title: Has anyone else thought about gravity in this way?

text: Picture yourself standing on a ball that is expanding at such a rate that it makes you stick to the ball. Everything in the universe is expanding at this same rate. To escape the earths gravitational pull we would need to jet upward faster than the expansion of the earth. Each object expands at a different rate on its surface according to its size. Thus different gravity affects for different size planets. When in space we are subject to being affected by the most distant body if we stand in its way. I just can not explain the reaction of our tides with our moon. Have any scientists seriously considered an idea like this? Follow up July 23 I am no scientist, but I think someone with more knowledge might explore this idea a little further. At the very lease the idea that every thing in the total universe is expanding, including all parts of the atom can be used as a simple way to see formulas and the same results to the effects of gravity of anything on the surface of a sphere planet or a donut shaped planet. The area of mass will grow but the density will remain the same. The idea can be cross referenced by light shifting etc, to see if it falls in line with the known action of planet gravity and the known expansion affect of the whole universe. Maybe the gravity affect of a planet on its surface dweller is a completely different force than is the force that maintains the orbits of planets. keplers law I believe. What happens when we have an eclipse of the moon?, does the earths orbit around the sun change for time of this eclipse? My summary is that if all scientists can not explain gravity totally, then maybe the common thought for all these years is not completely a correct one.


title: What is the physical meaning of diffusion coefficient?

text: In Fick's first law, the diffusion coefficient is velocity, but I do not understand the two-dimensional concept of this velocity. Imagine that solutes are diffusing from one side of a tube to another (this would be the same as persons running from one side of a street) to unify the concentration across the tube. Here we have a one-dimensional flow in x direction. The diffusion coefficient should define the velocity of solutes or persons across the tube or street direction. How the two-dimensional velocity does this? I wish to understand the concept to imagine the actual meaning of the diffusion coefficient.


title: Define Pressure at A point. Why is it a Scalar?

text: I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector. I am thinking pressure at a point is $P=\lim_{\delta A \to 0} \frac{\delta F}{\delta A}$. Please let me know if I am wrong. But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \mathbf{F}=-Pd \mathbf{A}$


title: Commutator of $L^2$ and $X^2$, $P^2$

text: In our quantum mechanics script, it states that $[L^2, X^2] = 0$ and $[L^2, P^2] = 0$, therefore for the following Hamiltonan $$H = \frac{P^2}{2m} + V(X^2)$$ it is that $[H, L^2] = 0$ therefore $H$ and $L^2$ have the same eigenvectors, and then it continues to calculate orbitals of the hydrogen atom. My question is: Is there a nice and simple proof of $[L^2, X^2] = 0$ and $[L^2, P^2] = 0$?


title: Force applied to two blocks

text: > Due to application of force tow blocks of mass 1Kg and 0.5Kg move together. > Each block exerts a force of 6N on each other. What is the acceleration by > which both the blocks move?


title: statistics for photon counts in photomultiplier

text: It seems that a common statistical model for the count numbers of a photomultiplier is a Poisson distribution whose parameter $\lambda$ equals to the square-root of the number of counts.(e.g. http://www.sciencedirect.com/science/article/pii/S1350448711005750). This in particular, implies that the variance of the resulting statistic   1. increases with the number of photons to detect,   2. is not directly dependent from the duration of the counting process. I did not manage to find the basis of this modeling choice. If somebody has some intuitive idea or a good reference I will apreciate. (I am not physicist and maybe I make a bad interpretation of the modeling applied to PM)


title: What Pauli matrices should I use for this two spin system?

text: Consider the Hamiltonian $H = -J_\text{F}S^{(1)}_zS^{(2)}_z + J_{AF}S^{(1)}_zS^{(2)}_z$, describing the graph ![enter image description here](http://i.stack.imgur.com/3lg1R.png) Here, F means ferromagnetic and AF means antiferromagnetic interactions. I am having problem with the value of $S^{(1)}_zS^{(2)}_z$.Someone suggested to me that $$S^{(1)}_z=\frac{1}{2}\begin{pmatrix} -1 & 0 &0 &0 \\\ 0&-1 &0 &0 \\\ 0 &0 &1 &0 \\\ 0 &0 &0 &1 \end{pmatrix},\quad S^{(2)}_z=\frac{1}{2}\begin{pmatrix} -1 & 0 &0 &0 \\\ 0&1 &0 &0 \\\ 0 &0 &-1 &0 \\\ 0 &0 &0 &1 \end{pmatrix},$$ and therefore $$S^{(1)}_z\cdot S^{(1)}_z=\frac{1}{4}\begin{pmatrix} 1 & 0 &0 &0 \\\ 0&-1 &0 &0 \\\ 0 &0 &-1 &0 \\\ 0 &0 &0 &1 \end{pmatrix}.$$ On the other hand, from page 7 of these notes on Pauli spin matrices, I know that for two spin systems $$\Sigma_z = \begin{pmatrix} 2 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & -2 \\\ \end{pmatrix}.$$ I asked the person but never got a reply. I don't see what is the difference between $S^{(1)}_z\cdot S^{(1)}_z$ and $\Sigma_z$; when I should use what?


title: Intercept a moving object

text: Object A can move at 50km/h, wants to intercept object B (currently $15^{\circ}$, east of north from A) moving at 26km/h, $40^{\circ}$ east of north. What angle should A take to intercept B? AB is 20km apart ![enter image description here](http://i.stack.imgur.com/Kcm8I.png) The provided answer looks like: * * * Choose x axis along 20km distance. $26t \sin{(40-15)} = 50t \sin{\theta}$ $\theta = \sin^{-1}{\frac{11}{50}} = 12.7$ $15 + 12.7 = 27.7$ * * * I took a different approach and used $\cos$ and got a different answer ... why is that? $26t \cos{(40-15)} = 50t \cos{\theta}$


title: Nonuniform circular motion

text: A ball rocks around an arc. In the following illustration, the ball reaches the end of the arc (its velocity magnitude is zero at that particular moment). ![Illustration](http://i45.tinypic.com/16ixldw.png) Now, I want to know which forces are acting on that ball at that particular moment. We have the tension force $\vec T_2$ acting on the ball, which is the centripetal force. We also have the gravity which consists of two Cartesian components: radial $mg \cos\alpha$ and tangent $mg \sin \alpha$. In the radial axis our net force ($\vec T_2 - mg \cos \alpha = mv^2 / R $) is zero because $v = 0$. However, in the tangent axis our net force is not zero - $mg \sin \alpha$. My question is - how it could be if the ball at that particular moment is not moving because he reaches the edge of his trajectory? If it doesn't move then there should be some opposite force acting on it. My intuition says that that tangent force $mg \sin \alpha$ is forcing the body to slow down in that direction, so it slowly "cancels" the force which caused the body to move initially. But how can I describe it with formulas and/or illustrate it from the point of view of inertial system (i.e., Earth)?


