title: d=2 O(3) sigma model becomes "conformal antiferromagnet"

text: In Advanced topic in quantum field theory / M. Shifman on page 251 the author discusses the fact that the theta term is topological and does not affect the equations of motion. Then he said: > "In particular, at $\theta=\pi$ the O(3) sigma model becomes conformal and > describes a ferromagnet rather than antiferromagnet. Unfortunately, I cannot > dwell on the aspect in this text" Where he refers to the term $\frac{\theta}{8\pi}S^{a}(\partial_{\mu}S^{b})(\partial_{\nu}S^{c})\varepsilon^{\mu\nu}\varepsilon_{abc}$. I would like to see here more details about this point. Is it a uniqe phenomena for $\theta=\pi$?


title: Problems related to Green's function?

text: My teacher told me to do a research studying some physics problems that has connection with Green's function on solving differential equations (with programmed numerical solutions) in my final year project, can you give me such problems to work on as an undergraduate? I found this but I need more interesting problems to work on! any help?


title: What does quantum phenomena exist as prior to observation?

text: It's been said that according to the Schrodinger equation, independent of observation, particles exist in a state of a wave function, which is a series of potentialities rather than actual objects. Then the act of observing, causes a wave of potentiality to collapse to a state of matter. Is this true? Also, are these quantum experiments anomalies, and not how nature works, or do these quantum experiments, like double slit and quantum erasers, actually reflect how nature works even in the absence of human manipulations.


title: Measurement of observables with continuous spectrum: State of the system afterwards

text: Suppose my system, described by a separable Hilbert space $H$, is in the state $\Psi$ when I measure an observable that has only continuous spectrum. What is the state of the system after the measurement? Say, for the sake of definiteness, that we are measuring the momentum of a particle whose state is $\Psi \in L^{2}(\mathbb{R}^{3})$. Will it "leave" the space $L^{2}(\mathbb{R}^{3})$ and "become" a wave? If, instead, my measurement is only partly accurate and says that the momentum of the particle is in a set $\Delta =(a_x,b_x)\times(a_y,b_y)\times(a_z,b_z)$, will the measurement collapse the wave function into $P\Psi$ (where $P$ is the spectral projector of the momentum operator on the set $\Delta$)? What is the correct description of the system after one such measure?


Is the Big Bang defined as before or after Inflation? Seems like a simple enough question to answer right? And if just yesterday I were to encounter this, I'd have given a definite answer. But I've been doing some reading while writing up my thesis and I'm finding conflicting definitions of the Big Bang. Everyone agrees that in standard Big Bang cosmology, the Big Bang is defined as the singularity; the moment in time when the scale factor goes to zero. Okay, but when you include the theory of inflation, it gets a bit murky. So here's what I mean by conflicting definitions. As an example, in _The Primordial Density Perturbation_ by Lythe and Liddle, they define the Big Bang as the beginning of the era of attractive gravity after inflation. However, _Modern Cosmology_ by Dodelson defines the Big Bang as coming before inflation; it effectively uses the old definition that the Big Bang is the moment when the scale factor approaches zero. This contradiction is evident in multiple places. When doing a google search for it, one can find many persuasive explanations for both definitions. All definitions agree that we cannot any longer define it as the singularity where $a=0$. But every one makes sense in its own way and so, I become more and more confused about which is right the more of them I read. The argument for the Big Bang coming after is that inflationary theory diverges from the standard Big Bang cosmology around $10^{-30}s$ before we'd expect to run into the singularity, when inflation ended, and that we have no evidence to anything coming before that, thus the big bang is now defined as the initial conditions for the hot, expanding universe that are set up by and at the end of inflation. The argument for the Big Bang coming before seems to be that inflation is still a period where the scale factor grows and as such, the Big Bang can be defined as the closest value to zero (which is before inflation), or rather, the earliest time as the scale factor _approaches_ zero. This essentially seems to be based on saying "well, we defined it as the moment when the scale factor was smallest before inflation was added. Why would we not continue to have that as the definition after inflation is added?" The former argument has merit because it defines the start of the epoch where the universe is describable (practically) by the standard Big Bang cosmology. But the latter argument has merit because of its simplicity and that it uses the spirit of the original definition; the smallest scale factor and the moment when the expansion of the universe seems to begin. Thus, my root question: **Which definition is correct? Do we say the Big Bang came before or after inflation?** P.S. I realize that asking this here only serves to add one or more additional persuasive arguments to an already crowded debate. However, this is Physics.SE, so I figure whatever we decide here can be definitive. Even if we can't find a truly correct answer, this can set the record straight, or at least, firmly crooked.


title: Cases in which angular velocity and angular momentum point into same direction

text: I know that angular momentum $\vec{L}$ and angular velocity $\vec{\omega}$ of a rigid body doesn't point into the same direction in general. However if your body spins around a principal axis, $\vec{L}$ and $\vec{\omega}$ point into the same direction. In which other situations is this is exactly or at least approximately true? Please give reasons or a mathematical derivations why it is true in a given situation.


title: Are the laws of physics objective or subjective?

text: The laws of physics have often been taken to be objective statements about reality, but are they? Suppose Bob is a smart computer programmer with access to a super-duper-supercomputer. In it, he simulates a world operating under different laws from that of his own. In that simulation emerges a super-duper- smart scientist Alice. Alice deduces the laws governing her simulation not realizing it is a simulation and calls them the laws of physics. What Bob calls the laws of physics is something else entirely. Without an ontological commitment to wild metaphysical speculations, Alice will have to conclude what she deduced to be the laws of physics, almost certainly have to be the laws of physics for her. According to some ontological speculations, the shortest description which fits observational data is the actual ontological reality. Certainly a law like Schmidhuber's computer program which multitasks over all possible programs, one of which simulates our universe, will have a shorter description than a program which only describes the laws of physics of our universe? By this "kolmogorov" ontology, wouldn't Schmidhuber's program be more likely to be ontologically real as the laws of physics? If there is some measure over observers, it would also means the laws of physics (not the ultimate) will have to be typically generic while giving rise to sentient observers. What about a quantum computer running a quantum superposition of all possible programs, with eventual decoherence between the different programs? That would also have a shorter description.


title: How can the date of lunar eclipses be calculated by hand?

text: How can the date of lunar eclipses be calculated? Especially without the aid of a computer.


title: Operators and Commutator Definitions

text: I have several problems with General Definitions of an Operator and Commutator :   1. the product of operators is **generally** not commutative: $$\hat A \hat B \not= \hat B\hat A .$$ what is this means that **generally** is not commutative? and , is the product of two operators $\hat A$ Hermitian Adjoint $\dagger$ then $\hat B$ (because in my book it is just $\hat A \hat B$ )? $$\hat A^{\dagger} \hat B \not= \hat B^{\dagger}\hat A .$$ **Remark** 2\. Note that the Hermitian adjoint of an operator is not, **in general** , equal to its complex conjugate: $$\hat A^{\dagger} \not= \hat A^* .$$ What happens if the Hermitian adjoint of an operator is, **in special** , equal to its complex conjugate!? $$\hat A^{\dagger} = \hat A^* .$$ **Commutator Algebra**   1. the commutator of two operators $\hat A$ and $\hat B$, denoted by: $$[\hat A , \hat B]=\hat A \hat B - \hat B \hat A .$$ again i do not know that, is the product of two operators $\hat A$ Hermitian Adjoint $\dagger$ then $\hat B$? $$[\hat A , \hat B]=\hat A^{\dagger} \hat B - \hat B^{\dagger} \hat A$$


title: Intuitive explanation for the de Broglie / Planck relations

text: A friend asked me to explain "why" a particle's energy is proportional to it's frequency, i.e: $$E=h\nu$$ The reason this result is so un-intuitive, is that in the macroscopic world, A wave's energy (e.g electric power dissipated through a resistor) is not frequency dependent but rather amplitude dependent. Of course a "pure" photon doesn't really have an amplitude due to it's description as a plane wave, but that's beside the point. **So in short** : Is there an easy explanation of "why" the energy is only dependent on frequency? can this be "derived" from other principles (say, the particle-wave duality)? * * * **N.B.** I'm aware there are no real "reasons" for any law of physics other then the fact that they agree with experiment, but laying those laws on a smaller number of basic principles is always nice when possible.


title: State collapse in Heisenberg picture

text: I've been studying quantum mechanics and quantum field theory for a few years now and one question continues to bother me. The Schrodinger picture allows for an evolving state, which evolves through a unitary, reversible evolution (Schrodinger's equation, represented by the Hamiltonian operator) and an irreversible evolution (wave function collapse, represented by a projection operator). The Heisenberg picture holds the states constant and evolves the operators instead. It provides an equivalent representation of the unitary evolution on operators, but I haven't yet seen an equivalent Heisenberg representation of wave function collapse. Is there any accepted explanation for how to represent state collapse in the Heisenberg picture? Thanks!


title: Electric potential due to circular disk

text: Relevant diagram is available here. The circular disk of radius $a$ lies in the $xy$ plane and carries surface charge density of $\sigma (s, \phi) = s^{2}cos\phi $, where $(s,\phi)$ are in cylindrical co-ordinates. The problem is to find potential at a point which is slightly displaced from the $z$ axis at position $ r = z \hat z + \delta s \hat s = z\hat z + \delta x \hat x + \delta x \hat y$, and $ r' = s cos\phi \hat x + s sin\phi \hat y $ Since potential is given by $V(r) = \frac{1}{4 \pi \epsilon_{0}} \int \frac{1}{\bf {r} - \bf {r'}} dq$ Here, $dq = \sigma da = \sigma dl_{s} dl_{\phi} = \sigma sdsd\phi = s^3 cos\phi dsd\phi$ and ${r-r'} = z\hat z + \delta x\hat x + \delta x \hat y - (s cos\phi \hat x + s sin\phi \hat y) $ Therefore, $|r-r'| = \sqrt{z^2 + s^2 - 2\delta x s(cos\phi + sin\phi)} $. It can be assumed that $(\delta x)^2 = 0$ since $\delta x$ is infinitesimally small. This means that the final integral for potential is given by $V(r) = \frac{1}{4 \pi \epsilon_0} \int_S \frac{s^3 cos\phi}{\sqrt{z^2 + s^2 - 2\delta x s(cos\phi + sin\phi)}} dsd\phi$. Any suggestions on how to proceed with evaluating this integral would be very much appreciated. Thanks.


title: When will Enceladus run out of water?

text: So its geyser eruptions are the primary source of water in Saturn's atmosphere (from this thread). And they even manage to contaminate Titan's atmosphere with it. But has the outgassing rate decreased over time? Could it eventually run out of water?


title: Vesta dwarf planet status

text: Now that we have close-up photos of Vesta, which the IAU had previously said was a candidate dwarf planet, when is the IAU going to decide the issue? Personally, Vesta doesn't look round enough to me.


