down only mention them tangentially, but I have also found no refutations of these papers and no suggestions that Kron had gone off the rails. The papers were published in respectable journals. **I am looking for an understandable exposition or refutation of these ideas, or pointers to such. Also pointers to follow on research by others, possibly using different terminology.** I am _not_ looking for explanations of Kron's other ideas like diakoptics and tensor analysis of networks. Some of the relevant papers are:   * G. Kron, Multi-dimensional space filters. Matrix and Tensor Quarterly, 9, 40 - 43 (1958).   * G. Kron, Basic concepts of multi-dimensional space filters. AIEE Transactions, 78, 554 - 561 (1959).   * G. Kron, Self-organizing, dynamo-type automata. Matrix and Tensor Quarterly, 11, 42 - 52 (1960).   * G. Kron, Power-system type self-organizing automata. RAAG Memoirs, III, 392 - 417 (1962).   * G. Kron, Multi-dimensional curve-fitting with self-organizing automata. Journal of Mathematical Analysis and Applications, 5, 46 - 49 (1962). I have mainly looked at the last one and material at the end of   * Diakoptics; the piecewise solution of large-scale systems. MacDonald, London, 1963. 166 pp.


title: High-spin neutron star

text: Suppose a large neutron star were to be spun-up by a particular pattern of mass-accretion. The increased centrifugal force could presumably mitigate the increased gravity thus delaying gravitational collapse. How far could this process be taken?


title: about laser wavelength and wave form

text: I watched a short introduction video online. There are few concepts which are a bit confusing. 1) The video said the laser is single mode (monochromatic), the wavelength is 780nm, so what is that wavelength really refer to? What I am asking is what's the wavelength related to the medium creating that laser? Is that the atom level of the medium generating that laser (sorry, I don't have much background on that) 2) In some book, it said the profile of the laser is Gaussian, but when laser is in transmission, what's the wave look like? Is it sinusoidal wave of the form $y = \sin(kx)$ where $k=2\pi/\lambda$, if so, why they said it is Gaussian? 3) I the undergraduate course of physics, I remember it said the plane wave have the form $\sin(kx - \omega t)$, so the speed of the wave is $v=\omega/k$. But if the laser wave has the form $y=\sin(kx)$ and transmit in the air, why there is no frequency term? So the speed is ZERO?


title: Why do some air-conditioned stores blast you with jets of air as you enter?

text: I went to a grocery store on a hot day that was very well air-conditioned, and I noticed as I went through the open entrance that there seemed to be a very powerful downward air current right at the doorway. After crossing the invisible threshold, the temperature immediately dropped a good 15 degrees or so. How does this process work?


title: is the Z boson one entity or are there as many entities as decay pairs, but they are equivalent and lumped together

text: just wondering if it is a distinction without a difference - it seeming a bit weird that one thing can decay into different things.


title: Gravitational Field from Irregular Object

text: How does one calculate the gravitational field of an irregularly shaped object? I was thinking specifically of the gravitational field due to a human, and tried to see if modeling a human as an infinite wire would work, and of course it's a horrible approximation*. Is there anyway to do it without relying on a computer, or is numerical approximation the only option? And even with a computer, what process does it actually do to get an answer? I would assume that it uses superposition with $\vec{F}= \frac{GMm}{ \vec{r}^{2}} \hat{r}$, which would work for an irregularly body acting on a point particle, but how would you calculator the field of two irregularly shaped bodies acting on each other?   * I used Gauss's law for Gravitation, and ended up with a distance of 289 Å, which would make the acceleration equal to $g$, which sounds really wrong, considering that the distance between atoms is only on the order of a few Å.


title: Holes in a P-type semiconductor under external force E

text: Basically in almost every semiconductor texts, there will be all these concepts concerning electrons, holes, dopants, fermi-levels. However, I have been always confused about the picture of hole transport in semiconductor device, say, a simple PN junction. With a specific acceptor level and dopant concentration, we have some "holes" in the valence bands, which are in fact the absence of some electrons having gone to the acceptor levels, then all theses books seem to assume in the valence bands of the p-type region, there will be only holes that conduct current? My questions:   * aren't there still many electrons in the valence band? though they have some negative effective masses, but still do contribute to the transport?   * under a certain external force(say,E), the electron and holes in the valence band are moving in the same directions since electrons have negative effective mass and holes have positive one, so the corresponding currents cancel with each other in a P-type semiconductor?


title: Metric of following spacetime and refractive index

text: Let's have metrics $$ ds^{2} = f(\mathbf r)dt^{2} - h(\mathbf r )\delta_{ij}dx^{i}dx^{j}. $$ Hot to show that motion of light in spacetime with this metrics is equal to motion in continuous media with refractive index $n = \sqrt{\frac{h}{f}}$? Is it logical to start from Maxwell equations in curved spacetime, $$ D_{\mu}D^{\mu}A^{\nu} - R^{\nu}_{\quad \sigma}A^{\sigma} = 4\pi J^{\nu}, $$ and then show that it is equal to the Maxwell's equations for continuous media (by the way, how to write them down for this case?)? Edit. Hmm, the task is almost solved. One more edit. It is solved. First, I wrote Maxwell's equation for media in a form of $$ \partial_{\alpha}F_{\beta \gamma} + \partial_{\beta}F_{\gamma \alpha} + \partial_{\gamma}F_{\alpha \beta} = 0, \qquad (1) $$ $$ \quad \partial_{\beta}H^{\beta \alpha} = 4\pi j^{\alpha}, \qquad (2) $$ where $H^{\alpha \beta} = \varepsilon \eta^{\alpha \mu}\eta^{\beta \nu}F_{\mu \nu}$, $$ \eta^{\alpha \mu} = diag \left(\sqrt{\mu \varepsilon}, -\frac{1}{\sqrt{\mu \varepsilon}}, -\frac{1}{\sqrt{\mu \varepsilon}}, -\frac{1}{\sqrt{\mu \varepsilon}} \right). $$ This connection will obvious if write down connection $$ F(\mathbf E , \mathbf B ) \to F(\mathbf D , \mathbf H ) = H(\mathbf D , \mathbf H ). $$ Second, I wrote down the equations for curved spacetime: $$ D_{\alpha}f_{\beta \gamma} + D_{\beta}f_{\gamma \alpha} + D_{\gamma}f_{\alpha \beta} = \partial_{\alpha}f_{\beta \gamma} + \partial_{\beta}f_{\gamma \alpha} + \partial_{\gamma}f_{\alpha \beta} = 0, \qquad (3) $$ $$ D_{\beta}h^{\beta \alpha} = \frac{1}{\sqrt{-g}}\partial_{\beta}(\sqrt{-g}h^{\beta \alpha}) = 4 \pi J^{\alpha}, \qquad (4) $$ where $h^{\beta \alpha} = g^{\beta \mu}g^{\alpha \nu}f_{\mu \nu}$, and where I used the equality for covariant derivative of antisymmetric tensor when there is the convolution. So if use substitution $$ F_{\alpha \beta} = f_{\alpha \beta}, \quad H^{\beta \alpha} = \sqrt{-g}g^{\alpha \mu}g^{\beta \nu}f_{\mu \nu}, \quad j^{\alpha} = \sqrt{-g}J^{\alpha}, $$ the equations $(3), (4)$ will be "reduced" to $(1), (2)$. Then all that remains is to find connection between metrics and $\varepsilon , \mu$. It is not hard when metrics is diagonal: $$ D^{i} = g^{i \alpha}g^{0\beta }F_{\alpha \beta} = -g^{i0}g^{0j}F_{j0} + g^{ij}g^{0l}F_{jl} + g^{ij}g^{00}F_{j0} = g^{ii}g^{00}F_{i0} = \frac{1}{fh}E^{i} = \varepsilon E^{i}, $$ $$ F^{ij} = g^{i \alpha}g^{j \beta }F_{\alpha \beta} = -g^{0i}g^{jk}F_{k0} + g^{ik}g^{j0}F_{k0} + g^{ik}g^{jl}F_{kl} = g^{ii}g^{jj}F_{ij} = \frac{1}{h^{2}}F_{ij} \Rightarrow $$ $$ H_{m} = -\frac{1}{2}\varepsilon_{m ij}F^{ij} = \frac{B_{m}}{h^{2}} = \frac{B_{m}}{\mu}, $$ so $$ n = \sqrt{\varepsilon \mu} = \sqrt{\frac{h^{2}}{hf}} = \sqrt{\frac{h}{f}}. $$


title: How sodium atoms in lamp are energized to produce light?

text: In a recent spectroscopy lab I performed for one of my classes we used a Lambda Scientific LLE-1/2 mercury/sodium lamp (to look at the sodium doublet) and a Lambda Scientific LLE-8 hydrogen/deuterium lamp (to look at the hydrogen/deuterium line). I have been trying to find exactly how these lamps excite the atoms inside to produce light but have failed thus far. From what I have been able to find, I believe it is due to collisional excitations. My main question though is whether the atoms are just excited to a higher level before emitting the light we observed or if the atoms were ionized (rather than just being excited to a higher energy level) and the light we observed in the lab was emitted from the electron's subsequent recombination and cascade back to the ground level?


title: Confusion about duality transformation in 1+1D Ising model in a transverse field

text: In 1+1D Ising model with a transverse field defined by the Hamiltonian \begin{equation} H(J,h)=-J\sum_i\sigma^z_i\sigma_{i+1}^z-h\sum_i\sigma_i^x \end{equation} There is a duality transformation which defines new Pauli operators $\mu^x_i$ and $\mu^z_i$ in a dual lattice \begin{equation} \mu_i^z=\prod_{j\leq i}\sigma^x_j \qquad \mu_i^x=\sigma^z_{i+1}\sigma^z_{i} \end{equation} then these $\mu_i^x$ and $\mu_i^z$ satisfy the same commutation and anti-commutation relations of $\sigma^x_i$ and $\sigma^z_i$, and the original Hamiltonian can be written in terms of $\mu_i^x$ and $\mu_i^z$ as \begin{equation} H(J,h)=-J\sum_i\mu_i^x-h\sum_i\mu_i^z\mu_{i+1}^z \end{equation} At this stage, many textbooks will tell us since $\sigma$'s and $\mu$'s have the same algebra relations, the right hand side of the last equation is nothing but $H(h,J)$. My confusions are:   1. Does that the operators having the same algebra really imply that $H(J,h)$ and $H(h,J)$ have the same spectrum? We know for a given algebra we can have different representations and these different representations may give different results. For example, the angular momentum algebra is always the same, but we can have different eigenvalues of spin operators.   2. This is related to the first confusion. Instead of looking at the algebra of the new operators, we can also look at how the states transform under this duality transformation. In the eigenbasis of $\mu_i^x$, if I really consider it as a simple Pauli matrix, the state $|\rightarrow\rangle$ corresponds to two states in the original picture, i.e. $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$. The same for state $|\leftarrow\rangle$. In the $\mu_i^z$ basis, the correspondence is more complicated. A state corresponds to many states in the original picture, and the number of the corresponding states depend on the position of this state. Therefore, this duality transformation is not unitary, which makes me doubt whether $H(J,h)$ and $H(h,J)$ should have the same spectrum. Further, what other implication may this observation lead to? For example, doing one duality transformation is a many-to-one correspondence, then doing it back should still be a many-to-one correspondence, then can we recover the original spectrum?    3. Another observation is that in the above $\mu_i^z$ involves a string of operators on the left side, we can equally define it in terms of a string of operators on the right side, so it seems there is an unobservable string. What implication can this observation lead to? Is this unobservable string related to the unobservable strings in Levin-Wen model?