title: Does negative temperature in Carnot cycle yield a counterexample of the second law of thermodynamics?

text: By Carnot Theorem, the efficiency of Carnot cycle is$$\eta=1-\frac{T_C}{T_H}$$ where $T_C$,$T_H$ are the absolute temperature of the cold reservoir and hot reservoir respectively. Since $T_C > 0$, that means $\eta < 1$, so it concludes the Kelvin statement: "no process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work" However, I think in that period they don't know the negative temperature, if we let $T_C=300K$ and $T_H=-300K$, it is easy to get $\eta=2$. That seems a counterexample of the second law of thermodynamics.


title: Buoyancy fluxes in a stratififed fluid and units

text: I am calculating the buoyancy flux ($B$) for a stratified fluid as follows: $$ B=\frac{g\alpha S}{C_{pw}\rho_0} $$ where $g = 9.81$ $m/s$; $\alpha = 1.6 t\times10^{-5} + 9.6\times10^{-6} \times (20 \text{ degC})$; $S = 100\text{ }Wm^{-2}$, $\rho_0 = 1000$ $kg/m^3$, and $C_{pw}$ is the specific heat of water. The question I have is: Is $C{pw} = 4200$ or $4.2$ I've seen it used both ways, and I am unsure which I would use with the units of the other terms in the equation.


title: Excess charge on an insulator and conductor

text: So I was recently wondering what happens to the excess charge when it is placed on an insulator or conductor e.g. rubbing two objects together. I know in the conductor the electrons are free to move whereas in the insulator they in general are not very mobile so the charge stays in a small region. But why does this happen? I know about the differences in insulator and conductor band structure, how is this related? Do the excess electrons become part of the band structure? Thanks!


title: How many unequivalent Seifert surfaces appear in a AdS/CFT extension?

text: When introducing the 't Hooft diagrams from Feynman diagrams on a torus has there been a classification in terms of knots and Seifert surfaces?


title: Why are geons unstable? Are there other problems with geons?

text: I read in various places geons are "generally considered unstable." Why? How solid is this reasoning? Is the reason geons are not studied much anymore because we can't make more progress without better GR solutions or a better theory of quantum gravity, or is it because it really is a failed theory with fundamental problems (other than the unproven stability question)?


title: How far could one fire a charged particle in space?

text: I've wondered how far would be able to send a concentrated beam of electrons in space. The reason is would we be able to launch a magnetic ring with an electron absorbing material on one side and just fire electrons through the ring to create a push?


title: How energy conservation is not violated here?

text: If I lift a book of mass $m$ from the ground and put it at height $h$,then initial and final energies with respect to ground are 0 and $mgh$ respectively. So how energy is conserved here?


title: Treatment of boundary terms when applying the variational principle

text: One of the main sources of subtlety in the AdS/CFT correspondence is the role played by boundary terms in the action. For example, for a scalar field in AdS there is range of masses just above the Breitenlohner-Freedman bound where there are two possible quantizations and which one you get depends on what boundary terms you add to the action. Boundary terms are also essential in the treatment of first-order Lagrangians for fermions and self-dual tensor fields. These all involve the "UV" boundary as $z \rightarrow 0$ in Poincare coordinates. Then there are dual models of QCD like the hard-wall model where one imposes an IR cutoff and imposes boundary conditions at the IR boundary and/or adds IR boundary terms to the action. My question is a bit vague, but basically I would like references to reviews, books or papers that give a good general treatment of the variational principle when one has to be careful about boundary terms. It would help if they clearly distinguish the requirements that follow from mathematical consistency from those that are imposed because of a desire to model the physics in a certain way.


title: on understanding tunnel effect

text: i was reading a specific book on quantum mechanics, partiicularly on the tunnel effect section. I am trying to understand how the probability of an eletron passing a barrier of energy V0 is calculated for both $E>V0$ and $E<V0$, i.e., the electron having more energy and less energy, respectively. For simplification, the book assumes that, for x>0, there is a potential barrier of intensity V0. Applying Shroedinger EDP, it is concluded that: $f_1(x)=Ae^{ikx}+Be^{-ikx}, x<=0$ and $f_2(x)=Ce^{\gamma x}+De^{-\gamma x}, x>0$, where $\gamma=\sqrt{2m(V_0-E)}/h$, where the functions represent the electron wavefunctions in space. After applying boundary conditions on x=0 and x=+infinity, it is concluded that $f_2(x)=De^{-\gamma x}, x>0, V0>E, \gamma \in R$. Then assuming a barrier of length $a$, the book concludes that the probability of the electron going through the barrier is $e^{-2\gamma a}$. Question 1: could someone explain me exactly how to get this value? How are the wavefunctions normalized in this case? Question 2: what is the probability when $E>V0$?


title: Confront Order Of magnitudes

text: Is it correct to say that 9.0 is one order of magnitude smaller than 10.0? Has anyone a link/source about confronting order of magnitudes, apart from wikipedia?


title: Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization?

text: From what I understand, the uncertainty principle states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon. However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics. Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions? What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation!


title: Is the uncertainty principle a property of elementary particles or a result of our measurement tools?

text: In many physics divulgation books I've read, this seems to be a commonly accepted point of view (I'm making this quote up, as I don't remember the exact words, but this should give you an idea): > Heisenberg's uncertainty principle is not a result of our lack of proper > measurement tools. The fact that we can't precisely know both the position > and momentum of an elementary particle is, indeed, a property of the > particle itself. It is an intrinsic property of the Universe we live in. Then this video came out: Heisenberg's Microscope - Sixty Symbols (skip to 2:38, if you're already familiar with the uncertainty principle). So, correct me if I'm wrong, what we may claim according to the video is: > the only way to measure an elementary particle is to make it interact with > another elementary particle: it is therefore **incorrect** to say that an > elementary particle **doesn't have** a well defined momentum/position before > we make our measurement. We cannot access this data (momentum/position) > without changing it, therefore it is **correct** to say that our ignorance > about this data is not an intrinsic property of the Universe (but, rather, > an important limit of how we can measure it). Please tell me **how** can both of the highlighted paragraphs be true or how they should be corrected.