title: Why is the universe described in terms of Euclidean space and not Minkowski spacetime?

text: The universe is described as an infinite Euclidean space in cosmology. Why isn't it treated as Minkowski spacetime?


title: Electrical force vs gravitational force

text: Given that the electrical force is so much stronger than gravitational force at atomic levels, why is it that it's the gravitational force between you and the earth that keeps you on the ground rather than the electrical force between you and the earth?


title: Why is $L^2$ norm of the gradient called kinetic energy?

text: I'm reading Lieb-Loss's book 'Analysis', chapter 7. The authors refer to the following integral: $$\tag{1} \lVert \nabla f\rVert_2^2=\int_{\Omega}\lvert \nabla f(x)\rvert^2\, d^nx $$ as the _kinetic energy_ (see pag. 172), without explanation. To what physical system are they referring to? My intuitions says that (1) would be more appropriately called _potential_ energy, as we have discussed here. * * * (Note: In a later paragraph the authors introduce a magnetic potential $\mathbf{A}$ and the so-called _covariant derivative_ $\nabla+i\mathbf{A}$, remarking that after this introduction the kinetic energy integral must be replaced with $$\int_{\mathbb{R}^n}\lvert (\nabla + i \mathbf{A})f(x)\rvert^2\, d^n x.$$ This induces me to think that they take as a model some kind of electromagnetic system.) Thank you for your attention.


title: Magnetic flying engine

text: I invented a flying engine moving entirely by magnetic forces (such as the force related with magnetic field of the Earth). See http://porton.wordpress.com/2011/12/23/magnetic-vehicle/ The question is, can we build an engine of this type which is enough powerful to actually fly? The question is both about the mechanical engine described in the post above and about a hypothetical (not developed by me in details) purely electronic version. My question is: Describe operating characteristics of these kinds of engines.


title: Is it possible to generate an instrument which can generate wireless electricity?

text: Is it possible to build an instrument which can generate wireless electricity? I ask this since at present the electricity was passes through wires and i thought about wireless electricity I am curious to know about this:               Is it possible to build an instrument which can generate wireless electricity ?


title: Difference between charge density wave and charge distribution

text: We can always see modulated charge density, the Friedel Oscillation, around an probe charge due to other electrons' response. Can this be called charge density wave (I believe not)? If not, what is the fundamental difference between this situation and charge density wave? Is Peierls Transition a precondition to charge density wave? If so, why? If not, what is the condition? Thanks so much!


title: Can two electrons never touch each other?

text: The Coulomb's force is given by $$ F = {k q^2 \over r^2} $$ When $ r \rightarrow 0 $, $ F \rightarrow \infty $ Does this mean two electrons never touch each other?


title: Vibrations after polarization of light

text: When we polarize a light, do we get electric vibrations, magnetic vibrations or the mixture of both. If both, then how can both electric and magnetic vibrations occur in single plane because polarization actually means to confine these vibrations into one plane?


title: What determines the speed required to pull a table cloth?

text: I was watching this show "Street Genius" on National Geographic and the host Tim Shaw demonstrated an experiment about Inertia, What he did was, He tied one end of a table cloth to a car through a long rope and started driving the car, when the rope was taut, the cloth was pulled and the things on the table were still on the table. What are the calculations involved in it and what determines the speed required to pull the table cloth without disturbing the things on it?


title: Probability density of detection of collinearly emitted photons in two detectors

text: **Update** :   As proposed by @dmckee, I added equation numbers and improved the display of some equations. The answer by @Trimok inspired me to look at coordinate systems which are not specific to the detector locations, but more general, and this indeed made things clearer since $\delta(\cos \theta + 1)$ is a $\delta(\cos \theta + \cos 0)$ in disguise. We have a relation between a Dirac delta of a ray in spherical coordinates, $\delta^2$, and a Dirac delta of a point, $\delta^3 = \delta(r-r_0)\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta)$, such that $\delta^2 = \int_0^\infty \delta^3 \operatorname{d}r_0 = \delta(\theta-\theta_0) \delta(\phi-\phi_0) / (r^2 \sin\theta) \tag{E1}$ (Using this definition, one can easily show that the sifting property of $\delta^2$ holds and the result is a line integral of some $f$.) The major problem (of understanding) which I am having is now reduced to this: We have two expressions for $\delta^3$, which are $\delta(r-r_0)\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta) \tag{E2}$ and $\delta(r-r_0)\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r_0^2 \sin \theta_0) \tag{E3}$ These are totally symmetric in $x$ and $x_0$ (note that $\delta$ is an even function). However, for $\delta^2$ we have $\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta) \tag{E4}$ and $\delta(\theta-\theta_0)\delta(\phi-\phi_0)/(r^2 \sin \theta_0) \tag{E5}$ which are not symmetric any longer. So what makes the $r$ stand out compared to $r_0$? One explanation I worked out with the help of a colleague is directed towards the fact that there is no $r_0$ after integration, so having any $r_0$ would "overdetermine" the set of rays through the origin. Does anyone have a better (for example, more intuitive) explanation why the symmetry is lost here? Thank you! **Original question** :   This is a long one. I have been working on this for quite a while now, so I summarized most of what I found out. I am stuck at a certain point, however, and it would be great if someone could help me out. Assume the following situation: Two perfectly collinear photons $p_1$ and $p_2$ are emitted in the origin of some Cartesian coordinate system. We consider two infinitesimal detectors $d_1$ and $d_2$, with surfaces $\operatorname{d}\\!A_1$ and $\operatorname{d}\\!A_2$ directed towards the origin so that we do not have to deal with angles. The solid angles of these detectors be $\operatorname{d}\\!\Omega_1$ and $\operatorname{d}\\!\Omega_2$. I now express some basic detection probability densities, where $p_id_i$ means that "$p_i$ is emitted towards $d_i$".   $\frac{\operatorname{d}\\!P(p_1d_1)}{\operatorname{d}\\!\Omega_1}=\frac{1}{4\pi} \tag{1}$ $\frac{\operatorname{d}\\!P(p_1d_1)}{\operatorname{d}\\!A_1}=\frac{1}{4\pi (r_1)^2} \tag{2}$ $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!\Omega_2}=\frac{1}{4\pi} \tag{3}$ $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!A_2}=\frac{1}{4\pi (r_2)^2} \tag{4}$ I think there is no doubt that these probabilities are well-defined and the integrated detection probabilities in a spherical $4\pi$-detector equals 1, no matter what its radius is. So a photon is surely emitted towards a detector surrounding the origin. Now, for some reason we want to calculate $P(p_2d_2 | p_1d_1)$, the conditional probability of $p_2$ emitted towards $d_2$, given that $p_1$ has been emitted towards $d_1$. Due to collinearity of the photons, the individual events are not independent: $P(p_2d_2 | p_1d_1) \neq P(p_2d_2)$. I now propose that $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2} = \frac{\delta(\cos \theta_2(d_1)+1)}{2\pi}, \tag{5}$ where $\theta_2(d_1)$ is the polar angle of the position of $d_2$ expressed in spherical coordinates with polar axis from the origin through $d_1$. $\delta$ is the Dirac function, so obviously, this expression equals 0 if $\cos \theta_2(d_1) \neq -1$, that is, when $\theta_2(d_1) \neq \pi$. We can verify that the integral of this probability over a spherical $4\pi$-detector $d_2$ equals 1: with $\operatorname{d}\\!\Omega_2 = \operatorname{d}\\!\cos \theta_2(d_1) \operatorname{d}\\!\phi_2(d_1)$, we find $\int \frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2} \operatorname{d}\\!\Omega_2 = \int \frac{\delta(\cos \theta_2(d_1)+1)}{2\pi} \operatorname{d}\\!\Omega_2 \tag{6}$ $= \int_0^{2\pi} \frac{1}{2\pi} \operatorname{d}\\!\phi_2(d_1) \cdot \int_0^\pi \delta(\cos \theta_2(d_1)+1) \operatorname{d}\\!\cos \theta_2(d_1) = 1. \tag{6}$ So $p_2$ is still surely emitted towards a detector surrounding the origin, no matter where $p_1$ is emitted towards. Further, we would like to verify that the law of total probability is fulfilled by integrating $P(p_1d_1 \wedge p_2d_2) = P(p_2d_2 | p_1d_1) \cdot P(p_1d_1)$ over a spherical $4\pi$-detector $d_1$. This works regardless of which formulation of $\operatorname{d}\\!P(p_1d_1)$ (with respect to $\operatorname{d}\\!\Omega_1$ or $\operatorname{d}\\!A_1$) we use -- I chose $\operatorname{d}\\!A_1 = (r_1)^2 \operatorname{d}\\!\cos \theta_1(d_2) \operatorname{d}\\!\phi_1(d_2)$ here. Note, however, that integration over $d_1$ changes the polar axis used in the formulation of $\operatorname{d}\\!P(p_2d_2 | p_1d_1)$ above, so we need a different, but equivalent formulation. This is more than a change of coordinate system (from one polar axis to another), because $\theta_2(d_2)$ would not help us a lot (it's 0). Instead, one can use $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2}=\frac{\delta(\cos \theta_1(d_2)+1)}{2\pi}, \tag{7}$ which uses both a different variable in the argument of the $\delta$ as well as a different coordinate system: $\theta_1(d_2)$ is the polar angle of detector $d_1$'s location in spherical coordinates with polar axis from the origin through $d_2$. **Note that both expressions of $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2}$ are symmetric in the indices 1 and 2.** Finally, we arrive at $\int \frac{\operatorname{d}\\!P(p_1d_1 \wedge p_2d_2)}{\operatorname{d}\\!A_1 \operatorname{d}\\!\Omega_2} \operatorname{d}\\!A_1 = \int_0^{2\pi} \int_0^\pi \frac{1}{4\pi (r_1)^2} \cdot \frac{\delta(\cos \theta_1(d_2)+1)}{2\pi} (r_1)^2 \operatorname{d}\\!\cos \theta_1(d_2) \operatorname{d}\\!\phi_1(d_2) \tag{8}$ $= \frac{1}{4\pi} \int_0^{2\pi} \frac{1}{2\pi} \operatorname{d}\\!\phi_1(d_2) \cdot \int_0^\pi \delta(\cos \theta_1(d_2)-1) \operatorname{d}\\!\cos \theta_1(d_2) = \frac{1}{4\pi} = \frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!\Omega_2} \tag{8}$ So, the law of total probability is fulfilled: the probability of $p_2$ being emitted towards $d_2$ can be partitioned into all conditional probabilities of $p_2$ being emitted towards $d_2$ _given that_ $p_1$ has been emitted towards $d_1$, weighted by the probability that $p_1$ is _actually_ emitted towards $d_1$. So far, so good. Congratulations for reading up to this point :) As expressed above (and verifiable relatively easily), it does not matter which formulation of $\operatorname{d}\\!P(p_1d_1)$ I use -- both $\operatorname{d}\\!\Omega_1$ and $\operatorname{d}\\!A_1$ work fine. However, when I try to base my calculations on $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!A_2}$ instead of $\frac{\operatorname{d}\\!P(p_2d_2)}{\operatorname{d}\\!\Omega_2}$, hence, I try to calculate $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!A_2}$ instead of $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!\Omega_2}$, I arrive at the following problem. So I started with $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!A_2}=\frac{\delta(\cos \theta_2(d_1)+1)}{2\pi (r_2)^2}, \tag{9}$ and (6) is still easily verified (integrating over $\operatorname{d}\\!A_2$) since the two additional $(r_2)^2$ simply cancel out. However, working out (8) is now more complex when it comes to changing from one variable/coordinate system to another. I figured that $\frac{\operatorname{d}\\!P(p_2d_2 | p_1d_1)}{\operatorname{d}\\!A_2}=\frac{\delta(\cos \theta_1(d_2)+1)}{2\pi (r_2)^2} \tag{10}$ is the expression which fulfills total probability (I can show this, if necessary), but now I wonder why this is not symmetric with the first expression in the indices 1 and 2 (as it was above). Specifically, I wonder where the connection to a Dirac of a ray along Oz in spherical coordinates is, which is $\frac{\delta(\cos \theta - 1)}{2\pi r^2}$ (compare http://www.physicsforums.com/showthread.php?t=215263) More generally, I am working on expressing the probability density in 3D, so I expect to obtain something like $\operatorname{d}\\!\Omega_2 \cdot \delta_{L(x_1, x_2)}(x_0)$ or $\operatorname{d}\\!A_2 \cdot \delta_{L(x_1, x_2)}(x_0)$, where $L(x_1, x_2)$ is the line segment from $x_1$ and $x_2$, and $x_0$ is the point of emission. So my specific questions are:   * Why are the expressions involving $\operatorname{d}\\!A_2$ not symmetric, while those involving $\operatorname{d}\\!\Omega_2$ are?   * Specifically, where is the connection to the Dirac of a ray in spherical coordinates, which involves an $r^2$ and which I would expect to be changed when converting from one coordinate system to another.   * How (else) can I obtain a 3D probability density involving the three locations, which can be used to integrate over all three volumes in order to describe the number of coincidences detected from emissions from a volume in two finite detectors?