title: How do I start learning particle physics?

text: I am 16 at the moment. I am really interested in physics. Especially particle physics. Can someone please tell me how to start learning the subject. like what to learn first. like which fundamental theories and concepts, the math needed in it, etc, etc.


title: Probability of being in the same initial state

text: If the Hamiltonian has basis of eigenvectors $\phi _1, \phi_2,..$ with corresponding eigenvalues $E_1,E_2,...$. I then define an observable $A$ by: $$A\phi_1 = \cos(\beta)\phi_1 + \sin(\beta)\phi_2$$ $$A\phi_2 = \sin(\beta)\phi_1 -\cos(\beta)\phi_2$$ And $A\phi_n=0$ for all $n=3,4...$ Now I can calculate that this observable has eigenvalues $\lambda = -1,1$ and corresponding eigenvectors $(1, \frac{-1-\cos(\beta)}{\sin(\beta)})$ and $(1, \frac{1-\cos(\beta)}{\sin(\beta)})$. If initially the system is in the state corresponding to $\lambda = -1$, so the wave function looks like: $$\psi(t) = \exp \left(\frac{-iE_1t}{h} \right)\phi_1 + \frac{-1-\cos(\beta)}{\sin(\beta)}\exp \left(\frac{-iE_2t}{h} \right)\phi_2$$. How would I go about finding the probability that at a later time, the particle is in the state corresponding to $\lambda = 1$? And in general, how do you find the probability that a wave function is in a certain state at a later time? Any help is much appreciated P.s Apologies if I have used any strange notation, I am studying mathematics and am not entirely sure on the correct way to write things in physics!


What is the motivation for the definition of concurrence in quantum information? On the surface, the definition looks pretty ad hoc. The definition is often given for the case of 2 qubits only. What is the generalization to higher dimensional spaces?


title: What is going on in front of and behind a fan?

text: Why is it that when you drop paper behind a fan, it drops, and is not blown/sucked into the fan, whereas if you drop paper in front of a fan, it is blown away?


title: How can a drone simulator help a physics classes?

text: I'm working on a drone simulator to make the physics classes more interesting for the students, especially for ones do not like this subject so much. What kind of information can help the teachers in their classes? If you are a teacher, I will appreciate so much your contribution Thank you


title: Can someone explain LO-TO Splitting?

text: LO-TO splitting occurs in an _ionic_ (i.e. polar) solid such as GaAs or NaCl. What happens is that the degeneracy of the transverse optical (TO) and longitudinal optical (LO) phonons at $k=0$ is broken and the LO phonon has a greater energy. From a physical point of view, in the limit that the wavelength is infinitely long (i.e. $k\rightarrow0$ or $\Gamma$-point), how is one supposed to tell the difference between a longitudinal and transverse excitation (i.e. from a fundamental physics point of view how is it possible that the LO and TO are non-degenerate)? My other question concerning this problem is that if the bonds were not ionic but instead covalent, then this splitting would not occur. However, the symmetry of the lattice has not changed. How is this possible? For reference, GaAs has a phonon dispersion spectrum that looks like so: ![GaAs phonon dispersion](http://i.stack.imgur.com/QfRmK.jpg) while Ge has the following phonon dispersion: ![Ge pohonon dispersion](http://i.stack.imgur.com/btejb.gif)


title: Can all systems be put in equilibrium?

text: I'm in a first year statics course. We have spent the whole semester solving for forces and moments so that the system is in equilibrium. When we are given a system, we immediately begin solving for all variables for the system to be in equilibrium. But how do we know that the system _can_ be put in equilibrium? I asked my prof about this and he said that all systems can be solved to be put in equilibrium, but some systems might possibly have multiple solutions. Why can all systems be put in equilibrium? If I misunderstood the prof and all systems _can't_ be put in equilibrium, is there a quick way to check if a system can be put in equilibrium? (Other than the obvious way of solving the system). Thanks.


title: Disappearance of moduli for condensate of open strings

text: Consider a Dp-brane. Compactify $d$ spatial dimensions over a torus $T^d$. Suppose $d\geqslant p$, and that the Dp-brane is completely wrapped around the compactified dimensions. Look at the open string modes ending on this wrapped D-brane. There is a zero energy open string mode associated with each spatial dimension. That corresponds to the orientation of the worldsheet field excitation. If the direction is along an uncompactified spatial dimension, that corresponds to quanta of brane displacements along that direction. These cases needn't concern us. If the direction is normal to the brane but lies along a compactified dimension, that corresponds to quanta of brane displacements along that direction. If it's tangent to the brane, it corresponds to quanta of the Wilson line of the brane gauge field along that wrapped direction. Here's the question. Suppose the wrapped D-brane has a total mass of $M$. Suppose there is a compactified spatial dimension of radius $R$ along which the brane isn't wrapped, i.e. $p<d$. The brane has Kaluza-Klein momenta along that direction of value $n/R$ where n is integral. The energy spectrum is given by $$\sqrt {M^2 +n^2/R^2} ~\approx~ M + \frac{n^2}{2MR^2} +\mathcal{O}(M^{-3}).$$ A condensate of zero energy open strings with orientation along that direction ought to give a continuous moduli? Why is there no moduli then, and why is the energy spectrum discretized? Or consider a direction in which the brane is wrapped. We ought to have a continuous modulus of Wilson line of the brane gauge field along that dimension? Once again, we have discretization. Why? Anyway, how can a completely wrapped D-brane have KK momentum? In the string worldsheet picture, we have open strings ending at the D-brane background at a fixed position. Along those compactified dimensions along which the brane isn't wrapped, we have Dirichlet boundary conditions. Such open strings can only have winding numbers, but no KK momentum either. Even then, we're dealing with a condensate of open strings with zero winding number. This discretization doesn't occur if the brane remains unwrapped under at least two uncompactified spatial dimensions because the brane now has infinite mass. If it remains unwrapped only along one uncompactified spatial dimension, there's the Mermin-Wagner theorem, meaning there’s no fixed brane position or Wilson line. PS: Maybe this question can be rephrased in terms of BPS. We have a wrapped BPS brane. But nonperturbatively, somehow, the BPS state has to be delocalized along the compactified dimension or its Wilson line in a superposition over all possible values? Open strings with no energy are also BPS. So, we can have any condensate of them and still remain BPS? This clearly isn't the case with a lifted modulus, and a discretization of the energy spectrum. PPS: How do you even express the KK modes of the D-brane, or the dual to its Wilson lines, in terms of a condensate of open strings? Suppose you have the lowest energy state with zero KK momentum. Then, disregarding transverse displacements along the uncompactified spatial dimensions, there is an energy gap to the next energy state. However, a condensate of open strings with internal mode excitations along the compactified dimensions would naively give no energy gap. PPPS: Is the number of open string modes corresponding to these "disappearing moduli" even a well-defined operator? This is precisely because at the perturbative level, such open string modes have no energy. If it's not a well defined operator, just how do you even express this in terms of open string worldsheets?


title: Anyonic phase factors when encircling one about another

text: The question Statistics of bound states of anyons with order pq, and its answer inspires this question. Suppose you have an anyonic particle with nonintegral spin s. Presumably, if there's an (unstable) bound state between this anyon and its antianyonic particle (an analog of a positronium), the total angular momentum (plus relative orbital angular momentum) of this bound state will have to be integral. This is because this bound state would presumably decay away entirely into say photons or phonons which are bosonic. What this means is if you encircle an antianyon around an anyon of the same species counterclockwise by $2\pi$, you pick up no phase factor of the sort $e^{i\theta}$, or rather, $\theta=0$. Otherwise, you pick up a nonintegral contribution to the spin of the bound state coming from the relative orbital angular momentum. OK, but now consider the following three particle system. In region A, you have an anyon. In region B, you have a localized bound state of an anyon and its antianyon of the same species. The bound state is bosonic. Presumably, that means if you encircle the bound state once counterclockwise around the anyon at A by $2\pi$, you pick up no phase factor? At any rate, once it decays away entirely into photons and phonons, it can't pick up any phase factor, and by continuity over time, this means it can't pick up any phase factor before it decays either. But won't this phase factor be the sum of the phase factors you pick up by encircling an anyon around the anyon at A, and encircling an antianyon around the anyon at A? According to the second paragraph, the latter phase has to be zero? So, the former phase also has to be zero? But when you encircle an anyon about another anyon, you pick up a nontrivial phase factor, right? What am I missing here? This issue doesn't crop up for fermions because a single encirclement leads to a phase factor of $(-1)^2$, which is just 1.


title: Why does mass in the universe have no limit in large size, but has a limit in small size?

text: We found VY Canis Majoris, a star so big it can't even be seen in scaled illustratations with the sun itself. However, we stop at particle physics, or quantum mechanics, i.e. particles, subatomic, etc. If stars keep getting bigger, and we keep finding things bigger, and the size of the universe still is unknown, why does the size (oppositely) have a limit? Like, why is the atom the "smallest mass", but the "largest mass" is indeterminate? I hope you get my point. If not, please tell me how I should improve.


title: If there is no gravity, does that mean there's no mass as well?

text: I have demonstrated that weight only is measured based on the gravitational pull of where you live. For example, the gravity on the surface of Mars is three times smaller than the gravity on the surface of Earth. So my question is simple: If we take into account that weight is only measured based on gravitational pull on that mass, if there was no gravity how would we define mass? That is, would mass weigh anything if no gravitational pull acted on it? Then what would be mass?


title: Non-uniqueness of solutions in Newtonian mechanics

text: In _The Variational Principles of Mechanics_ by Lanczos, in section 1 of Chapter 1, Lanczos states that for a complicated situation, the Newtonian approach fails to give a unique answer to the problem, in contrast to the analytical mechanics approach. Can anyone provide an example where the Newtonian approach cannot give a unique solution to a mechanics problem? I don't mean a trivial response like simply re-expressing a solution in a different inertial frame, but a "complicated" situation that Lanczos had in mind. I cannot think of any.


title: Why do we care about the maximally extended versions of spacetimes?

text: One can take a spacetime and maximally extend it, so that geodesics end only on singularities, where they have to end -- not on coordinate singularities, which are not physically significant. But when we do this, we seem to get _spacetimes_ that are not physically significant. For example, the maximally extended Schwarzschild spacetime includes all kinds of crazy stuff, including a white hole and a second copy of Minkowski space that is not actually observable even if you do happen to find yourself living inside this spacetime. So are there strong physical or practical motivations for forming the maximally extended versions of spacetimes? Do they help us to understand things about the actual, physical universe that would otherwise be hard to understand, or are they just pure math? I've given my own possible answer below, but I don't know if it's optimal or even correct. Other answers would be welcome.


title: What is the importance of the Odderon?