title: Uncertainty principle implies the non-deterministic universe?

text: Does the uncertainty principle imply the non-deterministic universe, or just the fact that our model of the universe, the one based on observation, can be at most non-deterministic, since we will not be able to measure with perfect accuracy, even if such would exist (and therefore we will have to provide probability distributions rather than an event which should happen with overwhelming probability (term used by Terence Tao))?


title: About Heisenberg uncertainty principle

text: What would happen if someone invented a way to measure both position and momentum precisely? If it is impossible why?


title: Is the uncertainty principle just saying something about what an observer can know or is it a fundamental property of nature?

text: I ask this question because I have read two different quotes on the uncertainty principle that don't seem to match very well. There are similar questions around here but I would like an explanation that reconciles these two interpretations specifically:   1. Feynman talks about the uncertainty principle in one of his lectures and mentions it as the reason why electrons don't crash into the atom's nucleus: If they did they would have an exact location and momentum which is not allowed by the uncertainty principle. In saying this it is clear that the uncertainty principle is a fundamental property of nature because it has an effect on where an electron can reside.   2. Recently I read - somewhere else but I forgot where exactly - an account of the uncertainty principle where there was explained how we can measure position of a particle by firing another particle into it, the collision disturbs the velocity of the observed particle therefore we can not know its momentum anymore. Now, 2) very much seems like a limitation of what the observer can know, while 1) attributes a fundamental property of nature to it (electrons don't crash into the nucleus). What is the correct way to think about this?


title: A list of inconveniences between quantum mechanics and (general) relativity?

text: It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them? E.g. relativity theory uses a space-time _continuum_ , while quantum theory uses _discrete_ states. I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest.


title: What are the Conflicting Predictions of General Relativity & Quantum Mechanics?

text: I see a lot of questions in various sites about why the 2 theories are or aren't incompatible, I'm satisfied as to why that's the case. However it has been mentioned that both theories make predictions about phenomena that contradict or are incompatible, and I've been unable to find any examples. What are the conflicting/contrary/incompatible predictions made by General Relativity versus Quantum Mechanics? Or are the claims false?


title: Incompatibility of GR and QM

text: I am told that the theories of General Relativity and Quantum Mechanics are fundamentally incompatible... Why is that? Someone explained that it had to do with the fact that quantum particles such As quarks and photons are points without volume... Leading to singularities. Do people really believe that ^ What data do we have to suggest that quarks are really zero dimensional points?


title: What is an example of a situation where Quantum Mechanics and Relativity do not work together?

text: I've learned special relativity in school last semester, and this semester we began learning about Quantum Mechanics, and my teacher told us that there was a Relativistic Schrodinger equation. I was under the impression that you couldn't use Quantum Mechanics and Relativity together, so why is it appropriate to have a Relativistic Schrodinger equation, and what is an example of a situation where QM and Relativity cannot be used together?


title: Quantum Mechanics and General Relativity in Macroscopic Level

text: Hi I read a book yesterday.The book was Brian Greene's The Elegant Universe. I learned that uncertainty principle affects space-time very microscopic levels and this affection makes conflict in General Relativity and Quantum Mechanics. I wanted to know this is the only reason that this two theoris can not be conbined or there are more reasons ?


title: General relativity && quantum mechanics "incompatibillity"

text: Now this may be utterly weird layman-physics-question, but anyways... I have read recently following: "The fundamental universe laws are everywhere the same. It's just that the manifestation (!) of (these) laws is different (!) on a specific "scale"." Which brings me to the question: Could it be that GR and QM are one and the same thing on a "different scale"? If yes (LOL), that would "explain" why it is not possible to "merge" GR and QM. Was question like this ever considered?


title: Sound as a use to separate molecular structures

text: Sound can be a destructive force. However, could it be used to separate say the Hydrogen atom from the Oxygen atoms?


title: From Navier–Stokes equations to Euler, Bernoulli, etc

text: How can one go from the 3D compressible Navier-Stokes equations to the simpler Euler equations, Bernoulli's equation and other fluid dynamic equations?


title: Zero modes ~ zero eigenvalue modes ~ zero energy modes?

text: There have been several Phys.SE questions on the topic of zero modes. Such as, e.g.,   * **zero-modes** (What are zero modes?, Can massive fermions have zero modes?),    * **majorana-zero-modes** (Majorana zero mode in quantum field theory),   * **path-integral-with-zero-energy-modes** (Path integral with zero energy modes), etc. Here I would like to understand further **whether "Zero Modes" may have physically different interpretations** and **what their consequences are** , or **how these issues really are the same, related or different.** There at least 3 relevant issues I can come up with: ## (1) **Zero eigenvalue modes** By definition, **Zero Modes** means zero eigenvalue modes, which are modes $\Psi_j$ with zero eigenvalue for some operator $O$. Say, $$O \Psi_j = \lambda_j \Psi_j,$$ with some $\lambda_a=0$ for some $a$. This can be Dirac operator of some fermion fields, such as $$(i\gamma^\mu D^\mu(A,\phi)-m)\Psi_j = \lambda_j \Psi_j$$ here there may be nontrivial gauge profile $A$ and soliton profile $\phi$ in spacetime. If zero mode exists then with $\lambda_a=0$ for some $a$. **In this case, however, as far as I understand, the energy of the zero modes may not be zero.** This zero mode contributes nontrivially to the path integral as $$\int [D\Psi][D\bar{\Psi}] e^{iS[\Psi]}=\int [D\Psi][D\bar{\Psi}] e^{i\bar{\Psi}(i\gamma^\mu D^\mu(A,\phi)-m)\Psi } =\det(i\gamma^\mu D^\mu(A,\phi)-m)=\prod_j \lambda_j$$ In this case, if there exists $\lambda_a=0$, then we need to be very careful about the possible long range correlation of $\Psi_a$, seen from the path integral partition function ( **any comments at this point?** ). ## (2) **Zero energy modes** If said the operator $O$ is precisely the hamiltonian $H$, i.e. the $\lambda_j$ become energy eigenvalues, then the zero modes becomes zero energy modes: $$ H \Psi_j= \lambda_j \Psi_j $$ if there exists some $\lambda_a=0$. ## (3) **Zero modes $\phi_0$ and conjugate momentum winding modes $P_{\phi}$** In the chiral boson theory or heterotic string theory, the bosonic field $\Phi(x)$ $$ \Phi(x) ={\phi_{0}}+ P_{\phi} \frac{2\pi}{L}x+i \sum_{n\neq 0} \frac{1}{n} \alpha_{n} e^{-in x \frac{2\pi}{L}} $$ contains zero mode $\phi_0$. * * * Thus: **Are the issues (1),(2) and (3) the same, related or different physical issues?** If they are the same, why there are the same? If they're different, how they are different? I also like to know when people consider various context, which issues they are really dealing with: such as the **Jackiw-Rebbi** model, the **Jackiw- Rossi** model and **Goldstone-Wilczek** current computing induced quantum number under soliton profile, **Majorana zero energy modes**, such as the Fu- Kane model (arXiv:0707.1692), Ivanov half-quantum vortices in p-wave superconductors (arXiv:cond-mat/0005069), or the issue with **fermion zero modes under QCD instanton** as discussed in Sidney Coleman's book ``Aspects of symmetry''. ps. since this question may be a bit too broad, it is totally welcomed that anyone attempts to firstly answer the question partly and add more thoughts later.