title: Components of the Ricci Tensor

text: Is there any interpretation of what each of the components of the Ricci tensor corresponds to? For example, for the stress-energy tensor, $T_{00}$ corresponds to energy density, $T_{0i}$ is the momentum flux in the $i$ direction, etc. Is there something similar for the Ricci tensor?


title: Energy Required To Push Up a Slope

text: I'm having a bit of trouble with this problem: A $5.0\:\text{kg}$ block is moved up a slope with constant velocity of $2.5\:\text{m/s}$. The amount of force required to push the block up the slope is $20.0\:\text{N}$. The slope is $45.0\:\text{cm}$ high, $150\:\text{cm}$ horizontally on the ground, and its hypotenuse/slope is $125\:\text{cm}$. How much energy is required to push the block up the slope? Angles are not supposed to be used in order to obtain the answer. (I initially thought that the energy required would just be equal to the potential energy at the top of the slope, i.e. $mgh$. However, I'm not sure if there are other factors that are needed to be taken into consideration?)


title: Bragg's diffraction and Simon's problem

text: In Preskill's notes, John Preskill goes as > If we scatter a photon off of a periodic array of needles , the photon is > likely to be scattered in one of a set of preferred directions , where the > Bragg scattering condition is satisfied. and further > These preferred directions depend on the spacing between the needles , so by > scattering just one photon , we can already collect some useful information > about the spacing. How can the analogy with Bragg's diffraction help me understand Simon's problem ?


title: Sharpest camera shoot within a mirror

text: In order to take the sharpest camera shoot from the reflected object image in the mirror, the camera should be placed $x\,m$ farther from the reflected image object. ![enter image description here](http://i.stack.imgur.com/vgV20.jpg) How can I determine $x$ ?


title: Is the idea of dividing the universe into particles anything more than an untrue convenience?

text: In theory, we speak of a particle as having properties. In reality, the measurement of any property is just an interaction between the target to be measured, and the measuring apparatus, where the measurement affects both the target and the apparatus. So any property we can measure about a particle, is described in terms of all other particles and their properties. (It's all rather circular.) In effect, we have a way of describing a universe of n particles, with n-1 particles. Since it was only through changes in the n-1 particles that we determined we had an nth. Continue this 'rewriting' and inductively you will end with 1 'particle' with several properties that are functions of themselves. So can we say the universe is just 1 thing, and we have artificially subdivided it for a more convenient representation?


title: New focal point of converging light through a concave lens

text: Suppose I pass rays of already converging light through a concave lens. The rays are equally centered on the lens such that the original focal point is through the lens and centered on the lens axis. My understanding is that the lens will bend the incoming rays, such that they all focus on a new focal point closer to the lens. Is there a formula I can use to determine the location of the new focal point, given the focal point of the lens and the original focal point?


title: Time for Light to travel from most distant objects in universe

text: I understand that it is possible to detect the most distant objects in space over 13 billion light years away from us and that the universe is 13.75 billion years old. Does this imply that there were some objects many billions of light years away from us even in the earliest stages of the expansion of the universe, has the universe at some point expanded faster than the speed of light or is my mental image of the expansion process incorrect?


title: Why does observation collapse the wave function?

text: In one of the first lectures on QM we are always taught about Young's experiment and how particles behave either as waves or as particles depending on whether or not they are being observed. I want to know what about observation causes this change?


title: Composition of kinetic energy

text: As kinetic energy is the "energy of motion", it cannot exist without matter. However, the energy in an E/M wave can. I understand that E/M waves have been described in structure as electrical and magnetic waves in a sinusoidal configuration. This means that kinetic energy cannot be made of EM waves or it would be mass-independent. So my question is, what is kinetic energy made of?


title: Example of Torque, Center of Precession

text: So here's the set up, we have a fence of length $2L$, and a support strut a distance $l$ from the axis (think of a railroad crossing gate). We need to find the best position for the support rod, so that the pivot takes minimum wear. We analyze the forces on the gate, and find that there are three vertical forces (we are ignoring the radial components). The force $\mathbf F _s$ is the force due to the support, $\mathbf W$, the weight, acting at center of mass (position $L$) and the force at the axis of rotation, which we will call $\mathbf F _{\rho} $ ![enter image description here](http://i.stack.imgur.com/vf6RA.jpg) So for torques, taking the origin being the AoR (axis of rotation) are $$\tau = MgL-F_s l=I_{\rho} \ddot{\theta}.$$ Now we integrate from $t$ to $t+\Delta t$ ( _angular impulse?_ ) which gives us $I_{\rho} \dot{\theta}=-F_s l \Delta t$, where we have neglected the torque due to the weight. Next step is where we apply $\mathbf F = m\mathbf a$, and which case: $$m\frac{dV}{dt}=-(F_s + F_{\rho})+Mg$$ ( _This step seems to be arbitrary, what is our motivation for including both torque and Newton's law?_ ). We have again neglected the weight, and integrate $mv=-ML\dot{\theta}=(F_s + F_{\rho}) $ This step also confuses me, how in the world does this make sense $$\int ^{t+\Delta t}_{t} (F_s + F_{\rho}) dt = -(F_s + F_{\rho}).$$ Shouldn't it be $(F_s + F_{\rho})\Delta t$? The next step says we can solve our equation $$I_{\rho} \dot{\theta}=-F_s l \Delta t$$ and get $$F_s \Delta t=I_{\rho}\dot{\theta}/l$$ and plug this into our last equation, $$ML\dot{\theta}=(F_s + F_{\rho}) .$$ The only way this works out to what my book says, is if we have $(F_s + F_{\rho})\Delta t$. We were supposed to end up with $$F_{\rho}\Delta t = \dot{\theta}(ML-\frac{I_{\rho}}{l}).$$ From this point on, it just talks about the conditions for the impulse from $F_{\rho} = 0$.


title: How to simulate temperature change of oven?

text: I am trying to write a software, which will model the oven temperature change when turning on/off. The data I can get is graph, by taking temperature reading each second from T0 time up to some temperature. Then I can take data for cooling graph. So I will have two non linear curves. Having this data, how can I simulate temperature change? Should I take current temperature at any given moment, see if oven is on/off, then follow respective curve to increase or decrease the temperature with calculated speed? Mostly I'm not clear on how to calculate the speed from curve. What will be precision of such modelling? took some readings from oven as sample data. I've turned off oven at 80 degree. ![enter image description here](http://i.stack.imgur.com/YyIsY.png)


title: Auxiliary fields in supersymmetry

text: I know that auxiliary fields can be used to close the supersymmetry algebra in case the bosonic and fermionic on-shell degrees of freedom do not match. Could somebody please elaborate on this concept and explain in which context auxiliary fields play a role?


title: How to track the visual path of a LEO satellite as seen from the ground

text: I have been struggling with this problem for a while so I decided to ask. I'm new here and I'm not sure where this type of question belongs, so forgive me if this isn't the right section. I am working on a satellite tracking program. I have it using TLE data from Celestrak to track a variety of satellites. I have the basic information for a pass, including the AOS azimuth, LOS azimuth, and Max Elevation, as well as the total time it takes to cross the sky as seen from the observer on the ground. I need some way to translate this info into a list of azimuth and elevation values that represent the path the satellite will take across the sky. I've searched quite a bit and haven't been able to solve it yet. In other words, given the starting azimuth, ending azimuth, max elevation, and total time, how can I create a function to tell me the azimuth and elevation at any particular moment in time?


title: Is angular momentum always conserved in the absence of an external torque?

text: Consider either the angular momentum of the earth around the sun or equivalently swinging a ball horizontally on a string. I know that with respect to the point of rotation of the swinging ball, angular momentum is constant. Consider now an origin outside of the 'orbit'. It seems to me that the angular momentum is different as the orbit progresses but there is no external torque (the physical scenario is unaltered!). The easiest way to see this is to consider when the ball is at opposite ends of the circle - the radius changes but the velocity only flips sign. In addition to this, how about we consider an event where we cut the string. Now place our reference point on the line of motion when the ball flies away in a straight line. By definition the ball will have zero angular momentum with reference to any point on this line ($m|r||v|sin\theta=0$) but certainly before the event it had some angular momentum. The main question is in the title, but essentially what am I missing conceptually about this problem?