text: From hep-ph/0001149v1: (1) an Odderon contribution is absolutely necessary to reproduce quantitatively well the data; while its presence is not explicitly needed at $t = 0$, its inclusion is necessary to have a good fit of the other $|t|$ data, specially in the dip region and in the high-$|t|$ domain. (2) hints are found that secondary structures (diffraction-like) develop in angular distributions with increasing energies at intermediate $|t|$ values in both $pp$ and $\bar{p}p$ angular distributions. * * * The questions that not clear to me: a. How the does the Pomeron contribution help us to explain the results (in specifically the $p\bar{p}$ dip)? b. Why is it necessary for high-$|t|$ domain (while less important for $t = 0$)? c. What are the "secondary structures" mentioned in (b)? better explanation is welcomed.


title: Why does this object periodically turn itself?

text: See this video about 30 sec in. http://www.youtube.com/watch?v=dL6Pt1O_gSE Is this a real effect? Why does it seem to turn periodically? Can it be explained by classical mechanics alone? Is there a simple equation that models this behaviour?


title: Air Regeneration in Closed Systems

text: I wonder what's the way to regenerate O2 in air without using consumable chemicals (where one can use electricity through electrolisis or using UV lamps)? We can dissolve water into O2 & H2, but it won't remove CO2 from air. Any ideas?


title: What exactly is electric field screening?

text: I've run into this concept of 'screening' several times before, but I admit I've never sat down and really thought about why the process works. My understanding of the effect is that mobile electric charge carriers (say in a fluid) are able to arrange themselves in such a way that reduces or effectively cancels out the charge from a particular particle at a large distance. To quote wikipedia: "For example, consider a fluid composed of electrons. Each electron possesses an electric field which repels other electrons. As a result, it is surrounded by a region in which the density of electrons is lower than usual. This region can be treated as a positively-charged "screening hole". Viewed from a large distance, this screening hole has the effect of an overlaid positive charge which cancels the electric field produced by the electron. Only at short distances, inside the hole region, can the electron's field be detected." I really struggle to see what is happening here: how can a region of relative neutrality 'cancel out' the field of an electron? Any help, insights, or suggestions for further reading would be greatly appreciated. Thanks


title: What are some good resources for learning how to apply vectors in physics?

text: Although I don't have any problems with vectors when using them in Mathematics but I am having a hard time using them in physics. It is really frustrating me.   Can you please recommend me some good resources for learning how to use vectors in physics?


title: Use of Operators in Quantum Mechanics

text: I understand the form of operators in use for quantum mechanics such as the momentum operator: $$\hat{\text{P}}=-ih\frac{d}{dx}$$ My question is in what ways can I use it and what am I getting back? For example: if I simply apply th momentum operator to the wave function $$-ih\frac{d}{dx}\Psi$$ Will I get an Equation that will provide the momentum for a given position? Or is that a useless mathematical thing I just did? If I use it to get an "expected value" by $$\langle \Psi | \hat{\text{ P }} | \Psi \rangle =\int_{-\infty}^\infty \Psi^* \hat{\text{ P } }\Psi$$ am I getting a number representing the probable momentum of that area integrated for? Or the percentage of total momentum there? Essentially is it a probability (if so of what kind?) or a value for the momentum? I'm trying to understand these basic things because it has always remained unclear. I'm using it to find the momentum of and electron INSIDE the potential energy barrier as it is tunneling, i.e. the electrons between a surface and a Scanning Tunneling Microscope.


title: What is the physical meaning of the affine parameter for null geodesic?

text: For time-like geodesic, the affine parameter is the proper time $\tau$ or its linear transform, and the geodesic equation is $$\frac{d^{2}x^{\mu}}{d\tau^{2}}+\Gamma_{\rho\sigma}^{\mu}\frac{dx^{\rho}}{d\tau}\frac{dx^{\sigma}}{d\tau}=0. $$ But proper time $\Delta\tau=0$ for null paths, so what the physical meaning of is the affine parameter for null geodesic?


title: Relativistic mass and imaginary mass

text: The (relativistic) mass of an object measured by an observer in the $xyz$-frame is given by $$m = \frac{m_{rest}}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}.$$ Mathematically $v$ could be greater than the speed of light, but the mass $m$ would become imaginary. Physically we would have to get to the speed of light first i.e. $v = c$, which gives us an undefined value for $m$. So we believe that nothing moves faster than the speed of light because we do not like observables to be imaginary?


title: Why does the law of increasing entropy, a law arising from statistics of many particles, underpin modern physics?

text: As far as I interpret it, the law of ever increasing entropy states that "a system will always move towards the most disordered state, never in the other direction". Now, I understand why it would be virtually impossible for a system to decrease it's entropy, just as it is virtually impossible for me to solve a Rubik's cube by making random twists. However the (ever so small) probability remains. Why does this law underpin so much of modern physics? Why is a theory that breaks this law useless, and why was Maxwell's demon such a problem? Does this law not just describe what is _most likely_ to happen in complex systems, not what _has_ to happen in all systems?


title: Inhomogeneous Effective Mass in a 2D Lattice

text: Consider a tight-binding square lattice in 2D. This lattice has two different nearest neighbor tunneling rates along the x and y directions; call them $J_{x}$ and $J_{y}$. All longer range tunneling rates are zero. How can I compute the response of the expected momentum of a very low momentum particle to a force applied along a vector n? Is the rate of change of momentum parallel to n? Where is the excess momentum going? I think the excess momentum is being lost to heating the environment. I think the rate of change of momentum IS parallel to the force applied along the vector n.


title: Expression of $\kappa_S$ in photon gas

text: I'm calculating some properties of the photon gas. That gas is defined by $$U=aVT^4$$ $$P=\dfrac{1}{3}aT^4$$ I calculated photon gas entropy: $$S = \dfrac{4}{3}aT^3V $$ I could calculate $c_V$. Also, I know that $c_P$ and $\kappa_T$ are not defined because it's impossible to change temperature at a constant pressure. Now I'm trying to calculate $\kappa_S$, defined as $$\kappa_S = -\dfrac{1}{V}\left(\dfrac{\partial V}{\partial P}\right)_S $$ This parameter looks defined, because, in principle, I can variate $P$ at constant $S$, that will have effect on $V$. It's difficult to write $V=V(P,S)$ so I'm trying to calculate it using: $$\kappa_S = -\dfrac{1}{V}\left(\dfrac{\partial V}{\partial P}\right)_S = -\dfrac{1}{V}\left(\dfrac{\partial V}{\partial T}\right)_S \left(\dfrac{\partial T} {\partial P}\right)_S = -\dfrac{1}{V}\left(\dfrac{\partial V}{\partial T}\right)_S \left(\dfrac{\partial P} {\partial T}\right)_S ^{-1} $$ That gives me a value of $$\kappa_S = \dfrac{27}{16} \dfrac{S}{Va^2T^7}$$ which looks really strange. I think I'm not calculating it correctly, and my reasoning in partial derivatives is not correct. My question is, is $\kappa_S$ really defined in the photon gas and which is its expression? What am I doing bad?


title: A question about T duality

text: Normally T-duality is introduced perturbatively by computing world-sheet spectrum of fundamental strings, and one of the conclusions is that it switches between momentum mode and winding mode of fundamental strings. My question is that does this switching extend beyond fundamental strings, namely, is it true that T-duality simply turns any momentum modes (for example, those carried by D-strings) into winding fundamental strings? If yes, is there a convenient way to see why is this the case?


title: Complex impedance triangle, why is there a complex part?

text: Why is the inductive reactance written as jωL? I know that multiplying by j rotates the inductive reactance on the j axis, but why do we do that? Why is it not on the same axis as the resistance?


title: Showing a fourth rank tensor in $\epsilon$'s reduces to one in the metric $g$

text: Consider the fourth rank tensor $$S_{\mu \nu \rho \sigma} = a(\epsilon_{\mu \sigma}\epsilon_{\nu \rho} + \epsilon_{\mu \rho}\epsilon_{\nu \sigma})f(x^2),$$ in 2D where $a$ is a constant and $f(x^2)$ is a Lorentz invariant function (not important for this question). I want to show this reduces to a linear combination of $g's$, that is to say $$S_{\mu \nu \rho \sigma} \equiv (a_1 g_{\mu \nu}g_{\rho \sigma} + a_2 (g_{\mu \rho}g_{\nu \sigma} + g_{\mu \sigma}g_{\nu \rho}))f(x^2)$$ This is in a context of 2D Conformal field theory, so each index takes only the value 1 or 2. To do this explicitly, write out all possibilities. When $\mu = 1,2$ we have that the original expression reduces to $$\epsilon_{1 \sigma}\epsilon_{\nu \rho} + \epsilon_{1 \rho}\epsilon_{\nu \sigma} + \epsilon_{2 \sigma}\epsilon_{\nu \rho} + \epsilon_{2 \rho}\epsilon_{\nu \sigma}.$$ Now I want to consider the values of $\nu=1,2$. I have been told that I should get another four terms by expanding the above taking into consideration the values of $\nu$. But it seems to me that I would have six terms. To illustrate my problem, I could obtain the additional terms $$\epsilon_{1 \sigma}\epsilon_{1 \rho} + \epsilon_{1 \rho}\epsilon_{1 \sigma} + \epsilon_{2 \sigma}\epsilon_{1\rho} + \epsilon_{2 \rho}\epsilon_{1 \sigma}$$ or $$ \epsilon_{1 \sigma}\epsilon_{2 \rho} + \epsilon_{1 \rho}\epsilon_{2 \sigma} + \epsilon_{2 \sigma}\epsilon_{2 \rho} + \epsilon_{2 \rho}\epsilon_{2 \sigma}.$$ But I notice that two of the terms are common in these expressions, so should I not have an expansion of the form $$\epsilon_{1 \sigma}\epsilon_{1 \rho} + \epsilon_{1 \rho}\epsilon_{1 \sigma} + \epsilon_{2 \sigma}\epsilon_{1\rho} + \epsilon_{2 \rho}\epsilon_{1 \sigma} + \epsilon_{2 \sigma}\epsilon_{2 \rho} + \epsilon_{2 \rho}\epsilon_{2 \sigma}?$$


title: What will happen if a plane trys to take off whilst on a treadmill?

text: So this has puzzled me for many a year... I still am no closer to coming to a conclusion, after many arguments that is. I don't think it can, others 100% think it will. If you have a plane trying to take off whilst on a tread mill which will run at the same speed as whatever the planes tyres rotation speed is will it take off? **[edited to be more clear]** The question is simple. Will a plane take off if you put this plane onto a treadmill that will equity whatever speed the plane wheels are moving at. So the plane should not be able to move. This is a hypothetical situation of course. But i am very interested.