title: Is entangled state after measurement still an entangled state?

text: Suppose there is an entangled state of two electrons, the spin part is $$| \downarrow \uparrow \rangle - | \uparrow \downarrow \rangle \tag{1} $$. If I add the spatial part of the wavefunction as two Gaussians, it should be something like $$ ( e^{- (r_1-R_a)^2- (r_2-R_b)^2} + e^{- (r_1-R_b)^2- (r_2-R_a)^2})( | \downarrow \uparrow \rangle - | \uparrow \downarrow \rangle ) \tag{2} $$. Now I measure the spin of the electron at position $R_a$, and I get down result. The wavefunction should be $$ e^{- (r_1-R_a)^2- (r_2-R_b)^2} | \downarrow \uparrow \rangle - e^{- (r_1-R_b)^2- (r_2-R_a)^2} | \uparrow \downarrow \rangle \tag{3} $$ , which is still indistinguishable and antisymmetric. ( I cannot get just $ e^{- (r_1-R_a)^2- (r_2-R_b)^2} | \downarrow \uparrow \rangle $, which breaks the antisymmetry) Since the entanglement is defined by anything more than simple product, both wavefunctions (2) and (3) are entangled. However, is wavefunction (3) too trivial to be called entangled? It looks antisymmetrization itself will automatically produce entanglement.


title: Why consciousness should not be a reason for quantum decoherence?

text: As postulated by Stuart Hameroff in his article "quantum consciousness",that it is one of the reasons for reduction of quantum superposition. Roger Penrose suggested that consciousness is not necessary for reduction but a system can undergo self reduction or objective reduction due to intrinsic feature of spacetime. But he too added that consciousness is also responsible for decoherence. Their ideas were intensively criticized. So i want to know why consciousness should not be a reason?


title: How to prove the Hubble law is the unique expansion law compatible with homogeneity and isotropy?

text: In the book _physical foundations of cosmology_ , it saids that Hubble law is unique and a problem seems to be a hint of proving that. > In order for a general expansion law, **v** =f( **r** ,t), to be the same > for all observers, the function f must satisfy the relation > $$f(\bf{r_{CA}}−\bf{r_{BA}}, t) = f(\bf{r_{CA}},t)−f(\bf{r_{BA}},t),$$ where > ABC are three points in space. Show that the only solution of this equation > is given by the Hubble law. With a little help of Taylor approximation, I can convince myself $f$ should be linear and then a linear function without constant. But it seems to me that's not good enough for a proof...So how can one prove it in a more mathy way? Thanks !


title: Orientifold Plane, Anti-Dp Brane and SUSY breaking

text: In "String Theory and M-Theory" by K. Becker, M. Becker and J.H. Schwarz, page 222, they give a brief introduction about the (space-filling) Orientifold Plane $O9$ as an object needs to be add in the theory to make the it consistence (or at least, that what I thought). However, I still don't really get their arguments.   1. What are the physical properties of the Orientifold Plane $Op$ in String Theory?    2. How are they described in the context of targer space? How do they couple with objects in String theory (string + brane)?    3. Is there any low-effective description for it? What're the theoretical evidences of their existence in String theory?   4. Moreover, they also say that (on the same page) the presence of $\bar{Dp}$ branes breaks all the SUSY. How can I see that?


title: Negative probabilities in quantum physics

text: Negative probabilities are naturally found in the Wigner function (both the original one and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the Klein-Gordon equation. The question is if there is a general treatment of quasi-probability distributions, besides naively using 'legit' probabilistic formulas? For example, is there a theory saying which measurements are allowed, so to screen negative probabilities? _Additionally_ , is there an intuition behind negative probabilities? (Providing other examples than ones mentioned in the question can illuminate the issue.)


title: Classic mass predictions from Left-Right models with discrete symmetries?

text: I am covering the classic literature on predictions of Cabibbo angle or other relationships in the mass matrix. As you may remember, this research was a rage in the late seventies, after noticing that $\tan^2 \theta_c \approx m_d/m_s$. A typical paper of that age was Wilczek and Zee Phys Lett 70B, p 418-420. The technique was to use a $SU(2)_L \times SU(2)_R \times \dots $ model and set some discrete symmetry in the Right multiplets. Most papers got to predict the $\theta_c$ and some models with three generations or more (remember the third generation was a new insight in the mid-late seventies) were able to producte additional phases in relationship with the masses. Now, what I am interested is on papers and models including also some prediction of mass relationships, alone, or cases where $\theta_c$ is fixed by the model and then some mass relationship follows. A typical case here is Harari-Haut-Weyers (spires) It puts a symmetry structure such that the masses or up, down and strange are fixed to: $m_u=0, {m_d\over m_s} = {2- \sqrt 3 \over 2 + \sqrt 3} $ Of course in such case $\theta_c$ is fixed to 15 degrees. But also $m_u=0$, which is an extra prediction even if the fixing of Cabibbo angle were ad-hoc. Ok, so my question is, are there other models in this theme containing predictions for quark masses? Or was Harari et al. an exception until the arrival of Koide models?