What is the history behind the factors of 3 in the classification of electromagnetic radiation? See e.g. http://en.wikipedia.org/wiki/Radio_spectrum#By_frequency Is this (just) inherited from the meter unit via the wavelenght frequency dependence or light?


title: How to tell if a complex exponential blows up

text: I'm following Griffiths' Introduction to Quantum Mechanics, where he's discussing the general solution to the delta-function potential problem. The solution he refers to is $$\psi(x)=Ae^{ikx}+Be^{-ikx}$$ Now, he says that neither of the the terms blows up. The case he is dealing with is $x<0$. I cannot see this, because it seems to me that the second term blows up if $x$ is negative. I suspect he is using some argument that involves the complex nature of the terms. I would like to know if indeed this is the case. If so, what is the argument to show that the terms do not blow up? Thanks.


title: Transfer of electron energy to atoms (heating up of matter by absorption of photons)

text: If an electron absorbs a photon to get exited to a higher energy level, it should either come back to same state or any other lower state by emitting the required photon. How then can there be a net transfer of energy to the atom? Heating up means increase in kinetic (vibrational) energy of the atoms. If the energy absorbed by a photon is re-emitted as a photon then how do atoms extract energy from incident photons?


title: Train construction-need of suspension

text: Suspension generally used to make ride comfortable. But rail tracks are smooth and even; intution says trains no need suspension. But they have it. Why do railway wagon/coach require suspension when the tracks are so smooth and flat?


title: Can one prove the full spin-statistics theorem from the spin 0, 1/2 and 1 cases?

text: Using second quantization for scalar field, spinor field and vector fields, we can get commutation and anticommutation relations for the birth and destruction operators of the fields, which leads us to the Bose or to Fermi statistics. Is it possible to expand these results on a field of arbitrary spin (integer or half-integer), using the basic idea that each field can be built by combination of spinor $\frac{\hbar }{2}$ fields?


title: What is a two-photon process?

text: I am reading some introductory materials on modern optics, in which they mention two-photon processes everywhere. I know fundamental optics and a bit on quantum mechanics. Can anyone explain in a plain way what is a two-photon process or any simple introduction to a related topic.


title: Speed of an object in air versus in vacuum

text: Okay, imagine that you are shooting a rocket down a tunnel that is 60 miles long and the rocket is travelling at 60mph, so the rocket should reach the end of the tunnel in a hour, right? Yep, in an hour. BUT if next to that tunnel you have the same length tunnel and that tunnel is a vacuum, similar to that of space. And you shoot a rocket that can fly propel it self in the vacuum, and it's travelling at 60 mph also. Which would reach the end first? Also imagine that the rocket is flying straight and not zagging around, I am just wondering how the resistance of air slows down something compared to the same thing in a vacuum.


title: How fast do electrons move through a conductor?

text: If I apply $1 \text{ V}$ across a $1 \text{ }\Omega$ resistance, I'd get $1 \text{ A}$ flowing. $1 \text{ A}$ is defined as $1 \frac{\text{C}}{\text{s}}$, and $1 \text{ C}$ is equivalent to $6.24150975 × 10^{18}$ electrons. Therefore, Ohm's Law describes only the number of electrons passing a given branch per second. How do we determine the speed of electrons as they move through a conductor? And what is that speed?


title: If I take a handful of salt and wait for an infinite time will it become a single crystal?

text: That pretty much says it. Suppose I have some powder of $NaCl$. It is kept in contact with itself in vacuum. You are free to remove all the disturbances that bother you. Is that true that, well, there exist a ($\mbox{very}^\mbox{very}$ large) amount of time $T$ that for every moment $t>T$ you will have a single beautiful crystal with 99% probability.


title: Inflation and string theory in general relativity

text: How can we relate inflation and string theory? We study some inflationary cosmological model or some string cosmological model under some specific assumptions or conditions. Is their any relation or any difference between these two concepts? My knowledge of inflation is that it is a theory or concept to resolve some standard problems of early universe. I already got the answer about strings in more questions on this site. I actually want to know the relation between these two terms?


title: Why does the light at the bottom of the pool form this awesome pattern?

text: Take a look at the following picture: ![enter image description here](http://i.stack.imgur.com/fkwo9.jpg) Why does the light at the bottom of the water form this pattern? I have also seen the same phenomenon in all the swimming pools I've ever visited, of whatever depth they may be. How does this happen?


title: intermetallics and solid solution

text: For some systems, discrete intermediate compounds rather than solid solutions may be found on the phase diagram, and these compounds have distinct chemical formulas; for metal–metal systems, they are called intermetallic compounds. e.g. Mg2Pb has 33 at% Pb For binary isomorphous system like copper-nickel, if we get 33 at% Ni, can we say it is Cu2Ni? How to distinguish intermetallics from other solid solution with same ratio?


title: Simple Acceleration Problem throwing me off

text: I am having trouble with this problem from my homework. I have gone over it many times, and I must have some concept wrong. A rocket is launched from the ground with constant acceleration. After 4 seconds, a bolt falls off and hits the ground 6 seconds later. Gravity is -9.8m/s^2. What was the rocket's acceleration? So here's my thoughts. ## First, let's get the height of the rocket when the bolt falls. Fell at 9.8m/s^2 for 6s, so the final velocity was 58.8m/s Average velocity was half of that, because of the constant acceleration, so 29.4m/s Total distance is that average, times 6 seconds, 176.4m. Good so far? ## Next, the rocket's acceleration We know it was 176.4m in the air after 4 seconds. So the average velocity 44.1 m/s, final velocity is 88.2 m/s It got to that in 4 seconds, so the acceleration must be 22.05 m/s^2 But it's not. The correct answer is 5.5 m/s^2, a quarter of what I got. ## So what went wrong? I've tried looking through all the steps a bunch of times, and I can only think of one step that doesn't seem quite right. The bolt falling off the rocket makes it sound like it just drops, but maybe it starts that drop with an upwards velocity? But I can't see any way of getting to that velocity. Without knowing how high the rocket is, we can't know how fast it's going, so we can't know how fast the bolt was moving upwards. And then we have two variables in the equation, and an infinite number of correct pairs. I must be missing some concept, some formula must be missing. Can you find it?


title: How do I build an alpha emitter?

text: I would like to build an alpha emitter. I suppose I had some vague idea of knocking the electrons off of some helium. But upon further examination the idea does strike me as...naive? My question is then, how do I go about building an alpha emitter, cheaply, and with household/hardware store items. (Besides springing for a smoke detector.)


title: Infinite acceleration?

text: Why is acceleration regulated by mass? In a frictionless environment, why doesn't an object move at infinite acceleration if force is applied on it? Force causes movement, so unless there is an opposing force there shouldn't be any reason for the force to cause infinite acceleration. Why does mass act like an opposing force that limits the acceleration caused by an force to less than infinity?


title: Does the earth's gravity affect horizontal movement?

text: In a frictionless environment, would the weight of an object affect how hard it is to move it horizontally? Since without friction, there is no horizontal force resisting your applied force, would the object be easy to move? I am aware of inertia but don't see any reason for it to play a role considering there is no opposing force to your applied force.


title: Why does the force of gravity get weaker as it travels through the dimensions?

text: Some theories predict that the graviton exists in a dimension that we of course can't see, and that is why the force of gravity is so weak. Because by the time gravity has got from the dimension in which the graviton exists to our dimension it has lost much of its strength. For a better understanding of this theory, I think this might help: ![](https://lh4.googleusercontent.com/-M3Z7HBllZys/T5JT- dGobZI/AAAAAAAAFIE/I_XJUA3M0pI/randall_750.jpg) Bigger: https://lh4.googleusercontent.com/-M3Z7HBllZys/T5JT- dGobZI/AAAAAAAAFIE/I_XJUA3M0pI/randall_750.jpg?imgmax=1600 Why does the force lose strength as is travels through the other dimensions?


title: Accelerating elevator?

text: If your in an elevator accelerating downwards faster than gravity, would you be pushed up to the ceiling? Since gravity cannot match the acceleration of the elevator, you're wouldn't be able to move at a constant velocity relative to the elevator.


title: Normal ordering

text: If I understood correctly there are two terms called normal ordering:   1. $:c c^\dagger: = c^\dagger c \hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the right.   2. $:c^\dagger c:=c^\dagger c- \langle c^\dagger c \rangle\hspace{.5cm}$ so subtracting the contraction Both of these are called normal ordering (I think) and use the same notation. But what is the connection? If I got something wrong, please correct and explain!


title: A closed surface, no charge enclosed, yet flux not 0?

text: ![](http://img213.imageshack.us/img213/5474/gausslaw.jpg)! The book says it is $E_0\pi r^2$ because the flux through the circle is equal to the curved part of the paraboloid. I don't understand this, shouldn't the total flux be 0 for the whole surface? IN fact, since the E field is constant, then the flux must also be 0.


title: A basic question about Heisenberg State Kets (in particular the simple harmonic oscillator)

text: I know base kets in the Heisenberg picture are $U^\dagger |{a}\rangle$ but if the base kets are the base of the hamiltonian, and the hamiltonian is independent of time, are all of the base kets stationary as well? Specifically my question is: Consider a particle subject to the simple harmonic oscillator potential with state $\exp[\frac{-ipa}{\hbar}]|0\rangle$, what is the probability that this state is found in the ground state at time t? I think it is independent of time, but I don't know a completely solid line of reasoning to say this is so, because I am confused as to what is a base ket and what is a state ket here, and why I can just say that it is time independent.


title: How do you determine the value of the degeneracy factor in the partition function?

text: In the partition function, expressed as $$Z = \sum_j g_je^{-\beta E_j}$$ I'm wondering what determines the $g_j$ factor. I've been trying to look around the internet for an explanation of it but I can't find one. I guess it is the number of degenerate states in a given energy level? How do you determine how many degenerate states there are? A simple example involving how to determine the $g_j$ factor would be great. Thanks for the help!


title: Tensors: relations between physics and linear algebra

text: In continuum mechanics we use finite deformation tensors to exprime deformations in a point. The 9 components of the tensor (in reality 6 because of its symmetry) are defined as $$ \sigma_{ik}(\vec{r}) = \frac{dF_k^{(i)}}{dS_i} $$ where $k$ is the component of the force that acts over the surface perpendicular to axe $i$. We put this components on a "table" and we use matrix calculus to manipulate the tensor. In linear algebra we saw only the definition of a tensor. It is a multilinear map $$ T: V\times \cdots \times V \times V^* \times \cdots \times V^* \to K. $$ In fact I really can't see that this definition correspond to my finite deformation tensors. Cans someone show me the relations between physics and linear algebra in this case? Why I can use a tensor as a matrix? I read that if we use orthogonal axis we can avoid tensor analysis. Why?


title: Is a volumetric rate frame-invariant in general relativity?