title: hydraulic scissor jack lifting capacity

text: Can you please assist with a really simple question? I have a hydraulic scissor jack table with the following: 2 hydraulic cylinders, each with a cylinder stroke of 240mm, a cylinder bore of 50mm and a hydraulic pressure of 15Mpa. I need to calculate the lifting capacity of this table (ignoring the mass of the components). The rams drive a center pin of the scissor mechanism. I'm trying to approach this from two directions, either Fx=w/tan (theta) or P=mgh/Al. I've hit a blank - any clarity? Thank you


title: Electomagnetic Field Quantization

text: From Quantum Field Theory by Franz Mandl and Graham Shaw page 4. When we are expanding the vector potential as a Fourier series; $\renewcommand{\vec}[1]{\mathbf{#1}}\vec{A}(\vec{x},t) = \sum\limits_{k}\sum\limits_{r}(\frac{\hbar c^2}{2V\omega_k})^{1/2}\vec{\varepsilon_r}(\vec{k})[a_r(\vec{k},t)e^{i\vec{k}\cdot\vec{x}}+a_r^*(\vec{k},t)e^{-i\vec{k}\cdot\vec{x}}]$ I did not understand how we can determine the constant term $(\frac{\hbar c^2}{2V\omega_k})^{1/2}$. Thanks in advance.


title: How could we travel to the nearest supermassive Black hole?

text: I have just watched a trailer for the upcoming movie _Interstellar_ and started to wonder about some physics involved. In the end of the trailer, they are obviously plunging into a Black hole hoping to travel into a different universe. Let us for a moment forget about the fact that most Black holes just probably are not worm-holes. But a large problem is, how do they get there since the closest Black holes such as Sagittarius A* are several thousand light-years away? The Black hole at Sgr A* is surely massive enough for tidal forces not to pose a larger problem for the starship. My first guess of transport would be a warp drive, since in the trailer there is a ring around the starship resembling the one usually included in warp drive designs (see the NASA concept below). But for a warp drive we need exotic matter and how much exotic matter would we need to get through 26 000 light-years in few tens of Earth years (using also hibernation of the pilots)? Is there possibly enough exotic matter in the Solar system for such a transport? Or is there something I am missing such as a very large Black hole nearby (i.e. large enough for tidal forces not to rip the spaceship apart)? * * * EDIT: Luboš Motl pointed rightly out that Sgr A* isn't the closest black hole, this doesn't change the argument that the closest known black holes are still too far away. I also edited some of the wording - even primordial black holes would not be good wormholes, only exotic matter allows them to be traversable. (Thanks again to Luboš for the point.) Everybody seems to be very perplexed by the concept of exotic matter. Yes, it is a highly speculative idea which might not have any physical significance, thus eliminating the possibility of both wormholes and warp drive. But, it is not completely ruled out. We could e.g. speculate that the negative energy density is actually an underdensity of dark energy. Etc. etc. Either way, both traversable wormholes and warp drives are serious lines of (speculative) research not so different in empirical substantiality from e.g. all quantum gravity theories including the very much stressed string theory, so let us consider them at least for a while. ![warp drive starship](http://i.stack.imgur.com/j9t5k.jpg)


title: Different phase in capacitively coupled RLC circuits

text: I was trying to work with some data for a lab report I'm writing about capacitively coupled RLC circuits. The setup is pretty simple and looks like that: ![Schematic](http://i.stack.imgur.com/gURa1.jpg) Where $C^{'}$ is the coupling capacitance. I took some data using an oscilloscope, I was using a very low frequency square wave in order to see the exponentially decaying wave. Then tried to fit it to the theoretical function using Matlab. First I've tried to fit it to $e^{-\gamma t +\varphi_{1}}\cos\left(\pi F_{+}t+\varphi_{1}\right)\cos\left(\pi f_{-}t+\varphi_{1}\right) $ where $F_{+}=f_{+}+f_{-},\ F_{-}=f_{+}-f_{-}$ Because I though that I probably would need to add some constant time-shift. So I've tried using the same in every place that I had $t$. That didn't work, but it was quite reasonable because $t$ in the exponent has a $\gamma$ coefficient so I would expect its phase to be different, but that didn't work either. What did work after trying several permutations of placing the phases in different places was this $e^{-\gamma t + \varphi{1}}\cos\left(\pi F_{+}t+\varphi_{1}\right)\cos\left(\pi f_{-}t+\varphi_{2}\right)$. But I don't get why would one cosine and the exponent share the same phase while the other is totally different, this is not really part of my lab report but it would be nice to understand why this happens. The "good" fit: ![fit](http://i.stack.imgur.com/DiXE0.png)


title: What is the derivation of the formula for the probability of transmission through a barrier?

text: This site  has a formula for the transmission probability over a barrier (in transistors): $$ T \propto \exp [-2(2m^*/\hbar ^2)^{1/2}(q\phi)^{1/2} d ]. $$ Where $T$ is the transmission probability over barrier, $m^*$ is the effective mass of silicon, $\phi$ is the barrier height and $d$ = barrier width. I'm assuming $q$ is charge (of an electron), but it is not specified. The article does not state its sources, so I'm wondering how this formula is obtained, can anyone explain this equation or give a source?


title: Why longitudinal accoustic and optical branch have the same energy (and thus frequency)?

text: **Context** : Solid state physics, lattice dynamics. **Question** : At point $X$ the LO and LA branches have the same energy. Is this something that can be expected before doing the experiment or is it just it ? ![enter image description here](http://i.stack.imgur.com/BL8S1.png)


title: How does compression of carbon atoms work?

text: From what I understand, it's extremely difficult/ impossible to compress atoms closer together. I'm wondering how this applies in the case of graphite and diamond, where the carbon atoms adopt a different structure under different temperatures and pressure.


title: The maths/physics in "The Amazing Spider Man"

text: > **Possible Duplicate:**   >  What does Peter Parkers formula represent? I just saw the trailer for the new Spider Man movie and couldn't help but notice the image after the guy says "How did you come up with this". So here is a transcript (the last part can't be seen): $$\eqalign{ & \frac{{d\log \Phi }}{{dt}} = \alpha \left( {1 - g{{\left( {\frac{\Phi }{k}} \right)}^\beta }} \right) \cr & \Phi = K\sum\limits_i {\prod\limits_j {\exp \left( {\frac{{{q^j}{{\left( {1 - {E_a}} \right)}^{j - 1}}}}{{\left( {j - 1} \right)!\left( {1 - {{\left( {1 - {E_a}} \right)}^{j - 1}}} \right)}}} \right)} } \log a \cdots \cr} $$ ![enter image description here](http://i.stack.imgur.com/vTnmp.png) Does it resemble, make sense, relate to any real math? (I dont really see any $i$ index inside the expression for example). I actually laughed at the sigma, pi, exp triad. I put this in pshysics.SE from a sugestion from math.SE > On the other hand, the notation looks awfully like that of a (thermodynamic) > partition function, and thus this might be more appropriate as a question on > physics.SE instead of here...


title: How can a human eye focus on a screen directly in front of it?

text: I am asking this question here because I think the answer has something to do with the way light is bent as it's captured through the eye. I saw a show a while ago about tiny screens on contact lenses to pull up data on objects you see in the real world, I also just saw this article about Google testing the same idea with screens in the lenses of sunglasses. The part I do not understand is how your eye would focus on a screen that is so close. My eyes (and I believe most others) cannot focus on anything closer than a couple inches away. Yet the lenses on glasses are much closer to the eye than a couple inches. So I would guess they would have to use some special technology to separate the light rays in a way that your brain could make an image from it. If that's correct, how would it work? if not, how would they get your eye to focus on a screen so close?


title: x-ray interaction with atmosphere

text: Why x-ray are stopped by atmosphere while they are more energetic than UV or IR? They certainly interact with atmosphere but I can't understand which phenomenon stop them.


title: What are "packets"?

text: Paraphrased from Wikipedia: > Infrared sensing in snakes depends on a kind of natural thermography, by > which tiny **packets** of cellular water are raised in temperature by the > infrared radiation. What is "packets" refering to in this is example?


title: Conduction, convection, & radiation examples (besides vacuum flask)

text: I am studying heat transfer And understanding the vacuum flask example that involves **conduction, convection, and radiation** Do you know other example that involves these 3 phases? we can name some by category: CONDUCTION:   * Touching a stove and being burned   * Ice cooling down your hand   * Boiling water by thrusting a red-hot piece of iron into it CONVECTION:   * Hot air rising, cooling, and falling (convection currents)   * An old-fashioned radiator (creates a convection cell in a room by emitting warm air at the top and drawing in cool air at the bottom). RADIATION:   * Heat from the sun warming your face   * Heat from a lightbulb   * Heat from a fire   * Heat from anything else which is warmer than its surroundings. but what example involves all of them?


title: How universal gravitation falls short

text: As a non physicist I can understand how Newtonian mechanics falls short in cases of high velocity etc. and is properly generalized by the special theory of relativity. What is not clear to me is how universal gravitation falls short: under what conditions does it fail to make accurate predictions? What are the cases where the inverse square law actually fails to give accurate results and the general relativity is needed? I do understand that Einstein's theory provides the mechanism, whereas Newton's did not. Assuming a hypothetical system involving a black hole and a planet like the earth revolving around it (far beyond the horizon, of course) would universal gravitation fail to accurately predict the behavior of the system?


title: The interior of a cylinder as an Einstein manifold

text: The interior of a curved cylinder is an Einstein manifold (the Ricci Curvature Tensor is proportional to the Metric $R_{\mu\nu}=kg_{\mu\nu}$) since it has a constant curvature. Using the metric $$ {g_{\mu \nu }} = \left[ {\matrix{ { - 1} & 0 & 0 & 0 \cr 0 & {{{{C^2}} \over {{C^2} - 2{\pi ^2}{x^2}}}} & 0 & 0 \cr 0 & 0 & {{{{C^2}} \over {{C^2} - 2{\pi ^2}{y^2}}}} & 0 \cr 0 & 0 & 0 & {{{{C^2}} \over {{C^2} - 2{\pi ^2}{z^2}}}} \cr } } \right] $$ and where $C$ is the circumference of the cylinder. However, I was unable to derive the constant of proportionality $k$. May I know how one may find this constant?


title: Perfect elastic collision and velocity transfer

text: So my teacher told me that when you have two identical balls in a perfectly elastic collision, the first ball A will collide with B and afterwards A will stop and B continue. Why is this? Doesn't Newton's 3rd law imply both balls would get an equal force into opposite direction during the collision? And if A was heavier than B, does A continue in the same direction after elastically colliding with B (that's the only logical result I can think of if this is true).


title: Does gravitation of a sphere equal gravitation of a point?

text: Under Newtonian model of gravity, a perfect sphere creates the same gravitation field as a point mass in its center. General Relativity describes gravitation differently. How much this difference affect the above equivalence? If it does not hold, what kind of difference (qualitatively) there is?


title: Will a wheel rolling down a hill ever fall over?

text: Say I have a detached bicycle wheel and I roll it down an infinite, flat, asphalt inclined plane, in air (ie there will be a terminal velocity). When, if ever, will the wheel fall over? I've been wanting to try this and see, but I don't know of a suitable hill near my house. My second question is if someone can recommend such a hill in the Trenton, NJ, USA area.