title: Which exact solutions of the classical Yang-Mills equations are known?

text: I'm interested in the pure gauge (no matter fields) case on Minkowski spacetime with simple gauge groups. It would be nice if someone can find a review article discussing all such solutions EDIT: I think these are relevant to the physics of corresponding QFTs in the high energy / small scale regime. This is because the path integral for a pure gauge Yang-Mills theory is of the form $$\int \exp\left(\frac{iS[A]}{ \hbar}\right) \, \mathcal{D}A$$ In high energies we have the renormalization group behavior $g \to 0$ (asymptotic freedom) which can be equivalently described by fixing $g$ and letting $\hbar \to 0$. EDIT: For the purpose of this question, an "exact" solution is a solution in closed form modulo single variable functions defined by certain ODEs and initial / boundary conditions.


title: Center of mass and gravitational force

text: Say we have a two star system with both stars of equal mass $M$. The center of mass of this system is by definition in the center of the two stars. There is a small asteroid with mass m in orbit around the center of mass of the two stars. (What is known as a halo orbit apparently) The orbital distance is x. The question is why the gravitational force on the asteroid cannot be calculated using the center of mass. ie $F=Gm(2M)/x^2$? I can only get the force on the asteroid if I resolve the centripetal component of force on the asteroid due to each star individually and sum them. https://www.dropbox.com/s/ft5ipvknczqsr0a/Photo%2029-12-13%207%2028%2013%20pm.jpg


title: How can I determine the magnitude of an electric field based off of the magnitude of a dipole in the field?

text: Point charges q1=− 5.00nC and q2=+ 5.00nC are separated by a distance of 3.50mm , forming an electric dipole. The torque exerted on the dipole has a magnitude of 7.60*10^-9 N*m at an angle of 37∘ with the horizontal. Using this information, I calculated that the magnitude of the electric dipole moment would be 1.75 * 10^-11 C*m. From here, I attempted to calculate the magnitude of the field by using the formula T = p * E sinθ, getting a result of 674.8 N/C, but this is not correct. Is the error with the formula I am using or one of my calculations?


title: Smaller free remote control telescopes?

text: There are several online services that let you control a large telescope (eg, lightbuckets.com and slooh.com), even some that are free (eg, telescope.org). Unfortunately, the pay services are expensive, and you get very little reserved time on the scopes. The free services are painfully slow: I had to wait several months for a picture of Jupiter I wanted. Has anyone set up automated (ideally free) remote access to a smaller telescope? I'd much rather play with a 10" telescope in real-time than a larger one in limited/delayed time. In fact, even a CCD or camera would be nice for wide-angle shots.


title: Stepping into a pool of water - shock - why?

text: If I put my feet into a pool of water that contains a live wire, why do i get shocked? For electrons to flow through my body, there has to be a E-field/wave that passes through my body along a certain path, that motivates the electrons to flow through me. Now given that the live-wire is in the water and on the ground, aren't both my feet at the same potential: 1\. Lets say i hop into the pool (with both feet hitting the water surface at the same time. 2\. Let's say i place one foot in and keep the other out and on dry land. Both my feet are in contact with the earth.. so why does electricity have to flow through my upper body, or beyond my knee? At the most, I should loose a few toes but why? There's no incentive for a cable lying on the ground to send its current through me when it's already on the ground. Am i not like a bird standing on a transmission line - both feet on the transmission line? Agreed that's not an ideal 'earth-ground' representation - you'd need salt and a proper copper plate for that.. even so..


title: What's the best way to watch meteor showers?

text: I have read somewhere that the best/easiest way to watch meteor showers is to lie on the ground or other horizontal surface with your feet oriented towards the "apparent point of origin" (what was that called again?) of the shower and just relax while having eyes open to the heavens. Last November I actually tried this method to watch the Leonids and I did see some nice meteors where I have not really consciously seen any before. (Which was probably because I had not watched a meteor shower before purposefully, I guess.) Is this really the best way for amateurs to watch a meteor shower with the naked eye? What might be other/better ways? How about when you use binoculars? Or would that just limit your field of vision, making you be better of to just use eyes alone?


title: Uncertainty Principle for a Totally Localized Particle

text: If a particle is totally localized at $x=0$, its wave function $\Psi(x,t)$ should be a Dirac delta function $\delta(x)$. Accordingly, its Fourier transform $\Phi(p,t)$ would be a constant for all $p$, thus the particle's momentum is totally uncertain. I guess this is what the Uncertainty Principle told us. But in another hand, since the particle is totally localized at $x=0$, it is not moving and therefore static. Its velocity should be 0, so is its momentum, which contradicts the conclusion above. So, what's wrong with my thinking? Is it that there shouldn't be a TOTALLY STATIC particle, or it's in a non- normalizable state, so the uncertainty principle is not applicable?


title: Calculate/Estimate the fractal dimention of the logistic map

text: This is the logistic map:   . ![Image of bifurcation diagram of the logistic map](http://i.stack.imgur.com/ooYe8.png)   It is a fractal, as some might know here.   It has a Hausdorff fractal dimension of 0.538.   Is it possible to calculate/measure its fractal dimension using the box counting method?   A "hand waving calculation" is good enough.      **Update:** I understand there is another way to calculate the logistic map using the Kaplan-Yorke Conjecture. Can anyone explain that and how it can help calculating the fractal dimension of the logistic map? **Update2:** Seems like the way to go around this is not Kaplan-Yorke Conjecture (which is a unproven Conjecture still), but use the correlation dimension. There is a paper with the solution here, hope I'll know more as I read it.


title: What is the intensity of this light?