text: Imagine that I have a radioactive material with a long half life. The atoms in this material decay at a certain rate $R$. The rate is the decay constant times the number density $R = \lambda N $. It has dimensionality: $$ \left( \frac{ \text{decays} }{m^3 s} \right) $$ Imagine that the material is on board a spaceship traveling at some significant fraction of the speed of light. Length is contracted and time is dilated. $$ \Delta t' = \Delta t \gamma = \frac{\Delta t}{\sqrt{1-v^2/c^2}} $$ $$ L'=\frac{L}{\gamma}=L\sqrt{1-v^{2}/c^{2}} $$ The volumetric decay rate according to the lab reference frame is found by correcting for both the increased density (due to length contraction) and the decreased decay constant (due to time dilation). $$ N' = L' A = \frac{N}{\gamma} $$ $$ R' = \frac{N'}{N} \frac{\Delta t'}{\Delta t} R = R$$ It's the same volumetric decay rate! Amusingly, the $Q$ value of the decay would be greater, but that's aside the point. Question: What if the material was put in a large gravity well? If you use the coordinates from outside the gravity well, would you obtain this same result?


title: a positive potential as $ x \rightarrow \infty $

text: let us suppose i can calculate the asymptotic of any potential $ V(x) $ in one dimension , and that i manage to prove that $ V(x) \ge 0 $ as $ x \rightarrow \infty $ could i conclude taht if or big 'x' the potential is POSITIVE that for big 'n' the enegies will be also POSITIVE $ E_{n} \ge 0 $ ?? as $ n \rightarrow \infty $ the idea is that if we define a turning point $ E=V(a) $ and 'a' is a turning point then if a is very big for big energy E then V(a) will be also positive but this is just in the WKB approximation isn't it ??


title: Galilean relativity in projectile motion

text: Consider a reference frame $S^'$ moving in the initial direction of motion of a projectile launched at time, $t=0$. In the frame $S$ the projectile motion is: $$x=u(cos\theta)t$$ $$y=u(sin\theta)t-\frac{g}{2}t^2$$ I know that that at $y_{max}$, $\frac{dy}{dt}=0$ so using this I find that $$t_{y_{max}}=\frac{usin\theta}{g}$$ so therefore: $$y_{max}=\frac{2u^2sin\theta}{g}$$ I know that when the particle lands at $y_{bottom}=0$ the distance in the $x$ direction is $$x_{y_{bottom}}=\frac{2u^2sin^2\theta}{g}$$ but I am confused about how to describe the motion for the particle in $S'$ frame.


title: Weak isospin confinement?

text: According to the Wikipedia article on color confinement: > _The current theory is that confinement is due to the force-carrying gluons > having color charge [...],_ i.e. because the gauge group is non-abelian. But that is equally true for the weak force. Does that mean that above the electroweak scale we should expect to find stuff like electrons and neutrinos confined to meson-like states?


title: _Actually_ hearing the shape of a drum

text: The problem of hearing the shape of a drum is well known as a mathematical problem. But what about as a problem of (1) physics and (2) human perception? Suppose you have the kind of membrane used with a kettle drum but without the kettle. One musician tries to adjust the pitch and loudness, etc., so that the square one and the circular one sound as similar as possible; an opposing musician is allowed to hear but not to see, and tries to identify the round one and the square one by the sound. And so on with other shapes. And if unaided human perception doesn't work, how much analysis can be done with electronic gadgets? **Later note:** I find a comment below, saying this isn't actually a mathematical question. As I pointed out below, an answer might bear upon the mathematics. A simple-minded stereotype says mathematics can shed light on the things it's applied to, but not vice-versa. But that is known to be an error.


title: Neutrino and electromagnetic forces

text: I learned from Wikipedia that neutrinos "are not affected by the electromagnetic forces". How was this identified experimentally?


title: Local gauge invariance and fields

text: I have one question about local gauge invariance of the spinor and scalar theories. For the scalar complex field with lagrangian $L_{0}$ requirement of local gauge invariance leads us to the Lagrangian $$ L = L_{0} - J_{\mu}A^{\mu} + q^{2}\varphi \varphi^{*} A_{\mu}A^{\mu} - \frac{1}{4 }F_{\mu \nu}F^{\mu \nu} = L_{0} + L_{el.} + q^{2}\varphi \varphi^{*} A_{\mu}A^{\mu}, \qquad (.1) $$ where $J_{\mu}$ is the conserved quantity of the theory giving by $(.1)$, $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ is tensor of electromagnetic field, $A_{\mu}$ is the local gauge invariance electromagnetic field, $$ A_{\mu} \to A_{\mu} - iq\partial_{\mu}f \Rightarrow L_{el.} = inv. $$ So, my question: (after quantization) how the summand $J_{\mu}A^{\mu}$ describes interaction between two charges $Q, -Q$ of the scalar field? Maybe, the spinor case is analogical, because the lagrangian of local gauge invarince spinor field is very similar to $(.1)$, except the last summand.


title: Conservation of BRST current in QED

text: I am trying to understand the conservation of the BRST current in QED but am having some trouble. This is what I have so far, QED lagrangian density in Lorenz gauge is, $$L = \frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{1}{2\xi}(\partial_{\mu}A^{\mu})^2 + \partial^{\mu}\overline{c}\partial_{\mu} c$$ I have been working in $\xi=1$ gauge with the following BRST transformations, $$\delta A_{\mu} = \partial_{\mu}c$$ $$\delta c = 0$$ $$\delta \overline{c} = \partial_{\mu}A^{\mu}$$ Using Noether's Theorem, I think the BRST current should be $$j^{\mu} = \partial^{\mu}A^{\nu}\partial_{\nu} c + \partial^{\mu}c\partial_{\nu}A^{\nu}$$ I can not show that this current is conserved, using the equations of motion $\Box_x A_{\mu} = 0$ and $\Box_x c = 0$. I am left with, $$\partial_{\mu}j^{\mu} = \partial^{\mu}A^{\nu}\partial_{\mu}\partial_{\nu}c + \partial^{\mu}c\partial_{\mu}\partial_{\nu}A^{\nu}$$ which I don't think is equal to 0. I'm not sure what I have done wrong here so any help would be greatly appreciated. Thanks!


title: Does the mass of the damper affect the transfer function in a vertical mass-spring-damper?

text: Usually in system dynamics, I dealt with horizontal mass spring dampers. Now in my advanced class I am dealing with vertical mass spring dampers. So a spring is hanging from the ceiling with a mass connected, and then the damped is under the mass. The damper is in some sort of oil and that is creating the "damping" effect. In this problem, the spring is assumed to be negligible mass, but the damper has a mass, along with the weight that is acting as the mass. I know that for a mass spring damper system (when its horizontal), the transfer function is $$H(s)=\frac{\omega_{n}^{2}}{s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}}$$ But now the mass of the weight and damper are acting on this system. So does that affect the transfer function?


title: Do Maxwell's equations independently impose constraints on the speed of light?

text: My question is about the relations and equations that makes us to impose constraints on the velocity at which electromagnetic waves propagate.   * Do Maxwell's equations independently impose constraints on the speed of EM waves?    * Are these equations compatible with the two special relativity principles with no need to consider some constraints?    * Does exceeding the limitation for speed of light violate the implications of Maxwell's equations?    * Does considering unequal constant values of velocity of light for different inertial references violate what Maxwell's equations imply?   * Who imposed such a limitation theoretically at first? What motivated him/her to suppose there is a limitation for the group velocity of electromagnetic waves?  We know Lorentz transformations are constructed on the assumption of constant speed of light in moving frames. What made him consider such an assumption, if Einstein was not the first one to consider the second postulate of special relativity (i.e. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light)? * * * Added: Einstein assumed "constancy of the speed of light for all observers of all moving frames" to derive and use his "Lorentz transformation" like transformation! Then he constructed his special theory of relativity based on two principles which we all heard about. Did I get this right? Considering the transformation for a moving frame along the $x$ axis for a frame moving at the speed of $v$ you get $${x_2} = {{{x_1} - {v_1}{t_1}} \over {\sqrt {1 - {{({v \over c})}^2}} }}.$$ This transformation mathematically implies that no frame is allowed to move at a speed higher than $c$. So this assumption puts some constraints on the speed of any moving frame too! Summing all these up, say: A) "the maximum speed of light has an upper bound which is called $c$" B) "nothing travels faster than light" C) "the speed of light is measured to be the same by all observers" From those, some questions arise:   1. Considering the definition of a frame and the observer which could be at a quite arbitrary conditions of velocity and so on, what motivated him to accept such a limitation for the speed of frames? I mean such a precise assumption can't come from nowhere! Particularly when its consequences seems unbelievable!   2. Why light? How was he so sure that nothing else could be faster than light? Was there any evidence of light being the fastest thing ever exists?   3. Where does the role of Maxwell's equations rise up in the creation story of this assumption?


title: Microscopic Theory of superfluidity

text: In comparison with superconductivity what is the microscopic theory of superfluidity? Who has done it?


title: How do we generalize limitations on the propagation speed of sinusoidal waves to limitations for transmitting any kind of information?

text: Our calculations on EM waves basically deal with harmonic waves and through such calculations and relations, we derive some relations and limitations for wave like phenomena. Of these relations are group and phase velocity of a wave. We impose some limitations on how great this velocity (e.g. for an EM wave) could be. But how could (and why) we generalize such limitations ( like the one on the propagation speed of sinusoidal waves) to limitations for transmitting any kind of information? Plus: What is the precise definition of "information" (which we always say it can not travel faster than C )??


title: Schwinger representation of operators for n-particle 2-mode symmetric states

text: A bosonic (i.e. permutation-symmetric) state of $n$ particles in $2$ modes can be written as a homogenous polynomial in the creation operators, that is $$\left(c_0 \hat{a}^{\dagger n} + c_1 \hat{a}^{\dagger (n-1)} \hat{b}^{\dagger} +c_2 \hat{a}^{\dagger (n-2)} \hat{b}^{\dagger2} + \ldots+ c_n \hat{b}^{\dagger n}\right)|\Omega\rangle,$$ where $\hat{a}$ and $\hat{b}$ are the annihilation operators, $c_i$ are complex coefficients and $|\Omega\rangle$ is the vacuum state. Alternatively, one can express the same state as a state in the fully permutation symmetric subspace of $n$ qubits (equivalently - as a state of the maximal total angular momentum, that is, $n/2$). The question is the following - **for a general symmetric operator** $$\sum_{perm} (\mathbb{I})^{\otimes (n-n_x-n_y-n_z)} \otimes (\sigma^x)^{\otimes n_x} \otimes (\sigma^y)^{\otimes n_y} \otimes (\sigma^z)^{\otimes n_z},$$ **what is its equivalent in the terms of creation and annihilation operators?** Partial solution: For the simplest cases (i.e. $(n_x,n_y,n_z)\in\\{(1,0,0),(0,1,0),(0,0,1)\\}$ we get the following: $$ \sum_{i=1}^n \sigma^x_i \cong \hat{a}^\dagger \hat{b} + \hat{b}^\dagger \hat{a}$$ $$ \sum_{i=1}^n \sigma^y_i \cong -i\hat{a}^\dagger \hat{b} + i \hat{b}^\dagger \hat{a}$$ $$ \sum_{i=1}^n \sigma^z_i \cong \hat{a}^\dagger \hat{a} - \hat{b}^\dagger \hat{b}$$ (AFAIR it is called the Schwinger representation). It can be checked directly on Dicke states, i.e. (${n \choose k}^{-1/2}\hat{a}^{\dagger (n-k)} \hat{b}^{\dagger k}|\Omega\rangle$). For general case it _seems_ that we get $$: \left( \hat{a}^\dagger \hat{b} + \hat{b}^\dagger \hat{a} \right)^{n_x} \left( - i \hat{a}^\dagger \hat{b} + i \hat{b}^\dagger \hat{a} \right)^{n_y} \left( \hat{a}^\dagger \hat{a} - \hat{b}^\dagger \hat{b}\right)^{n_z} :,$$ where :expr: stands for the normal ordering, i.e. putting the creation operators on the left and the annihilation - on the right. However, it's neither checked (besides correlators for 1-2 particles) nor proven. Of course one can construct the operators recursively, e.g. $$ \sum_{i\neq j}^n \sigma^x_i \otimes \sigma^y_j = \left(\sum_{i=1}^n \sigma^x_i\right)\left( \sum_{i=1}^n \sigma^y_i\right) - i \sum_{i=1}^n \sigma^z_i \\\ \cong \left( \hat{a}^\dagger \hat{b} + \hat{b}^\dagger \hat{a} \right)\left( -i\hat{a}^\dagger \hat{b} + i \hat{b}^\dagger \hat{a} \right) \- i \left( \hat{a}^\dagger \hat{a} - \hat{b}^\dagger \hat{b} \right),$$ but the question is on a general closed-form result.