title: Question about complex emissivity and complex permeability

text: Consider a linear, isotropic material. We can then write: $$\textbf{D}(\textbf{r},\omega) = \epsilon(\textbf{r},\omega)\textbf{E}(\textbf{r},\omega)$$ In phasor notation. The complex emissivity $\epsilon'$is defined as: $$\epsilon' = \epsilon - \frac{j\sigma(\textbf{r})}{\omega}$$ We can also write: $$\textbf{B}(\textbf{r},\omega) = \mu(\textbf{r},\omega)\textbf{H}(\textbf{r},\omega)$$ Where I call $\mu$ the complex permeability. I know now that a medium is passive (i.e. always absorbs the magnetic and electric energy that are stored in the fields) if the imaginary part of $\epsilon$ and $\mu$ are negative or zero (but one of them must differ from zero). A medium is active when the imaginary parts are positive or zero. I wonder if it is also the other way around: if the medium is passive, will the imaginary parts be negative or zero (but not both at the same time zero)? Or are the cases in which the medium is passive but the imaginary parts are not both negative (or zero)? I have the same question for active media, but I guess that is analogous.


title: Refraction Vector (Ray tracing)

text: I am doing ray tracing and I do the refraction of the ray using the following relation (I got it from PDF called "Reﬂections and Refractions in Ray Tracing"): ![enter image description here](http://i.stack.imgur.com/LqEOB.png) ![enter image description here](http://i.stack.imgur.com/MA8ii.jpg) But I have seen it in another PDF as follows:![enter image description here](http://i.stack.imgur.com/lLNvY.jpg) Could you please explain for me why? And how can I reassure that my refraction vector that I calculated is correct? Thanks


title: Type of stationary point in Hamilton's principle

text: In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all follow the path with the least resistance. However, a stationary point can be a maximal or minimal extremum or even a point of inflexion (rising or falling). Can anybody give examples from theory or practice where the action integral takes on a maximal extremum or a point of inflexion?


title: Calculating forces efficiently in Lagrangian formalism

text: I will Illustrate the question using an example problem: > We have a mass $m$ connected to a mass $m$ by a rod of length $l$, and also > to a mass $4m$ by another rod of length $l$. The rods are massless, the > masses are pointlike. The outer masses are positioned on a flat, slippery > plane in such a way that the rods make a right angle, so one of the masses > is in the air. What is the acceleration of the mass $4m$ immediately after > the system is released? I solved this problem successfully using Newton's laws (the answer is $g/9$). I then solved it by setting up the Lagrangian, using the Euler-Lagrange equations, and finally plugging in some special values (namely, the initial velocities are zero). I noticed it took significantly more effort, because I had to calculate the general equations of motion before plugging in the initial values; this was not necessary in the Newtonian approach. Is there a way to solve problems like this using the Lagrangian formalism, and avoid doing more work than necessary (i.e. to avoid calculating the equations of motion explicitly)? I find the Lagrangian approach very useful for deriving the equations of motion, but many problems do not really require that. ASCII sketch: (the arrow denotes the direction of the desired acceleration) ` m / \ / \ m 4m -->`


title: When does the correlator of a string of fields and the current vanish "sufficiently fast" at infinity and Ward's identity?

text: One consequence of the Ward identity (cf. Di Francesco et al) is that it means variation of correlators under infinitesimal transformation is zero. This can be seen by integrating the ward identity, and using the Gauss divergence theorem to integrate over the surface at infinity, and setting the surface term to zero. $$ \delta_{\omega} \langle j^{\mu}_a \Phi(x_1) \ldots \Phi(x_n) \rangle =\int \frac{\partial}{\partial x^{\mu}}\langle j^{\mu}_a \Phi(x_1) \ldots \Phi(x_n) \rangle = \int_{\Sigma} ds_{\mu} \langle j^{\mu}_a \Phi(x_1) \ldots \Phi(x_n) \rangle \\\ =0$$. DiFrancesco says the integrand goes to zero at infinity because the divergence of the correlator vanishes at away from the points $x_1 \ldots x_n$. How does the vanishing of divergence mean vanishing of the correlator? **In general when can we assume a correlator to vanish at infinity "sufficiently fast"?** In this particular case, from familiar examples of the free boson and fermion, I know that $j$ is proportional to the gradient of the fields, so does this mean we are considering field solutions which attain a constant value? **How is it different from solitonic solutions?** I have seen surface term to be vanishing sufficiently fast in many places in QFT, cannot seem to remember them now ,but this seems to be quite an adhoc assumption, is it because you want the drama of the theory to happen only in a finite range, and you cannot consider particles at inifinity?


title: Inverting the equation for $T_{\mu\nu}$ in terms of $F_{\mu\nu}$

text: The Stress-Energy Tensor for electromagnetism is given by: $$ T_{\mu \nu} = F_{\mu}\,^{\alpha}F_{\nu\alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} $$ How can I find $F_{\mu\nu}$ in terms of $T_{\mu\nu}$? Rewriting the above equation using: $$ T_{\mu\nu}=- F_{\mu \alpha} g^{\alpha\beta} F_{\beta\nu} + \frac{1}{4} g_{\mu \nu}g^{\alpha\beta}F_{\beta\delta}g^{\delta\gamma} F_{\gamma\alpha}$$ from which we can write the following $4\times4$ matrix equation for the three matrices $T,\,F,\,g$, where $T$ is symmetric, $F$ is anti-symmetric and $g$ is symmetric and invertible: $$ T = -F g^{-1} F+\frac{1}{4}\left(\mathrm{Tr}\, \left[g^{-1}Fg^{-1}F\right]\right)\,g$$ The only way I can think of is writing down 10 equations (as there are free components in $T^{\mu\nu}$) and then trying to find the 6 unknowns (as there are free components of $F^{\mu\nu}$). Is there a better way to do this?


title: Contra-rotating propellers torques

text: Please look at the following mechanism for contra-rotating propellers: YouTube video When a CCW torque acts on the upper gear and a same torque acts on the lower gear (both seen from above), the torques act against each other. But: does the whole system then move CCW or does it stand still (because the torques both cancel each other out)?


title: Explanation for transistor as amplifier

text: In transistor as an amplifier, we have NPN transistor (base in the middle and emitter and collector at the sides). The collector has electrons in majority because it is N-type. similarly, holes are in majority in base, but as it is lightly doped, it has very little amount of hole. Now, when a small signal of alternating current is supplied to the base, the base current changes and this change is called input current. most of this input current moves into the collector (base has less holes so most of the current moves on when it fills the holes). Now, the collector has its own majority charge carriers (electrons) and some more electrons are added to it by the input signal. So when collector gives current to the load, it will be the combination of input signal and the already present electrons. Hence, small signal is amplified. Am I right or wrong?


title: Parallel and series circuit problem

text: When two resistors, each of resistance 4.0 ohms, are connected in parallel with a battery, the current leaving the battery is 3.0 A. When the same two resistors are connected in series with the battery, the total current in the circuit is 1.4 A. Calculate the internal resistance of the battery. The solution started with $1.4 = \frac{E}{8.0 + r}$ and $3.0= \frac{E}{2.0+r}$ I understand the first part because the current in series is always the same, and they are considering the total resistance. But for the second part, I'm confused on why 3 A is used. It's 3 A when it leaves the battery, but didn't it already go through the internal resistor so it's not an accurate representation of the total current in the parallel circuit?


title: Question concerning the Lindhard function

text: I'm having a question concerning the Lindhard function. The reference I'm using is the standard text "Quantum Theory of Solids" by Charles Kittel. I'm concerned with Chapter 6, subchapter "Method of the self-consistent field". The goal is to solve the quantum mechanical eigenvalue problem of the Hamiltonian $H=H_{0}+V(x,t)$ where $V(x,t)=V_{0}(t)+V_{s}$ is a Hartree-Fock- type potential consisting of a time-dependent external potential (e.g. the source of an external electric field) and a screening potential $V_{s}$. $H_{0}$ is a standard kinetic-energy Hamiltonian. The problem is solved based on quantum statistical mechanics. We assume that the eigenstates of $|m \rangle$ and the eigenstates of $H_{0}$ are plane waves $|k \rangle$. The overall idea seems to be to find the density matrix time evolution to linear order expressed in terms of the plane wave states. Without alaborating more on the details of Kittels calculation, the point is that I don't why the "eigenfrequencies" (what eigenfrequencies does he mean?? The diagonal elements of the density matrix when the density matrix is diagonalized) are the zeros of $\epsilon(\omega,q)$. I don't see why this is the case. Furthermore I don't understand why there are two types of eigenfrequencies. Obviously the Plasma frequency produces a zero of $\epsilon(q,w)$ at small $q$ but why is $\omega = \epsilon_{k+q} -\epsilon_{k}$ a zero. To me this gives a $2\omega$ in the the denominator of the dielectric function. I'm a bit confused. Can someone maybe address my question and summarize briefly the key idea of the whole thing? This would be really helpful!! :)) I'm looking forward to your responses!!


title: Do all the particles acquire mass in the Standard Model due to the Higgs mechanism only?

text: I know that a mass term for an intermediate boson is not compatible with the gauge symmetry. But in principle a mass term for the electron field does not violate a gauge symmetry. However to build an electroweak theory consistent with the observation of the non-conservation of the parity of the neutrino, the electron mass term could not be included and it also acquire mass due to the Higgs mechanism. Are there some Standard Model particles having an explicit mass term, or all acquire mass as a result of spontaneous breaking of the gauge symmetry and its coupling with the Higgs field?


Why is commutation relations the first step in quantization?


title: How to derive the scale factor for special conformal transformation?

text: By definition a conformal transformation of the coordinates is an invertible mapping $x\rightarrow x'$ which leaves the metric invariant upto a scale factor: \begin{equation} g_{\mu\nu}'(x') = \Lambda(x) g_{\mu\nu}(x) \end{equation} I am facing problems in deriving the scale factor for special conformal transformation (SCT). This transformation has the finite form given as: \begin{equation} {x'}^{\mu} = \frac{x^{\mu}-b^{\mu}x^2}{1-2(b.x)+b^2x^2} \end{equation} I was trying to derive the scale factor for this transformation using the metric condition: \begin{equation} g_{\mu\nu}'(x') =\frac{\partial x^\sigma}{\partial {x'}^\mu}\frac{\partial x^\rho}{\partial {x'}^\nu}g_{\sigma\rho}= \Lambda(x) g_{\mu\nu}(x) \end{equation} I was not able to find out $\Lambda(x)$, however the final form of scale factor $\Lambda(x)$ is given in the book as: \begin{equation} \Lambda(x) = (1-2b.x + b^2x^2)^2 \end{equation}


title: Are different frequencies of light lensed differently during gravitational lensing a bit like refraction?

text: So I was wondering about the event horizon on a black hole. And wondering if the point of no return for radio waves vs gamma rays would be different. I guess the logic being, since gamma rays have more energy than radio waves, their point of no return might be different. But I'm not sure if gravity has the same affect on all frequencies of electromagnetic radiation. I'm sure the answer for this would lie in gravitational lensing, ie are different colours of light lensed by the same amount? or do higher frequencies lens less?