text: I am struggling with a derivation that calculates the cross sections for Mie scattering and since the incident light is considered to be a x-polarized plane wave I thought that we would have $$I_i = \frac{1}{2} \sqrt{\frac{\epsilon}{\mu}} \vert E_0 \vert^2$$, but I do not understand this derivation then, since a factor $2 \pi $ seems to be missing. It starts with an expression for the scattered field, explains how they got this expression by using some orthogonality properties and then - in my opinion argue - that this $Re(g_n)=1$. But then I do not understand what they take as the incident intensity in order to get the expression $C_{sca}$. Does anybody have an idea? > $$ > W_s=\frac{\pi|E_0|^2}{k\omega\mu}\sum_{n=1}^\infty(2n+1)\mathcal{Re}\\{g_n\\}(|a_n|^2+|b_n|^2), > $$ where we have used (4.24) and the relation $$ > \int_0^\pi(\pi_n\pi_m+\tau_n\tau_m)\sin{\theta}\text{ > }d\theta=\delta_{n\text{ }m}\frac{2n^2(n+1)^2}{2n+1}, $$ which follows from > (4.27). The quantity $g_n$, is defined as - $i\xi_n^*\xi_n^{'}$, may be > written in form $$ > g_n=(\chi_n^*\psi_n^{'}-\psi_n^*\chi_n^{'})-i(\psi_n^*\psi_n^{'}+\chi_n^*\chi_n^{'}), > $$ where the Riccati-Bessel function $\chi_n$ is - $\rho y_n(\rho)$ and, > therefore, $\xi_n=\psi_n-i\chi_n$. The functions $\psi_n$ and $\chi_n$ are > real for real argument; therefore, if we use the Wronskian(Antosiewicz, > 1964) $$ \chi_n\psi_n^{'}-\psi_n\chi_n^{'}=1,\tag{4.60} $$ it follows that > the scattering cross section is $$ > C_{sca}=\frac{W_s}{I_i}=\frac{2\pi}{k^2}\sum_{n=1}^\infty(2n+1)(|a_n|^2+|b_n|^2).\tag{4.61} > $$


title: Need to weigh a sack of kittens

text: Hypothetical brain teaser here, no real kittens involved! - For some bizarre reason I have a sack of kittens, I need to find the total weight of the kittens and the sack but I'm only allowed to use a spring scale and I'm only allowed use my arms to hold the scale. My arms are unsteady and the kittens are constantly moving around in the sack, causing the weight display on the scale to dance around wildly.The kittens all weigh different weights and they are in constant motion. I am wondering if it is possible to very accurately get a reading from the scale, perhaps by incorporating some kind of accelerometer? This has been puzzling me for some time! Looking forward to hearing your input.


title: Is there any quantum analogs of three body problem?

text: IS there any quantum analogy where a three state (or three body) system shows chaotic dynamics as three body problem in classical mechanics?


title: Is it possible to recover Classical Mechanics from Schrödinger's equation?

text: Let me explain in details. Let $\Psi=\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\Psi(x,t)$ so that $|\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)?


title: Classical limit of quantum mechanics

text: I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\hbar$ goes to zero, the position wavefunction reduces to a delta function and that the Schrodinger equation/Feynman path integral reduces to the Newtonian/Lagrangian/Hamiltonian equations of motion.)


title: How to prove Newton's second law with quantum mechanics?

text: Newton's second law claims that $F=ma$. In terms of quantum mechanics, the equality can be written as $ \frac{d\langle p \rangle}{dt} = -\langle \nabla V(x) \rangle$. How can I prove this with non-relativistic Schrodinger Equation? I've calculated the expectation of $p=mv$, and tried to take the derivative under the integral. And there I'm stuck.


title: Classical Limit of the Feynman Path Integral

text: I understand that in the limit that h_bar goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation for the propagator which is a function of the classical trajectory (see http://www.blau.itp.unibe.ch/lecturesPI.pdf pg 46). I am under the impression that this further implies that the particle follows the classical trajectory but I don't understand how the above mentioned fact implies this. The propagator describes the time-evolution of the wavefunction, so I would think that this classical limit form of the propagator should give a time- evolution in which the wavefunction follows the classical trajectory, but I have not been able to find such work. Moreover, even this statement itself is problematic since the wavefunction describes a probability distribution and not a single trajectory. $\textbf{New Edit:}$ In section 7 of Feynman's paper introducing the path integral (see http://imotiro.org/repositorio/howto/artigoshistoricosordemcronologica/1948c%20-FEYNMAN%201948C%20Invention%20of%20the%20path%20integral%20formalism%20for%20quantum%20mechanics.pdf) he discusses the classical limit. It appears that the key to understanding why the fact that the classical path dominates the path integral further implies that the particle follows the classical trajectory may be found in Feynman's remark on pg 21: "Now we ask, as $\hbar → 0$ what values of the intermediate coordinates $x_i$ contribute most strongly to the integral? These will be the values most likely to be found by experiment and therefore will determine, in the limit, the classical path." However, I don't understand why "These will be the values most likely to be found by experiment" ?


title: Can Newton's laws be explained by Quantum Physics?

text: I have only basic knowledge of physics. Could you please explain to me if a "Quantum" laws can theoretically (perhaps in the future?) be used to explain everything in macro levels? I'm having problems to understand how we can have "two physics" at the same time. are Newton's laws only a simplification?


title: Classical limit of the path integral formulation of quantum mechanics

text: It is well-known that if $S \gg \hbar$, then the classical path dominates the Feynman path integral. But is there some to show that if $S\gg\hbar$, then the particle's trajectory will approach the classical path?


title: From Quantum Mechanics to Classical Mechanics

text: Is it possible, and has it been attempted, to use quantum mechanics to deduce Newtonian, macroscopic level mechanics laws as was the case of statistical mechanics deriving thermodynamic relations?


title: How are the Lagrange equation and Feynmann path integral related?

text: My question is, where could I get some more info on how the Euler-Lagrange equations are related $$ \delta S [y(x)] =0 $$ with the Feynmann path integral formulation $ \int D[y(X)]e^{iS[y(x)]/\hbar} $ I believe that when $ \hbar \to 0$ (semiclassical), only the points that satisfy (1) contribute to the path integral. If someone can give me more info about it :D thanks


title: How to go from Quantum World to Classical World?

text: > **Possible Duplicate:**   >  Is it possible to recover Classical Mechanics from Schrodinger’s equation?   >  Classical Limit of the Feynman Path Integral In the quantum world we don't have specific trajectories, the particle so to speak goes through all possible paths. In the classical and macroscopic world we have definite paths, and usually one specific trajectory is assigned to a body's motion. How would you go from a trajectoryless world to trajectoried world? Are there any theories about this bridge between the two worlds? I guess there should be such a theory, cause one world is the building block of the other.