title: Inertial Mass of a scalar field

text: Does it make sense to talk of the inertial mass of a scalar field? By the equivalence principle, it must be equal to its gravitational mass. We know that the scalar field contributes towards the stress-energy tensor, so, shouldn't it have an inertial mass too?


title: How could we define speed without time?

text: This is a bit of a brain twister but it's a very serious question. Please be precise in your answer. I do not believe time exists. I believe it is simply an illusion of our perception, the perception of seeing things around us change. The faster things change the more time seems to go by. In some sort, change is the result of the dissipation of energy. Thus what I believe is that what we are really observing is the energy being dissipated and created all around us and we call it time. Since time doesn't exist (for me). That would mean that the equation of [ speed = distance / time ] is impossible. But nevertheless the concept of speed is fundamental for physics. What could we replace this equation with to no longer have speed dependent of time but of energy?


Can a sound like a sonic boom change a object like a wall? And how the light on the walls will be changed after a sound like this?


title: Confusions relating mass defect

text: ![enter image description here](http://i.stack.imgur.com/CleaJ.png) I think I am confused between the mass defect of just splitting up a nucleus into its component nucleons and the mass defect involved between the two sides of a nuclear transmutation reaction. I assume the problem refers to the second, and the answer is B by just plugging in the numbers of the table. However can somehow please explain why the mass of the left side of the reaction is less than the right? I thought when you SPLIT up an atom (as in the Lithium to 2 Heliums) you have to put in energy therefore the mass-energy should be higher in the product side? Thanks in advance!


title: Vertical Loop like Anti-gravity

text: Let's take a bottle (no cap) with half its volume filled with water. If we rotate the bottle at some slower average velocity, water does not _drop out_ of it. The same principle works on Roller Coasters. Wikipedia says that the phenomenon of Vertical Loop is used here. What is the physics behind this? (I mean, Which opposes or **equilibrates** gravity during the vertical loop?)


title: What is ``thermal" about a thermal quotient of EdS and EAds?

text: This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of $dS^n$ and $AdS^n$ respectively. But I am not clear about the physical motivations of the quotients that they take. They take the following quotients,   * The sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$. Here they want to identify $\phi \sim \phi + \frac{2\pi}{n}$ for some $n \in \mathbb{N}$ where $\phi$ is the phase of $z_1$. And this quotient is given by $\mathbb{Z}_n\backslash SO(2n+2)/SO(2n+1)$   * On $\mathbb{H}^{2n+1} = SO_0(2n,1)/SO(2n)$ they take the metric as, $cosh^2\rho dt^2 + d\rho^2 + sinh^2 \rho d\Omega_{2n-1}$ and take the quotient as $t \sim t + \beta$ and then the quotient space is given as $\mathbb{Z} \backslash SO_0(2n,1)/SO(2n)$ . Here $\frac{2\pi}{n}$ and $\beta$ are connected by the same analytic continuation as what takes one from the metric on $S^{2n+1}$ to the metric on $\mathbb{H}^{2n+1}$ Clearly in the paper they keep referring to \beta as the inverse temperature but I can't clearly see what is the thermal picture which will correspond to evaluating the partition function on the these quotients as above. If there were a particle at inverse temperature $\beta$ on spatial $\mathbb{H}^{2n}$ then I would think that one would calculate the partition function on the manifold $\mathbb{H}^{2n}\times S^1$ such that the $S^1$ is of radius $\beta$. Is this somehow related to the quotient of $\mathbb{H}^{2n+1}$ described above in the second bulled point? (...the same could have been asked about $S^{2n} \times S^1$ also..)


title: Why do some materials reflect (metals) and other materials reflect and refract (glass) from the quantum perspective?

text: Recently I was asked to explain the difference between reflection and total internal reflection from a purely conceptual standpoint (no math). Let me explain what I already know. Reflection and refraction at the quantum level are the same thing. Light is a photon. A photon is a discrete particle that has wave characteristics (a finite wave traveling like a bullet). As the photon travels it collides with electrons in the matter of the medium it is traveling in. Depending on the energy of the photon and the allowed energy bands of the medium the photon cause the electron to jump up a level. If the photon is absorbed then the medium will increase its motion (at the macro scale increasing its temperature). If the photon is not absorbed it will be re-emitted (really as a new photon). I have read and watched Feynman's QED lectures and book and have a pretty good understanding of his process for determining how all these paths come together to give the net path of the photon. The general rule of thumb is that the photon wants to take the path which requires the least amount of time. I understand how this principle goes to explain refraction and reflection. What I don't seem to understand is why does one material seem to cause a higher percentage of refraction compared to another (metal vs. glass). What about the electron configuration of a the medium changes the net effect of the absorption and re-emissions of the photons? Is there a change in the probability of the photon being re-emitted in a reverse direction? Is there a farther distance the photon can travel before being incident onto an electron? This is the part where my understand breaks down. When you have hit the critical angle in a medium that refracts and the light completely reflects, are the photons moving is the same manner as they would be in a material that always reflects? How does this connect to the question in the previous paragraph? I know I have a bunch of mini questions embedded in answering this one larger question. Any help on any of the parts would be greatly appreciated?


title: Limits of laser spectroscopy for determining size of a molecule or particle

text: Say you have a number of particles respectively molecules suspended in air. The diameter of the particles $d_{particle}$ are of the order 1 to 100nm. Can you actually, i.e. in practice, measure the size (i.e. the diameter) of those particles using laser spectroscopy? The suspension in air may be a knock-out criterion for this measurement (I'm not sure). If so, would it change anything if the particles/molecules were placed in a vacuum chamber? What are the current experimental limits on doing such measurements with laser spectroscopy?


title: Why does water freeze at $0~^\circ\text{C}$ while poultry freezes at $-2.2~^\circ\text{C}$?

text: We all know the freezing point of water is $0~^\circ\text{C}$, however correct me if I'm wrong, but I believe the freezing point of poultry (which I'm guessing means water in poultry) is $-2.2~^\circ\text{C}$. If so, why is this the case?


title: At what point does everything become nothing?

text: I understand that the universe, which I'll call "everything", is expanding and it used to be much smaller. But I keep hearing assertions about a universe coming from nothing. If you rolled the clock back, at what point would everything become nothing?


title: Why will crushing a partially empty 2-liter bottle keep the soda more fizzy?

text: One of the curses of 2-liter bottles is that when carbonated beverages are stored in them for long periods (i.e. days between drinks) at least the last liter ends up being flat. However, if after every pour the bottle itself is crushed such that the remaining volume of air above the liquid is as small as possible, and such that the structure of the bottle will prevent it from deforming back to its original shape (easily done by crushing the dome first and then working your way down), then the soda will stay nice and fizzy down to pretty much the bitter end, when it becomes impossible to crush the bottle any more but a significant air volume remains. I know the answer has to do with vapor pressures and equilibrium, but when I try and articulate my thoughts to others their eyes glaze over. Can anyone come up with a broadly accessible, fairly pithy, and yet technically correct explanation? I realize this is perhaps slightly more on the chemistry side of things than physics, but (1) there's no chemistry Stack Exchange site, and (2) this post seemed reasonable precedence.


title: Brewster angle and perpendicular component

text: Fresnel's equations give that at Brewster's angle, the parallel component of the incident light is zero. The reflected perpendicular component is not zero, though. We can "remove" this component using multiple plates. Assume we have 'N' plates. How can I formulate a relation for the power of the perpendicular component as it passes through these N plates at Brewster's angle?


title: How can Planck units be consistent with conflicting dimensions of mass?

text: I suspect I'm missing something obvious, but I'm coming up blank. I've gotten pretty comfortable with so-called natural units over the years: in doing quantum mechanics/QFT, it's common to set $c = \hbar = 1$ and in GR, it's common to set $c = G = 1$. I'm told that quantum gravity researchers are fond of Planck units, where (according to that Wiki page) we set $c = \hbar = G = 1$, along with a couple other fundamental constants. I don't know what to make of this, though: it's dimensionally inconsistent in 3+1 dimensions. That same article gives the dimensions of each constant: $$\begin{align} [c] &= L T^{-1} \\\ [G] &= L^3 M^{−1} T^{−2} \\\ [\hbar] &= L^2 M T^{−1} \end{align}$$ Setting $c$ and $G$ to 1 implies $L=T=M$. Setting $c$ and $\hbar$ to 1 implies $L=T=M^{-1}$. Hence, making all three unity is internally inconsistent: mass cannot simultaneously have dimensions of length and reciprocal length. So, how do Planck units work?


title: Application of special relativity on sound waves

text: What is the difference between an electromagnetic wave and other waves that amounts to the EM wave following the Special Relativity. I have been reading about the Special Relativity for some time, and the arguments provided for light having a constant speed w.r.t. to all observers in all the inertial frames moving relative to each other seem to work on other waves too in my understanding. What is the point I am missing out on ? What separates the other waves from an EM wave regarding the 2nd postulate of Special Relativity ? If it is the EM nature, please elaborate.