title: Seeing The Light in a Black Hole

text: I assume if light can't escape a black hole, then light is in a black hole. Does the light 'shine' inside a black hole


title: Historical analysis of light interference - difference frequencies

text: It is well-known that light of two different frequencies illuminating a detector will produce an output with a component at the difference frequency. While such considerations are eminently useful (various heterodyne measurement techniques) it has not been practical until lasers came along. Regardless of practicality, the effect has been known for a long time. The question is: how long? And, specifically, can you provide a reference to an analysis which establishes it? A web-accessible reference, for preference. I'd expect it was determined shortly after Maxwell's equations were established, but I could be wrong. Ideally, this would be an acknowledged seminal paper, comparable to Skutt's explanation of the color of the sky.


title: Can a hydrogen atom sit still?

text: Can we take a single lone hydrogen atom (proton and electron?) and set it in one place and observe it staying there? How about just a proton? A neutron? How about just an electron?


title: Will an electron gun fire behind it?

text: If an electron travels as a wave, can it therefore not be aimed with any precision? If you fire an electron gun, can't you aim it at a particular slit? If the electron travels as a wave, will an electron gun hit anything at a 180 degree angle from the barrel? 360 degrees?


title: Effect of crystal growth on its scintillation

text: How does crystal growth affect its scintillation properties? What are possible ways in which growth of CaWO crystals grown via Chochralski method could be modified to avoid non-linearity in its scintillation output with respect to incident Gamma rays? Apparently this is relevant for rare event searches (eg. CRESST) using these scintillating crystals as detectors.


title: Can scattering amplitudes be simplified with 1PI diagrams?

text: I have been teaching myself quantum field theory, and need a little help connecting different pieces together. Specifically, I'm rather unsure how to tie in renormalization, functional methods, and the calculation of scattering amplitudes. Now, I understand how the LSZ formalism is used to link scattering amplitudes (i.e. $\langle f\vert i\rangle$) to the calculation of time ordered products, which can be calculated in two different ways:   1. In the interaction picture, with Wick's theorem and the Feynman rules, by expanding $T \exp(i\int dt \ H_I)$   2. Path Integrals methods, that is, taking functional derivatives of the generating functional $Z[J] = \int \mathcal{D}\phi \ e^{iS[\phi,J]}$ (collective index on $\phi ^a$). In particular, we may define a generating functional for 1PI diagrams by defining $Z[J] = \exp{W[J]}$, and then taking a Legendre transform to obtain the effective action $\Gamma [J]$. We then see that all the connected correlation functions can be built out of the 1PI correlation functions by constructing Legendre trees. But, if we get the correlation functions, then, can't we just employ the LSZ formula to find the scattering amplitudes, and so we need to calculate fewer diagrams? I cannot see how to do this myself. The other question, regarding renormalization, is similarly motivated. Now, I understand that, when considering higher corrections to scattering amplitudes, divergent integrals often arise, which can be regularized in a host of different methods (dimensional regularization, introducing a cutoff, etc.). If the the theory is renormalizable, we find that, with the addition of a finite number of counterterms to the Lagrangian (corresponding to the terms already present), we may "redefine" our fields and coupling constants so that everything is under control. This much I am comfortable with. What I don't understand is how this fits in with the general procedure of calculating scattering amplitudes. For instance, suppose I've calculated the amplitude for a typical, tree-level process for the exchange of one photon between two electrons. In the higher correction, I put a photon "wiggle" over an external electron line (that is, before the exchange process, an electron emits and reabsorbs a proton). Suppose I already regulated that electron propagator - which is 1PI - when it had that wiggle. For my higher order diagram, can I simply (in momentum space) multiply the original diagram with my regulated electron propagator? I do hope you can help me resolve these issues. Thanks in advance.


title: EM field from the electrons on the surface of a body and their influence to moving particles

text: Under this question quantum field theoretic models of decoherence @DanielSank gave a wonderful answer. Can I implement his answer to every material body? What about I'm thinking is that on the surface of every body are electrons which make allways a EM field. And thats way particles moving not far away from the bodies shap are influent with this EM field?


title: Gross-Neveu model analytic solution

text: I need to find an analytic solution via asymptotic expansion for the following system of equations: \begin{align} & i(u_t+u_x) + v = 0 \\\ & i(v_t-v_x) + u = 0 \end{align} \begin{equation} u(x,0) = Ae^{-x^2} \hspace{0.1 in} v(x,0) = -Ae^{-x^2} \end{equation} I uncoupled them \begin{align} & v_{tt}-v_{xx} + v = 0\\\ & u_{tt}-u_{xx} + u = 0 \end{align} Wrote the solutions in terms of fourier series \begin{align} & u(x,t) = \int_{-\infty}^{\infty}U(k,t)e^{-ikx}dk\\\ & v(x,t) = \int_{-\infty}^{\infty}V(k,t)e^{-ikx}dk \end{align} Came to the following differential equation \begin{align} & V_{tt} + V(1+k^2) = 0\\\ & U_{tt} + U(1+k^2) = 0 \end{align} found initial conditions for the derivatives by using the original equations and initial conditions \begin{align} u_t(x,0) = Ae^{-x^2}(2x-i) \hspace{0.2 in} v_t(x,0) = Ae^{-x^2}(2x+i) \end{align} Now I need to solve \begin{align} & u(x,t) = \int_{-\infty}^{\infty}\left[\left[\frac{-iAe^{-\frac{k^2}{4}}\sqrt{1+k^2}}{2k\sqrt{\pi}}\left[k -1\right]\right]\text{sin}\left(\sqrt{1+k^2}t\right) + \left[\frac{Ae^{-\frac{k^2}{4}}}{2\sqrt{\pi}}\right]\text{cos}\left(\sqrt{1+k^2}t\right)\right]e^{-ikx}dk \notag\\\ & v(x,t) = \int_{-\infty}^{\infty}\left[\left[\frac{iAe^{-\frac{k^2}{4}}\sqrt{1+k^2}}{2k\sqrt{\pi}}\left[k +1\right]\right]\text{sin}\left(\sqrt{1+k^2}t\right) + \left[\frac{-Ae^{-\frac{k^2}{4}}}{2\sqrt{\pi}}\right]\text{cos}\left(\sqrt{1+k^2}t\right)\right]e^{-ikx}dk \notag \end{align} I changed the sins and cosines to their exponential forms and tried to use the method of stationary phase to find a solution. However my solution only contributes to x = 0. Any idea how I would find the asymptotic expansion of this? I need to ultimately find the large t behaviour of this integral: \begin{equation} I = \int_{-\infty}^{\infty}F(k)e^{i\sqrt{1+k^2}t-ikx}dk \end{equation} Except the only point of stationary phase is at k = 0 which eliminates the x dependence.


title: Generator phase noise influence on its averaged signal

text: I need to estimate averaged signal of a generator with known phase noise and the amplitude noise can be neglected. The averaged signal of the generator is expressed as: $$ \langle A\exp{i(\Omega t + \varphi)}\rangle + \mathrm{c.c.} = Ae^{i\Omega t} \langle \exp{i\varphi} \rangle + \mathrm{c.c.}, $$ with $\Omega$ the generator carrier frequency and $\varphi$ the phase noise, $\langle\varphi\rangle = 0$. I suppose that phase noise is stationary. Its spectral density $ S_{\varphi\varphi} = \frac 1{2\pi} \int_{-\infty}^\infty\mathrm d\omega \, e^{-i\omega t} \langle \varphi(0)\varphi(t) \rangle$ is given. To obtain $\langle \exp{i\varphi} \rangle$ I suppose $\varphi$ to have Gaussian statistics. It is easy to show for that case that $$ \langle \exp{i\varphi} \rangle = e^{-\frac{\langle\varphi^2\rangle}2}. $$ Then, one finds $$\langle\varphi^2\rangle = \int_{-\infty}^\infty\mathrm d\omega \, S_{\varphi\varphi}(\omega). $$ I've taken $S_{\varphi\varphi}$ of the Agilent N5173B at 20GHz carrier, see the picture below and the generator's datasheet. (Note the two-sided phase noise spectral density is called $\mathcal L$ in industry, see the IEEE standart.) ![Agilent N5173B phase noise ](http://i.stack.imgur.com/qd2p6.png) I approximate the integral of the spectral density as the square under the connected points shown in the table: $$ \begin{array}{|r|r|} \hline f, \mathrm{Hz}& 1& 10& 100& 10^3& 10^4& 10^5& 10^6& 1.1\cdot10^6& 1.4\cdot10^6\\\ \hline S_{\varphi\varphi}, \mathrm{dBc}/\mathrm{Hz}& -40& -50& -80& -90& -95& -95& -125& -130& -135 \\\ \hline \end{array} $$ First plotting the spectral density and approximating its integral by hand, than using Python scipy trapz integration, I obtain $$\sqrt{\langle\varphi^2\rangle} \sim 10\,\mathrm{rad}$$ which is hard to believe for such a costly generator. And, according to the abovesaid, it will actually diminish the generator output, as $e^{-\frac{\langle\varphi^2\rangle}2} \sim 0.01$. So, 0) What the hell? 1) Probably I've misunderstood the IEEE definitions? I guess there should be some narrower frequency cutoffs? 2) Is it valid to assume the phase noise of a generator to be gaussian? 3) Probably better approaches exist for estimating the averaged signal of a generator with phase noise?


title: The Linear acceleration in Gyroscope

text: I am trying to understand theory behind the Gyroscope. I found this article that explains things in much detail. But I am unable to understand how that perpendicular acceleration components (a1, a2) are coming on both side of the disk. The text only says that it is coming from the combine effects of the two angular velocities. But I need to know how? And why is it perpendicular? Let us consider a point on the circumference on top half. Now where are the two acceleration components , resultant of which is a1? ![enter image description here](http://i.stack.imgur.com/L00pj.png)


title: Definition of a field line?

text: Ok so I finished by A-levels last year (english exams 18 year olds take) and we defined in my physics course we defined field lines (for an electric field) as: The path a free positive test charge would follow if acted upon no other force but the force due to the field itself. Reading more in to it more advanced text books (and the internet) seem to ignore this and not even mention positive charges or they paths they take. Saying that field lines are simply the direction of the electric field around the charge in question. So here is my question, is the first definition (the path a free positive ...) correct? If it is not correct what path would a free positive charge take in an electric field. Thanks.


title: Conformal compatification of Minkowski and AdS

text: How do I show that the compactification of Minkowski is given by the quadric $$uv-\eta_{ij}x^{i}x^{j}=0$$ with an overall scale equivalence in the coordinates.I get that for $v \neq 0$, the surface can be parametrized with the Minkowski coordinates. Now for $v=0$, I can have arbitrary values of $u$, which means basically two values, $ u=0 $ and $u\neq 0$. So are the infinities mapped to these points ? After that is it obvious that the conformal group acts on the space time defined by the quadric ? From, $$uv-\eta_{ij}x^{i}x^{j}=1$$ if I have to show that the boundary of $AdS_{d+1}$ is Minkowski in $d$ dimensions, how do I take the limit ?