title: What are the experimental observations behind the first principle of thermodynamics?

text: As far as I understand it, the first principle of thermodynamics is a mere definition of the quantity “Heat”: $$\text d Q: = \text d L + \text d U.$$ This is somewhat the point of view taken in Fermi's introductory book "Thermodynamics": > [...] $$\Delta U + L=0$$If the system is not thermally isolated, the first > member of [eqn.] will be generally not equal to zero [...] Substitute the > [eqn.] with the more general: $$\Delta U + L = Q.$$ [...] Now we will call > $Q$, by definition, the quantity of heat received by the system during the > transformation. (if you want to read the full text you might want to google “Fermi Thermodynamics"... pag. 17). I think that this point is logically sound and I have a quite good understanding of some of the above structure starting from here (e.g. the second principle). On the other hand I feel as I'm missing something. To give an example, from mechanics, this is how I understand Newton's equation: > It is a matter of fact that the positions and the velocities of a mechanical > system fully determine the accelerations of the system. Hence, the dynamic > of each system follows second order differential equations: $$\ddot x = > F(x,\dot x, t).$$ An other example might be the second law of thermodynamics, that (in Clausius' form) is simply the statement of the fact that heat doesn't flow spontaneously from a cold body to an hotter one. Since I find strange that something that is called a “principle” is a mere definition (after all, there's no assumption involved in making a definition), I ask: what are the experimental facts behind the first principle of thermodynamics? Note: I understand that this is really about my personal understanding, however I think that this question can be useful to others. Furthermore, if something isn't clear and if I can improve my question, let me know.


title: How is relativistic N-body simulation possible without knowing the entire histories of each particle?

text: In n-body simulation you need to know the positions of the particles in order to calculate the force between them. The new velocity of each particle can then be calculated given a simulation timestep dt. If the gravitational interaction propagates at the speed of light, do we need to specify the force between particles given their retarded positions (not instantaneous positions)? If not, why not? Note that I am interested in special relativity only (not GR), I am more concerned with what happens when bodies move quickly or are separated by distances larger than the distance light travels in one timestep. I am also interested in whether or not the electric force between charges is any different in this respect. Thanks.


title: Solving actual weight lifted

text: I was wondering if anyone knew the formula to calculate how much force is required to lift a tire or any object by the edge. For example if a tire weighs 500 lbs (or has a mass of 226.8 kg). I would just like to know the formula how to calculate how much of that 500 lbs we are actually lifting since it is only from the edge.


title: How do inflationary models predict the generation of gravitational waves during the inflationary period?

text: Recent results from the BICEP2 experiment have produced a lot of talk about the primordial gravitational waves produced during the inflationary period. I would like to have some explanation about how inflationary models predict the generation of these gravitational waves. Have these gravitational waves been described as metric perturbations around a de Sitter spacetime? Are they predicted using a semiclassical gravity formalism so as to take into account the quantization of a scalar field (the inflaton) in such "perturbed" spacetime?


title: How to prove Eq. (2.4.5) in Polchinski's string theory book?

text: I got one more stupid question in Polchinski's string theory book. In p. 44, it is said > _The currents_ $$j(z)=i v(z) T(z), \tilde{j}(\bar{z}) = i v(z)^* > \tilde{T}(\bar{z}) \tag{2.4.5}$$ _are conserved for any holomorphic $v(z)$._ Did he mean $$\partial_z j(z)=0, \qquad \partial_{\bar{z}} \tilde{j}({\bar{z}})=0? $$ How to show that? The previous page states $\bar{\partial} T_{zz} = \partial T_{\bar{z} \bar{z}} =0 $. It seems these relations are insufficient to derive Eq. (2.4.5).


title: What is the speed of electrical current in salt water?

text: I am wondering about a specific question regarding the speed at which an electrical current traverses through salt-water / saline. By this I do _not_ mean the electron drift speed - I mean, at what speed would a current travelling from an anode to a cathode immersed in salt water be? A ball park figure will do. Thank you.


title: Is Electromagnetic Mass Possible?