title: Pressure and Density Using a General Lagrangian

text: Given a lagrangian of a form: \begin{equation}\mathcal{L}=f(\phi,\partial_{\mu}\phi\partial^{\mu}\phi)\end{equation} where $f$ is a function, I need to derive pressure and density in a FLRW universe with $g^{\mu}_{\nu}=\delta^{\mu}_{\nu}$. My approach is using: \begin{equation}T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\partial}{\partial g^{\mu\nu}}(\sqrt{-g}\mathcal{L})\end{equation} \begin{equation}=g_{\mu\nu}\mathcal{L}-2\frac{\partial\mathcal{L}}{\partial g^{\mu\nu}}.\end{equation} And finally, \begin{equation}\rho=T^0_{ 0}\end{equation} \begin{equation}P=T^i_{ i}\end{equation}. The problem I am facing right now is how to explicitly use the form of lagrangian to simplify the energy momentum tensor. Can anyone please help me? Edit I: Just to clarify, $\phi$ depends only on $t$ and is independent of $x^i$. Edit II: Okay, I solved it. Here's the way to do it: \begin{equation}T_{\mu\nu}=g_{\mu\nu}\mathcal{L}-2\frac{\partial\mathcal{L}}{\partial X}\partial_{\mu}\phi\partial_{\nu}\phi.\end{equation} \begin{equation}\rho=T^0_{ 0}=\mathcal{L}-2\frac{\partial\mathcal{L}}{\partial X}\dot{\phi}^2\end{equation} \begin{equation}P=T^i_{ i}=\mathcal{L}\end{equation}. where $X=\partial_{\mu}\phi\partial_{\nu}\phi$.


title: Where's the mass?

text: When you look a the Polyakov or Nambu-Goto action, you see a T for tension, but no mass. When you look the action of a relativistic particle, you see the m. See here. So here's the question. Where is the mass in the String actions? For a classical relativistic string I would expect that the equations of motion have m appearing as a constant. In this case I am thinking that the mass is related to the tension. In the end, if particles are strings in different states, then I would expect the mass to depend on the state.


title: Is the "far" universe expanding more quickly?

text: I'm reading this silly Time article: http://www.time.com/time/health/article/0,8599,2044517,00.html And they say "Even at its best, the 20-year-old telescope never had the acuity to peer so far into space, where the rapid expansion of the universe causes light waves to shift to a deep red." Doesn't that imply that things in the far universe are expanding more quickly? Is that true? Isn't space expanding faster than the movement of objects within space? Do both cause redshift to be observed?


title: Implementing a black-body using Planck's Law in a 3D ray tracer

text: **1_** In one of my software projects, I need to implement Planck's Law in terms of wavelength. I can't be sure if it's right because I've seen different variations of it, and all the implementations that I've tried give different results (besides, it's been years since I've had to do physics). Could someone verify that it's correct? Have I got the units right? It looks something like this, and it's the implementation of Planck's Law in terms of wavelength found on this wikipedia page:               h = 6.6260695729e-34 J*s     k = 1.3806488e-23 J*K^-1     c = 2.998e8 m/s     c2 = pow(c, 2) m^2/s^2     h_c = h * c     _2_h_c2 = 2.0 * h * c2          planck(wavelength, temp) {       return _2_h_c2 / (pow(wavelength, 5.0) * (exp(h_c / (wavelength * k * temp)) - 1.0))     }      `temp` is the temperature in Kelvins, and `wavelength` is in nanometers. **2_** If I wanted it normalized about 555nm, similar to this wikipedia page, what would it look like? This is what I have so far, but, again, I can't be sure if it's right:               planck(wavelength, temp) {       Real norm = pow(555.0, 5.0) * (exp(h_c / (555.0 * k * temp)) - 1.0);       return norm / (pow(wavelength, 5.0) * (exp(h_c / (wavelength * k * temp)) - 1.0));     }      **3_** Say I have an array of wavelengths from 360nm to 830nm, in increments of 5nm, and I apply the `planck` function with temperature of 6504k to each wavelength, and store the results in a new array. If I use this new array (spectral power distribution) and calculate the XYZ values for it, it should match CIE Standard Illuminant D65, should it not?


title: Lepton masses in the Standard Model

text: Some simple questions regarding leptonic masses in the Standard Model (SM):   1. Why there is not an explicit mass term in addition to the effective mass term that arises from the Yukawa terms after spontaneous symmetry breaking (SSB).   2. The lepton mass term is: $$m_{l_i}\sim y_i\cdot v \sim y_i\cdot \frac{m_H}{\sqrt{\lambda}}$$ with $m_{l_i}$ the lepton mass, $y_i$ the Yukawa coupling, $v$ the vacuum expectation value of the Higgs field, $m_H$ the Higgs mass, and $\lambda$ the self-coupling of the Higgs.   So the running of leptonic masses in the SM are given by the running of $y_i$, $m_H$, and $\lambda$. Thus the lepton masses are not protected to radiative corrections $\beta(m_{l_i})\neq 0$ for $m_{l_i} = 0$. This is in contrast with explicit lepton mass terms in theories without SSB where Chiral symmetry protect fermions' masses from radiative corrections. Is this correct? If not, why?


title: Relation among anomaly, unitarity bound and renormalizability

text: There is something I'm not sure about that has come up in a comment to other question: Why do we not have spin greater than 2? > It's a good question--- the violation of renormalizability is linked > directly to violation of unitarity, which was exploited by Weinberg > (surprise, surprise) to give an upper bound of something like 800 GeV on the > fundamental Higgs mass from the W's and Z's unitarization. The breakdown of > renormalizability is a wrong one-loop propagator correction to the gauge > boson, and it leads to a violation of the ward-identity which keeps the non- > falling-off part of the propagator from contributing. It's in diagrammar (I > think), it's covered in some books, you can ask it as a question too. I know what unitarity bound is the user talking about, but I don't know what is the violation of the Ward identity that he mentions. I guess that it is the _global_ $SU_L(2)$ symmetry but I have never seen anything relating the unitarity bound and this anomaly. The general issue is the following: Assume a Yang-Mills term and the coupling of the charged vector field to a fermionic conserved current under a global symmetry. Then one adds an explicit mass term to the vector field so that one breaks the gauge symmetry by hand, but not the global part that gives the conserved current (the gauge symmetry that goes to the identity in the boundary entails constraints). Then, according to the user (at least what I understood), when one takes into account loops effect the global part is also broken. Therefore, the mass term is breaking the redundancy part of the symmetry by hand (at the classical level) and it is also breaking the global part at the quantum level. I would be grateful if somebody is able to clarify this to me. References are also welcome.


title: condensed matter physics must reads

text: > **Possible duplicate:**   >  Books for Condensed Matter Physics I'm looking to learn more about cutting edge research in condensed matter theory. I hope you'll help me find some recommended articles in the (admittedly very broad) subject. I have a background in mathematics and quantum field theory, so I want to find articles which are at a high level but do not assume too much condensed matter jargon and are very explicit in their derivations.


title: Why $\lambda\phi^4$ theory, where $\lambda>0$, is not bounded from below?

text: Why the following interaction, in QFT, $$\displaystyle{\cal L}_{\rm int} ~=~\frac{\lambda}{4!}\phi^4$$ where $\lambda$ is positive, represents a theory that is unstable (or unbounded from below as it is usually said in textbooks). How can can one show that explicitly?


title: Are virtual photons affected by effective gravity in non-linear quantum electrodynamics?

text: Quantum electrodynamics based upon Euler-Heisenberg or Born-Infeld Lagrangians predict photons to move according to an effective metric, which is dependent on the background electromagnetic field in a given space domain. In other words, photon trajectories are curved in presence of electromagnetic fields, meaning that an effective gravity is acting upon the photons. Charged fermions in the same space domain are obviously electromagnetically interacting; one thus wonder whether the interaction mediating virtual photons are also affected by this effective gravity. If the answer is yes, fermions will likely be affected, according to extended charge models, so that electromagnetic fields will show as another source of gravity for any kind of particles.


title: Proper and rigourous derivation of Gaussian beam?

text: Gaussian beams are known solutions to the Paraxial Wave Equation: $$ \frac{\partial^2 \Psi(x,y,t)}{\partial^2 x} + \frac{\partial^2 \Psi(x,y,t)}{\partial^2 y} = 2ik\frac{\partial \Psi(x,y,t)}{\partial t} $$ I am looking for a **proof** (or a reference to it) that the _general_ solution to this equation is a Gaussian beam? Since they usually just assume that a Gaussian beam is OK, they plug it in and show that it works.


title: When and how were relative distances to the planets first measured?

text: I understand that the absolute distance to a planet can be measured using earth-baseline (e.g., diurnal) parallax, and that the first reasonably accurate such measurement was made for Mars by Cassini (and his assistant Richer) in 1672, and then, famously (with some modifications) for Venus by Halley, Cook, and others and others during its 1769 transit; but when were earth-baseline _relative_ distance measurements first made for each of the planets? Interpretation of such measurements requires assumptions about the nature of the orbits of planets, but some of the most compelling evidence for the validity of these assumptions is Kepler's third law of planetary motion, the persuasiveness of which itself rests on observations of relative distances. In fact, all of my texts say, in effect, that the third law "fit" the observed pattern of orbital period and relative distance. Where did these distance measurements come from? What relative distance measurements were available to Kepler in 1619? When was earth-orbit-baseline parallax used to make these measurements for each of the planets?


title: How could Horrocks have measured the AU?

text: I have always understood that the great historical significance of the transits of Venus, and the reason for the expeditions mounted to observe it, were that, by observing it simultaneously from _two distant locations_ , the absolute distance to the Sun could be measured. But I read in several sources, including some of my texts and two articles in Wikipedia, that Jeremiah Horrocks's observations of the 1639 transit of Venus allowed him to > make an estimate of the distance between the Earth and the Sun, now known as > the astronomical unit (AU) Horrocks's made his observations from a _single location_ , and thus could not have been using parallax (as was done 130 years later) to arrive at his estimate. Without knowledge or the sizes of both the Sun and Venus, how could he have performed the necessary calculations? Was he simply using the " well-informed guess as to the size of Venus" mentioned in the Wikipedia article. If so, what made it "well-informed"


title: Is this phase right?

text: Hello at physics lectures we wrote a phase of a sine wave like this: $$\phi = kx - \omega t$$ Is this right? As I recall the phase of a wave should be written like this: $$\phi = \omega t - kx$$ And if a wave changes direction $(k \rightarrow -k)$ like this: $$\phi = \omega t + kx$$ Can someone explain to me if first usage is even possible and when if so.


title: Classical Wave Equation - Approximations

text: I don't understand the derivation of the wave equation given below - $$T \sin (\theta _1) - T \sin (\theta ) = T\tan (\theta _1 )-T\tan (\theta ) = T \left. \left(\frac{\partial f}{\partial z} \right|_{z + \Delta z} - \left. \frac{\partial f}{\partial z}\right| _z \right) = T \frac{\partial ^2 f}{\partial z^2} \Delta z$$ I understand that the small angle approximation was used, but I'm at a loss for figuring out we turned $\tan$ into a derivative, and then after made it become a second derivative. The derivative of $\tan \theta$ is of course $\sec \theta$ which is equal to $\frac{1}{cos \theta}$, which was taken with respect to $\theta$, maybe there's a way to use the chain rule to find $\partial _z f$?