title: Dual methods in structure optimization

text: I am studying _Structure Optimization_ , and the teacher took a few weeks to detail dual methods, but I don't get why it is useful. From an optimization problem with `n` design variables and `m` active constraints, the dual methods introduce an equivalent `dual problem` with `m` variables. If I understand correctly, these methods would allow faster resolution if `m` is far less than `n`. However, in `structural analysis` (this course is using FEM methods), I would expect to have one constraint per element, and only a few design variables. For instance, I think it is usual to have a large number of elements to mesh a piece, but productions methods may only allow a constant thickness in the piece, or other constraints may limit the number of actual design variables. Hence I would expect to have a much larger `m` than `n`. Is something wrong with this? Also, I tried to look for publications about dual methods in this field, but I found nothing related. Is there a paper I could read about this subject?


title: Circular polarization of variable-frequency light by 3D cinema glasses

text: A dominant method to obtain 3D images in the cinemas seems to be circular polarization. Separate pictures are projected with (alternating) circular polarization filters and passive glasses of the viewers block the wrong images for each eye (left-handed is OK for the left eye and right-handed is OK for the right eye, or vice versa). This allows the viewer to rotate his head, something impossible for linear polarizations, and it allows the glasses to be very cheap. As far as I understand, all methods responsible for creating or filtering circular polarization are based on linear polarization filters with some quarter-wavelength "gaps" in between. But this only seems to work well for a fixed wavelength. How do the glasses manage to properly filter circular polarized of any color, i.e. variable wavelength? Or are the images projected so that only three particular frequencies – "red, green, blue" – are projected for which it just happens that the glasses work with the right $N+1/4$ multiple of the wavelength?


title: How Can Anything Escape A Supermassive Black Hole?

text: Some of my questions haven't been well received, because I seem like I'm challenging well respected theories without having done the proper university training and study. I have been studying this online as a hobby for over five years though. I have a problem with the idea of Hawking radiation. If you have a black hole such as the one in the center of our galaxy, with the gravity of tens of thousands of suns, how can a photon or anything possibly escape that gravity? I've heard that Hawking radiation supposedly happens through quantum tunneling, because we can see sub-atomic particles popping up here and there in labs, but if a photon managed to "tunnel" it's way out of the event horizon of a black hole with such amazingly huge gravity, through some other dimension where gravity doesn't have any effect, then wouldn't it appear just outside the event horizon, and be sucked straight back in anyway? If it didn't go straight back in, it would be caught in some sort of orbit of the black hole, or bump into the material that was surrounding it, wouldn't it? I don't understand how it could be possible for anything to ever escape a black hole of that magnitude. How far have they observed quantum particles tunneling in the lab? The event horizon would be many kilometers, and anything that popped up in that space would go straight back in.


title: How to measure area under the waveform (electric current)?

text: I have a current waveform from my circuit which goes up and down the x-axis. x-axis represents time and my y axis represents current. How should I calculate total area under the curve? Should I take the absolute for negative part of current or just waveform integral will work?


title: History of the names "Feynman-gauge" & "Landau-gauge". How arised & how settled?

text: Warning: Students, stay away from antiquities. The aim to learn is to survive. Hi. Today the nomenclatures _Feynman gauge_ and _Landau gauge_ seem established, but could you explain the history? It's two-fold: 1. Who used first such gauges and how? 2. How did the terminology gain ground? It also inevitably asks some history of the Lorenz gauges in QFT. In particular, when was it realized that the freedom (and need) of the gauge parameter ξ lies in quantum theory? First I apologize that I have only limited access to literature on asking such questions. My preliminary studies revealed below. 1 **Terminology in literature** (Schweber 1961) is not gauge-parameter conscious, and "Lorentz[sic] gauge" (in Feynman gauge) is opposed to coulomb. But it also refers to today's Yennie gauge in sec 15g, saying (Fried and Yennie 1958) found it is possible to take gauge where the photon propagator ∝ $g + 2pp/p^2$. In (Nakanishi 1966), the word "Landau gauge" is seen, and it cites several near-past articles regarding Landau gauge quantization. (It is an important paper in canonical quantization in Landau gauge, together with (Lautrup 1967). Nakanishi was a strong proponent of the Landau gauge.) In p. 74 of (Nakanashi 1972), it reads "Feynman gauge or Fermi gauge" and "Landau-Khalatonikov[sic] gauge, or simply Landau gauge". (Landau and Khalatnikov, 1955) is listed in the bibliography section, but I couldn't find which part of Nakanishi actually cites it. (Nakanishi 1972) is a review article, one of whose main topics is canonical quantization of EM field in arbitrary Lorenz gauge, i.e., for any gauge parameter. In p 134 of (Itzykoson & Zuber, 1980), the words "Feynman gauge" and "Landau gauge" are used. Were the names settled at that time? Hmm, in p 389 of (Siegel 1999), "Fermi-Feynman gauge" is introduced. (Srednicki 2007) uses the word "$R_\xi$" for QED, remarking "[it] has been historically used only in the context of spontaneous symmetry broken [...] but we will use it here as well." 2 **Gauge parameter symbol** ξ is now usual. Is it due to (Fujikawa, Lee and Sanda 1972)? For other symbols, I mention $\alpha$. (Nakanishi 1972) uses it, and even after (Fujikawa, Lee and Sanda 1972) it is sometimes used, for example and in (Siegel 1999). 3 **Theory Timeline** 1930 - Fermi: P. 240 of (Schweber 1961) says Fermi proposed to add $-\frac{1}{2} (\partial A)^2$ to the Langrangian. (Fermi was the first to introduce a subsidiary condition, but it was not perfect. See also Gupta and Bleuler below.) Although I haven't checked Fermi's papers, it may be better to call "Fermi-Feynman(-'t Hooft) gauge." 1948 - Feynman: Feynman simply justifies the use of Feynman gauge in the section 8 of (Feynman 1949). Before Feynman, it wasn't Lorentz covariant, and transverse photons were separated. Feynman says it's not necessary, and it's ok to do $\gamma^\mu$...$\gamma_\mu$. 1950 - Gupta & Bleuler: They say Gupta and Bleuler succeeds in covariant canonical quantization in Feynman gauge, by discovering the correct subsidiary condition. 1956 - Landau & Khalatnikov: See (Nakanishi 1972) above. 1958 - Yennie gauge: It is said (Fried and Yennie 1958) uses the "Yennie gauge" of today, $\xi = 3$, in bound state problems. Early or mid 60's - Rise of interest in Landau gauge? See (Nakanishi 1966) above. 1966 - 67 Nakanishi & Lautrup: canonical quantization of EM field for any ξ. 1967 - Faddeev & Popov 1971 - 't Hooft: In 1971, 't Hooft used "Feynman-'t Hooft gauge" or simply "'t Hooft gauge" for broken gauge symmetry. (Fujikawa, Lee and Sanda 1972) generalized to any ξ. Its abstract uses the word "Feynman-'t Hooft gauge". (According to Weinberg. Haven't read both two.) 1972 - Still canonical quantization for arbitrary ξ is of interest, including massive vector field. See (Nakanishi 1972). 4 Loren't'z gauge (misspelling) You may know that in the 20th century, the common spelling was "Lorentz gauge", with extra "t". I couldn't find any exceptions at my hand. The turning point might be the errata of Peskin & Schroeder. Srednicki and Siegel spell it correctly. 5 **Bibliography**   * Fermi, E., Atti. Acad. Lincei. **9** (1929) 881, Atti. Acad. Lincei. **12** (1939) 431, Rev. Mod. Phys. **4** (1932) 87.   * Feynman, R., Phys. Rev. **76** (1949) 769.   * Fried H.M., Yennie, D.R., Phys. Rev. **112** (1958) 1391.   * Fujikawa, Lee and Sanda, PRD **6** (1972) 2923   * Itzykson & Zuber "Quantum field theory", 1980.   * Landau, L. D., Khalatnikov, I. M., J. Exper. Theor. Phys. USSR 29 (1955), 89 [English translation: Sov. Phys. J. E. T. P. 2 (1956), 69].   * Lautrup, B., Mat. Fys. Medel. Dan. Vid. Selsk. 35 (1967), No. 11.   * Nakanishi, N., Prog. Theor. Phys. **35** (1966) 1111 (Downloadable gratis)   * Nakanishi, N., Prog. Theor. Phys. Supple. **51** (1972) 1 (Downloadable gratis)   * Schweber, "An introduction to relativistic quantum field theory", 1961.   * Siegel, W., "Fields", arXiv:hep-th/9912205   * Srednicki, (2007) _Quantum field theory_ 6 **Revisions of this question** 26 Jun: Added the post 't Hooft era and the symbol ξ.


title: Forces on a simple flat slope

text: ![enter image description here](https://docs.google.com/drawings/pub?id=1cm2nq5OKck4zKphMYgMIJuCREZDO9JlNriRA0_4pUTc&w=400) This diagram is of a block (b) moving down a ramp unaided. Taking gravity (g) as $9.8ms^{-2}$, and the mass of the block (b) as $0.2kg$, how can I find the force acting along the slope (f). BTW: the ramp is inclined 1.8 degrees, not radians. I thought I could use basic trigonometry, and $F=ma$: $$ Sin(1.8) = \frac{0.2*9.8}{f} $$ $$ f = \frac{0.2*9.8}{sin(1.8)} $$ However this results in: $f = 62.40$ Seeing as force is a vector I thought I could use trig to split into it's components but obviously I'm doing something wrong. (the answer should be ~0.06N) Sorry for the noob question, but it's got me clueless.


title: Conformal/trace anomaly and index theorem

text: I am reading the chapters on characteristic classes and the index theorems in Nakahara. It is proven in the text that any chiral or gravitational anomaly $\mathcal{A}$ is given by $$\mathcal{A}=\int I^1_{2r}$$ with $I^1_{2r}$ given by the descent equation, $$I_{2r+2}=d I_{2r+1}$$ $$\delta_{\mathrm{gauge}}I_{2r+1}=dI_{2r}^1$$ The various $I$'s are related to the theory of the characteristic classes and Chern-Simons forms. On the other hand, the trace anomaly cannot be written in this way, at least I cannot see how. In $2$ and $4$ dimension we have (see for instance Duff), $$(T_2)^\mu_\mu=cR$$ $$(T_4)^\mu_\mu=cW^2+aE_4+fF_{\mu\nu}F^{\mu\nu}$$ where $W^2$ is the Weyl tensor squared and $E_4= R_{\mu\nu\rho\sigma}^2-4R_{\mu\nu}^2+R^2$ (with some numerical coefficient) is the Euler density. This got me thinking: is there a geometric way to describe the anomaly, i.e. to write a descent equation and the index theorem for those anomalies? Any pointers to papers, lecture notes, books or other resources are most welcome!


title: The diode of the lowest voltage

text: Which diode (semiconductor material) has the lowest voltage in the direction of conductivity?


title: What happens to the temperature of the water when you add salt to a bowl of melting ice?

text: If you have a bowl of ice that's melting, so the ambient temperature is just above 0 °C, what happens to the temperature of the water when you add salt? I know that the freezing point of salt water is less than 0 °C (it goes down to around -21 °C for a fully saturated solution), but that's not what this question is about. There's no change in the ambient temperature. Does adding the salt create a reaction that adds or removes heat? or does the temperature just remain the same?


title: Why don't different air masses mix immediately?