title: Are signal fronts in a beam not at rest to each other?

text: I'd like to investigate how the notion of "mutual rest" might be applied consistently, but distinctively, in the following thought experiment: Consider a light source ("$A$") which directs a beam towards a receiver ("$B$"). Source $A$ is modulating the beam, emitting discrete recognizable signals in rapid succession (i.e. with short durations from stating one particular recognizable signal modulation until stating the next; compared to $A$'s duration from stating one particular recognizable signal modulation until observing the corresponding echo from receiver $B$). Is there some definite, operational sense of "mutual rest" whereby in this case it could be said   * that source $A$ and receiver $B$ had been "at rest to each other" ("throughout the experiment", if applicable); but    * that a successive pair of signal fronts had **not** been "at rest to each other" ("while both were on their way")  ?


title: Interstellar dust/matter distribution

text: It is known that one of the main problems of interstellar flight is a presence of matter between stars in form of very fine dust and huge asteroids. Which can slowly (or fast) destroy any ship. What are the most recent estimations about the concentration and size distribution of interstellar matter? From very small to high scales. Particularly i am interested in the local group of stars. Update II: smallest object still named as a meteorite (Yamato8333) is about 12mg in mass and roughly 2mm in diameter. Update: Below is an image of an interplanetary dust particle (IDP, not Yamato8333). ![](http://i.stack.imgur.com/DmvhV.png) "IDP composed of nanometre-sized mineral grains and organic matter." (Credit: N. Spring) Original post here.


title: A book sits on a table. What is the net force of air pressure?

text: An elementary problem asks, > A book is at rest on a table top. In what direction is the net force of air > pressure on the book? Is this a meaningful question, and if so, what's the answer? If we imagine that the book and table are completely smooth, there's no air between the book and table. Then the air pressure from above the book creates a large net downward force of hundreds of pounds. (This will be countered by an upward force from the table.) On the other hand, if we imagine that due to the rough nature of the book and table, there is some air between the book and the table at most places, maybe there is enough air underneath the book to provide a net pressure force that is upwards. The scale height of the atmosphere is about 10^4m, so a 1cm book needs to have less than 1 part in 10^6 in contact with the table to have net upward force from air pressure. How realistic are these approaches? Do we need a molecular view of the book, table, and air to understand the situation? For a typical, everyday book and table, is it meaningful to ask what direction the net force of air pressure pushes?


title: Do topological superconductors exhibit symmetry-enriched topological order?

text: Gapped Hamiltonians with a ground-state having **long-range entanglement** (LRE), are said to have **topological order** (TO), while if the ground state is **short-range entangled** (SRE) they are in the trivial phase. The topological properties of a system with topological order (such as anyonic statistics, ground-state degeneracy etc.) are protected for any perturbations of the Hamiltonian that does not close the energy gap above the ground state. See more details here. If we further require that the system is protected by a symmetry $G$, the LRE states split further into several classes called **symmetry-enriched topological order** (SET). The SRE state (which is trivial without symmetry protection) is also split into several classes called **symmetry protected topological order** (SPT). The added physical features of these systems (such as gappless edge-states) are only protected for perturbations which do not close the gap _and do not break the symmetry $G$_. **Topological insulators** are know belong to SPT states, they are SRE and their topological properties are only protected by a symmetry. Related cousins to topological insulators are **topological superconductors**. In this context, one usually think of superconductors as insulators + particle-hole symmetry (which comes from the Bogoliubov de Gennes Hamiltonian + Nambu spinor notation). This might lead you to conclude that topological superconductors are also SPT states. However, it is known (by some) that superconductors cannot be described by a local gauge-invariant order-parameter as usual symmetry-breaking phases of matter (but a non-local gauge invariant order parameter exist.) A s-wave superconductor is actually topologically ordered (and thus LRE), and exhibits anyonic statistics, ground-state degeneracy on higher-genus manifolds, low- energy topological quantum field theory (TQFT) description and so on. It is topologically in the same phase as the famous Kitaev Toric code, the $\mathbb Z_2$ topological order. See details here. _**Now, my question is the following:_** is it wrong to consider topological superconductors (such as certain p-wave superconductors) as SPT states? Aren't they actually SET states?


title: What is Si-delta doping?

text: I want to know what the delta means in this case. I know the Si means the element used, by some way to doping. I guess the delta means that using some elements to create holes in semiconductor made of Si. Am I right?


title: Dielectric modification of electrostatic equations?

text: I have learnt that in cases of electrostatic fields inside a dielectric of any source charge, the field is reduced by a factor of K( if K, the dielectric "constant", is taken everywhere to be same). But that means that if there is, say, a fixed (by non-electric forces) charge outside a randomly shaped dielectric block, then the fields experienced by our test charge inside the dielectric will be weaker (by a factor of K) than what it would have experienced had it been outside the dielectric. But should the fields of other charges outside the block be unaffected?? If this is so, we take two opposite charges in vacuum separated by a distance and a dielectric block in between them. The lines of force would be passing through the slab and vacuum both. If the fields outside are not affected than there would be a contradiction since the closed line integral of field wouldn't be zero. But then how and why would the fields outside the dielectric be changed. The contribution of the induced dielectric charges should be zero at least in the idealised case where infinitely large plates (producing uniform fields) have a dielectric bock (smaller then the separation) between them??


title: Physical Interpretation of a Scalar Quantity Related to Currents/Conservation Laws

text: Let $Q_{ab} = (\psi_{;a})(\psi_{;b}) - (1/2)g_{ab}|\nabla \psi|^2$ be the energy-momentum tensor of the wave equation in some space time. I will use semicolons to refer to covariant differentiation and $\partial$s to refer to coordinate differentiation. Let $\pi_{ab}$ be the deformation tensor for some fixed vector field $X$. In deriving "almost conservation laws" one uses the identity. $(Q_{ab}X^b)^{;a} = (\psi^{;a}_{;a})(X^a\partial_a\psi) + (1/2)Q^{ab}\pi_{ab}$ Does there exist a physical interpretation of the scalar $Q^{ab}\pi_{ab}$ that is not based on the above formula? I am most interested in how one should think about this quantity in relativistic contexts, e.g. black hole geometries. P.S. Just in case anyone is tempted, I am looking for something more than "$Q^{ab}\pi_{ab}$ vanishes if the flows of $X$ are isometries."


title: Could a human run horizontally inside a Wall of Death?

text: A popular circus stunt is for a motorcycle rider to ride inside a bowl shaped depression called a "Wall of Death." The rider goes higher and higher up the wall until they are actually horizontal. I wonder if a human could do the same.


title: Sphere in homogenou magnetic field

text: I have to find out the magnetic dipole moment of sphere in homogenou magnetic field in the limit $\mu_r\rightarrow+\infty$. I don't know what does the condition $\mu_r\rightarrow+\infty$ mean. Does it mean, that the field inside the sphere will be 0 because it won't let any field in? In that case should I determine the surface currents on the sphere and determine the dipole moment from that?


title: When and how do you represent a two body state as a tensor product?

text: I have read that in quantum mechanics, compound systems are constructed as tensor products. But on page 177 of Griffith, for example, a two body wavefunction is introduced as Psi (x1,y1,z1,x2,y2,z2,t). (Six space dimensions plus time) This is clearly a direct sum, not a tensor product. If this is correct, when do you use a direct sum and when do you use a tensor product? Why isn’t the tensor product representing the two body system Psi(x1x2,x1y2,x1z2,y1x2,y1y2,y1z2,z1x2,z1y2,z1z2,t) or something similar?   (nine tensor dimensions plus time). What am I misunderstanding? And what are the tensor space dimensions?   Or how do these abstract dimensions relate to physical lab space dimensions?


title: Can one create mass from energy?

text: Due to $ E =m c^2 $, one can convert mass to energy. A classic example would be matter/anti-matter annihilation to produce energy (photons, etc.). Can one do the reverse? So could one do something to photons to create mass?


title: Does a constantly accelerating charged particle emit em radiation or not?

text: The Abraham-Lorentz force gives the recoil force, $\mathbf{F_{rad}}$, back on a charged particle $q$ when it emits electromagnetic radiation. It is given by: $$\mathbf{F_{rad}} = \frac{q^2}{6\pi \epsilon_0 c^3}\mathbf{\dot{a}},$$ where $\mathbf{\dot{a}}$ is the rate of change of acceleration. If a particle has a constant acceleration, $\mathbf{\dot{a}}=0$, then there is no reaction force acting on it. Therefore the particle does not lose energy. Does this mean that a constantly accelerating charged particle does not emit electromagnetic radiation?


title: What is electron density and its unit in plasma physics?

text: I can not find (through Google search engine) that what means electron density and what is its unit? and what is there relation between electron density and electrical conductivity?


title: How to find electric scalar potential of infinite wire with Poisson/Laplace equation?

text: I though it will be easier then calculating the electric field and then integrating, but I am stuck. lets say we have an infinite wire, charged $\lambda$ per unit of length and its located at the origin, infinite to $z$ axis. Then due to the symmetry in the problem we can say that $\partial\phi$ and $\partial z$ are $0$ so we are left with only the trivial solution - $$ C+D\ln r$$ now because the potential need to be 0 in $ r=\infty$ we get that $C=0$ but how to go from there? I though about using the charge density, maybe with $E$ (derivative) and say that $\rho=\delta(r)$ but I don't know how to go from there. help will be greatly appreciated.


title: Kinetic Energy vs. Potential energy with regards to creating particles

text: So I know that when you collide particles with high enough kinetic energy, (kinetic energy = at least the rest mass of the particles you are making), you get particles. How come potential energy cannot make particles? Say you have an electron held at a potential such that the amount of potential energy it has equals at least the rest mass of a particle you want to make. How come potential energy does not convert into rest mass energy directly? Would this require whatever is creating the potential to disappear because the potential energy turns into a particle? This shouldn't violate any conservation of energy laws does it? Another question that kind of relates to my previous question is, say we have a 512 keV photon. What prevents this photon from turning into an electron that then has kinetic energy of 1 kev?


title: Are there notable cases of anomalies in the mass-decay rate relationship?

text: As a general rule of thumb, massive particles (both composite and fundamental) tend to decay rapidly through the weak force, while less massive particles tend to be more stable. Hence, taus are shorter lived than muons, top quarks are shorter lived than charm quarks, and all mesons and baryons except protons and neutrons are highly unstable. My understanding is that this relationship is largely captured in the Standard Model electro-weak equations. Are there any notable cases where the decay rate and the mass of the particle appear experimentally to deviate from the expected relationship?


title: Quantum Zeno effect and unstable particles

text: Is it possible to increase indefinitely the lifetime of unstable particles by applying the quantum Zeno effect? Is there a bound from theoretical principles about the maximum extension one can get in a unstable energy level in general from continuous measurement alone?