text: In meteorology, the atmosphere is considered to be divided into air masses, regions of relatively uniform temperature and humidity with fronts on their borders. But why doesn't the air from different air masses mix together when they meet? I can't find the answer in any weather book, and nobody seems to ask. People just accept the answer that different densities don't readily mix.


title: Canonical momentum operators in curvilinear coordinates

text: What is the quantum canonical momentum operator corresponding to arbitrary canonical position. For example, in Cartesian coordinates ($x^i$), the canonical momentum operator with respect to each $x^i$ is $-i\hbar \frac{\partial}{\partial x^i}$. For arbitrary canonical coordinates $q^i$, would the corresponding canonical momentum operators just be $-i\hat{\mathbf q}^i\cdot\hbar \nabla$?


title: Can traditional steam (water vapour) exist in a vacuum?

text: Can steam (water vapour) exist in a vacuum, and if so does it look and behave the same as in air? Let's assume the actual case is a kettle boiling in a vacuum. Note that I'm talking about a gas-less vacuum, not truly empty space devoid of all particles except the water.


title: Is this simulation following real physics?

text: I am trying to simulate a game in Box2D(Physics engine). The game that I am trying to simulate is very simple and can be found here: http://www.makaimedia.com/#/speartoss What I want to know is that, is the spear in that game following real physics principles? In other words, if I went out and threw a spear, will it behave like the spear in this game? Specifically, I am taking about the rotation of the spear. In the game, the spearhead is point up first half of the trajectory and as soon as it hits the vertex, the spear head starts point down. I am simulating a spear in Box2d physics engine. The density of the spear is 15x more than the density of the handle. I set an angle (between 0, 90 degrees) and give it a linear velocity like this:                   b2Vec2 force;         force.Set(cos(angle) * 22.0f , sin(angle) * 22.0f);         _spearBody->SetLinearVelocity(force);      What happens is, the spearhead never points downwards. It stays at the exact same angle throughout the trajectory. Is this more realistic or the physics in speartoss game? Thanks.


title: High and low pressure area and raining

text: In the high-pressure area it is mostly likely that there is sun. In low pressure area it is mostly likely that rain will occur. Because of the law that `p * V = (c) * T` where `p` is pressure, `V` is volume, `T` is temperature and `(c)` is a constant value, for the same `T` temperature, lowering pressure `p` should lead to increase volume of the gas (air in this example). Ant vice versa: lowering volume `V` leads to increase `p` pressure. The gas should be guided to condensation and the rain. So using this basic formula high pressure area should be more likely to cause rain. Where am I thinking wrong? Are there too many simplifications?


title: Where do black holes go?

text: According to many physicist what ever goes near to a black hole is pulled by it sucked up in a parallel universe. As far as i know black holes are also made up of material particles as any other thing is made up of so why isn't it sucked up in the parallel universe. And if it is true that it sucks up everything and teleport's it to a parallel universe so it should be very bright on the other side as also pulls light towards it. So why don't our universe have those type of white holes?


title: Why aren't there compression waves in electromagnetic fields?

text: I just started learning about optics, and in the book I'm reading they explain how the electrical field caused by a single charged particle could be described by a series of field lines, and compare them to ropes, to provide an intuition of the concept. Then they say that and that if we wiggle the particle up and down, that would produce transversal waves in the horizontal field lines, but no waves in the vertical lines. I know that the physical analogy is not to be taken literally, but I don't understand why wouldn't that cause compression waves in the vertical lines. I mean, even though the direction of the field in the points directly above and below the particle doesn't change, the intensity does. And I assume it wouldn't instantly. So what am I missing?


title: When is the Hamiltonian of a system not equal to its total energy?

text: I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical interpretation in such a case?


title: Relativity in Real Life

text: I have no formal training at all in relativity. But I was just wondering if our perception of time is altered by different events. Here are some examples I had in mind:   * If I touch a stove for 10 seconds versus enjoying chocolate for 10 seconds would I experience a time distortion? An observer who records the 10 seconds would have an objective sense of time.    * If I ran "very quicky" for 10 minutes would I experience a "slowing down" of time since I am exerting more effort? Do these examples have nothing to do with relativity? In "real relativity" do things actually slow down when we approach the speed of light? For example, if I was going near the speed of light would I notice the hand on my watch go by slower?


title: Electric Field-Drift Velocity

text: Why is the net velocity of electrons (drift velocity) considered to be zero, before setting up any electric field? I mean in the derivation of the equation drift velocity: $$ v =-\frac{Eet}{m} $$


title: A question about a relation between time-like world lines

text: Given any Lorentzian manifold containing three distinct time-like world lines $L$, $A$ and $B$ such that   $L$ and $A$ have exactly one common coincidence event, $\mathcal{E}_{AL}$,   $L$ and $B$ have exactly one common coincidence event, $\mathcal{E}_{BL}$, and   $A$ and $B$ have no coincidence event in common at all (therefore $\mathcal{E}_{AL} \ne \mathcal{E}_{BL}$),   is the following conclusion correct?: There also exist four distinct time-like world lines $F$, $J$, $P$, and $U$ in the given Lorentzian manifold, which have the coincidence event $\mathcal{E}_{AL} \equiv \mathcal{E}_{ALFJPU}$ in common and   there exist another four distinct time-like world lines $G$, $K$, $Q$, and $V$ in the given Lorentzian manifold, such that (1)   $F$ and $G$ have the coincidence event $\mathcal{E}_{FG}$ in common,   $J$ and $K$ have the coincidence event $\mathcal{E}_{JK}$ in common,   $P$ and $Q$ have the coincidence event $\mathcal{E}_{PQ}$ in common, and   $U$ and $V$ have the coincidence event $\mathcal{E}_{UV}$ in common, (2)   there is no time-like world line at all having taken part in any two (or more) of the five coincidence events $\mathcal{E}_{FG}$, $\mathcal{E}_{JK}$, $\mathcal{E}_{PQ}$, $\mathcal{E}_{UV}$, or $\mathcal{E}_{BL}$   (therefore the coincidence events $\mathcal{E}_{FG}$, $\mathcal{E}_{JK}$, $\mathcal{E}_{PQ}$, $\mathcal{E}_{UV}$, and $\mathcal{E}_{BL}$ are all distinct; and the nine time-like world lines $F$, $G$, $J$, $K$, $P$, $Q$, $U$, $V$, and $L$ are all distinct from each other), and such that (3)   for any coincidence event $\mathcal{E}_{WX}$ in which any time-like world line $W$ took part which also took part in coincidence event $\mathcal{E}_{BL} \equiv \mathcal{E}_{BLW}$ there exists a time-like world line $Y$ which took part in coincidence event $\mathcal{E}_{WX} \equiv \mathcal{E}_{WXY}$ as well as in one of the five coincidence events $\mathcal{E}_{FG}$, $\mathcal{E}_{JK}$, $\mathcal{E}_{PQ}$, $\mathcal{E}_{UV}$, or $\mathcal{E}_{AL}$   ?


title: Why was the truncated icosahedron (i.e. soccer ball) geometry chosen for the implosive lenses in the “Fat Man” atomic bomb?

text: Quoting from Wolfram Mathworld: " It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195)." but further insights are not provided. (http://mathworld.wolfram.com/TruncatedIcosahedron.html) Why would the Manhattan project scientists choose this geometry for designing the implosive lenses of the Fat Man atomic bomb? Assuming the explosive charges were placed on the faces of the icosahedron, is this the optimal geometry for focusing the shock waves on a single point (presumably a spherical plutonium core), or was this mainly due to practical/engineering considerations?


title: Is there an analogous Gauss' law which is applicable for a gravitational field?

text: Consider the Earth to be a flat infinite plane having linear mass density equal to the mass density of the actual earth.   1. Can there be an analogous Gauss' law that can give the gravitational field for any point on a particular Gaussian surface.    2. If yes, what would be the substitute for the permittivity constant that appears in the Gauss' law for an electric field?


title: Is it possible to 3D print a mirror to create a high quality telescope?

text: Is it possible to 3D print a mirror with todays available materials? If so, would there be a reduction in image quality?


title: Which areas of physics are related to the act of playing drums?

text: I'm musician (drummer) and I'm trying to figure out what can I study (related to Physics) for better understanding of the drumsticks and wrist movements, the force applied and the better way to apply it and introduce some theory in my training. I'm really awful at physics, so I really don't know what areas are important (pressure, mechanics, acceleration?).


title: How does holographic voxel density scale with holographic film metrics?

text: I'm trying to understand how one can generate bounds on the effective number of voxels (volumetric pixels) in a hologram, or information density, provided various metrics for the two-dimensional holographic film being used. For example, I often come across film parameters for grain size, resolving power (usually referred to in terms of lines/mm), and emulsion thickness. Is there a straightforward way to calculate this, or perhaps good rules of thumb?


title: Tangential and radial acceleration in projectile motion

text: I'm currently learing kinematics, specifically projectile motion and as an example in my textbook is a bullet fired at some angle. I understand the derivation of formulas to describe that motion and that the only acceleration acting on the bullet is $g$ directed downwards. But then it states that $g$ could be "broken" into tangentail and radial component. There is no picture how that would look, but wouldn't that mean that $g$ is not the only acceleartion acting on the bullet, but instead there is its $x$ component and $y$ component? And if that is so why then $x$ component isn't being used when calculating the speed in the $x$ direction?


title: Finding out the velocity

text: > Displacement of an object is related to time $x=at+bt^2-ct^3$ where a, b and > c are constant motion.Then the velocity of an object when it's acceleration > is zero is given by > >   1. $a+\frac{b^2}{c}$ >   2. $a+\frac{b^2}{2c}$ >   3. $a+\frac{b^2}{3c}$ >   4. $a+\frac{b^2}{4c}$ > I have done the following things: $\frac{d}{dt}tx=a\frac{d}{dt}t+2b\frac{d}{dt}t^2-c\frac{d}{dt}t^3$ $\ v=a+2bt-3ct^2$ After this I am stuck I don't know how to move ahead can anyone help me?


title: New version of double-slit experiment

text: I am trying to find a definite answer whether the following version of the double-slit experiment has ever been performed. Calculate/observe what interference pattern should appear by emitting photons individually one-by-one through the double-slit barrier and onto a detector screen behind it, with all equipment being in a vacuum to make sure each photon does not interact with anything other than the barrier and the screen. Then perform the same experiment but keep shifting the double-slit barrier slightly after each photon has been emitted but before it is supposed to go through the slits. Will the interference pattern appear to look similar and in the same location as in the original static version of the experiment or will it be different? Could it be that such experiment cannot be performed at all because it is not possible to tell whether and when each photon is actually emitted to be able to shift the barrier after the emission?


title: If we say the universe is expanding, shouldn't it be expanding relative to something?

text: I don't understand, if everything in this world is relative to something else, then cannot we essentially say that nothing exists independently? We say that the universe is considered to be the ultimate 'background'. However, if we say the universe is expanding, shouldn't it be expanding relative to something?


title: What is the formula for the glug point?

text: When you pour water out of a bottle, normally you have a smooth stream. However, if you pour it too fast it glugs, which is to say, comes out in quantized bursts. What is the formula for calculating the glug point angle, and what is the formula for calculating the size of the glug quanta and glug quanta frequency?