text: If the sinusoidal electric component of a light wave were off-set to one side of the magnetic component and then the smaller "lobe" were to cancel out with much of the larger side, then where would the energy go? Would it not form a closed loop much like a mass-bearing string? Could the electrical energy not be converted into a gravitational field to bend space-time over the length of the wave to form a 1-D "string" along the junction of the perpendicular E and B fields? This would be much like folding a sheet of paper in half so that one edge protrudes past the other. The protruding edge being the electric component and the rest being electrical energy converted into mass/gravity (comparable to a mass-bearing string). The magnetic component could then arise from Lorentz symmetry as described by Lubos: > If you only start with the $E_z$ electric field, the component $F_{03}$ is > nonzero. However, when you boost the system in the $x$-direction, you mix > the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a > part of the $F_{03}$ field is transformed into the component $F_{13}$ which > is interpreted as the magnetic field $B_y$, up to a sign. **Is the described "string wave" model possible?** I realize that I haven't provided a mechanism to explain the proposed model. However "I do not reject food because I do not understand digestion." -Oliver Heavenside From here on the rest of this post is merely a compilation of the "evidence" that circumstantially supports the proposed idea. Please do not feel required to address these topics. They are here because I think electromagnetic mass is real. I realize this borders on "promoting unaccepted theories," but it is merely my way of assessing the possibility. I promise not to bring this up again on SE if it is refuted. **The concept seems compatible with the standard model:** a.) an origin of charge has been proposed. b.) eliminating one lobe has reduced the "spin" or magnetic field by 1/2 relative to a light wave. c.) Lepton number 1 applies due to the "strong interaction" of the remaining electric component d.) a mechanism of mass has been proposed. **Maxwells equations seem to fit:** The Laws of Electromagnetism have geometric relations incorporated into them that naturally arise from the proposed electron-as-an-EM-wave model: the dot product and the cross product specifically. Ampere's Law Ampere’s Law describes the magnetic field produced by the flow of electrons along a wire. The negative components of electrons flowing along the wire should repel each other, which would mean that the negative components should protrude from the wire like the dorsal fins of sharks swimming parallel and breaking the surface. Since the electrons are travelling in the same direction along the wire the magnetic components (in the direction of the side fins of the shark analogy) should be tangent to the surface of the wire, which results in a circular magnetic field around the wire just as Ampere’s Law and the Biot-Savart Law predict. Faraday's Law Faraday’s Law describes the electric voltage produced in a coil of wire as a magnetic field through it changes. The voltage is proportional to the number of loops of wire, which is counter-intuitive/non-conservative. Why should the voltage depend on the number of loops? An explanation naturally arises from the proposed model. Electrons inherently have velocity in the form of a "Poynting-like" vector. When the loop of wire encounters a changing magnetic field the "poyinting vectors" align and under the right orientations they align with the wire loops and thereby form electric current. ![enter image description here](http://i.stack.imgur.com/eyCkq.jpg) In the bottom orientation the magnet produces no EMF along the wire. In the top orientation the EMF is along the wire as seen in generators. Schrodinger's statistical model won because Maxwells equations had already divorced physics from first causes. Saying that the surface covered area of a loop was the cause of EMF instead of the poynting vectors of electrons adding over the distance of the circumference. Gauss's Laws Gauss’ laws of electricity and magnetism are easily integrated with the proposed model as the net magnetic flux is zero and the net charge is unchanged. However Coulombs Law does not hold true at the subatomic level. The force between two charged electrons is modified by the presence of other electrons if electrons are not point charges, but electric components of an EM-wave. Coulomb's law Coulomb's law for point charges: $$F=k\frac{q_1 q_2}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{qq_0}{r^2}\hat{u}$$ Does not hold at the subatomic level for all directions if a blip of negative charge sticks off one side of a mass-bearing electromagnetic wave. This naturally provides a classical explanation for "quantum tunnelling:" certain orientations of subatomic particles behave unlike point charges. **Atomic Orbitals** Take one electron and one proton and place them near each-other. The proposed model suggests an intrinsic and intuative reason why electrons don’t fall into the nucleus, which current theories lack. The velocity of an imbalanced EM wave is perpendicular to both the electric and magnetic components, which means that the radial acceleration of attraction towards the nucleus experienced by the electric component of an electron is always perpendicular to its velocity or the “Poynting vector” of an electron. All that is left for the electron to do is to set up the lowest energy standing wave possible. This also suggests that the proton rotates with the electron's orbit so that the "fins" constantly point toward eachother. Spherical harmonics should arise naturally from this arrangement and approximate to the schrodinger equation. **Relativistic Explanation of the Lorentz Force** Using relativistic tensors: "If you only start with the $E_z$ electric field, the component $F_{03}$ is nonzero. However, when you boost the system in the $x$-direction, you mix the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a part of the $F_{03}$ field is transformed into the component $F_{13}$ which is interpreted as the magnetic field $B_y$, up to a sign\cite{Lubos}." If the electron is an E-M wave as proposed and the magnetic components align with the extern magnetic field, then performing the reverse of the above transformation should convert the external magnetic field into an electric component in the $E_z$ direction so that the electron feels the equivalent to a charge perpendicular to its direction of motion, which explains the Lorentz force. **Numerous Other Explanations** There are many other phenomena that seem to fit with the proposed model. I'm just running out of steam! **A Little History** Electrons were viewed as "matter waves" in DeBrogli's model. "Matter waves" is a polite way of saying: something is waving but we don't know what. Heisenberg came along and said that despite not knowing what is waving we can assume that the wave doesn't have any undiscovered properties that would allow for knowing both position and momentum (an arrogant assertion!). Schrodinger then came along and noticed that tweaking spherical harmonics provided a reasonable model of atomic orbitals (statistically). That all led to the Copenhagen interpretation, which Einstein called the "Born-Heisenberg tranquilizing philosophy, or religion." Electrons have continued to be treated as point charges or matter waves as is convenient for interpreting experimental results ever since. QED and QCD have introduced "virtual photons" in explaining the interactions of "point charges" and light, etc. Surely "virtual photons" would be unnecessary if the point charges were instead modelled as EM waves themselves. Particle physicists have invented a "higgs field" which they propose vibrates to create mass. The higgs field seems like adding epicycles: sorta unnecessary when there is a simpler and better alternative (that I have not really understood yet!). It has been obvious for decades that physics has stalled while trying to model hadrons as point charges. String theory has been a lonely success story waiting to happen. Electromagnetic string-waves are the future! (unless someone refutes me :-).


title: Moving conductors in magnetic fields: is there electric field or not?

text: this is my first question on PhysicsSE (I'm already an user of MathSE). I'm a mathematics students trying to understand Faraday's law, that is $$\varepsilon= -\frac{d \Phi_B}{dt}$$ where $\varepsilon$ means electromotive force and $$\Phi_B=\iint \mathbf{B}\cdot d\mathbf{S}$$ means flux of magnetic field. As my textbook points out, there is an interpretation problem here: if the change in magnetic flux is due to movement of the conductor, then free charges in it are subject to Lorentz force, which then causes a current. On the contrary, if the conductor holds steady in a changing magnetic field, the induced current must be explained in terms of an electric field $\mathbf{E}$, described by the equations $$\begin{cases} \nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0} \\\ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}\end{cases}.$$ **Question** Do those equations hold if we have a moving conductor in a stationary magnetic field? I guess not: this would mean $\mathbf{E}=\mathbf{0}$. How to solve this?


title: The theory of moon creation when a Mars size planet hit Earth

text: As we know the predominant theory where does the moon come from is that a Mars size planet hit the earth and took a chunk out of it which eventually materialized into moon. My question is that if a Mars size object were to hit Earth, wouldn't it knock it off the orbit all together? What kind of collision is required to knock a planet of its orbit. By 'knock off' I mean it would alter the orbit of Earth and possibly speed so that it will not have stable orbit anymore so it will either (gradually) leave solar system or (gradually) collapse into sun.


How exactly does time slow down near a black hole? I have heard this as a possible way of time traveling, and I do understand that it is due in some way to the massive gravity around a black hole, but how exactly does that massive gravity slow down time?


title: Does a photon exert a gravitational pull?

text: I know a photon has zero rest mass, but it does have plenty of energy. Since energy and mass are equivalent does this mean that a photon (or more practically, a light beam) exerts a gravitational pull on other objects? If so, does it depend on the frequency of the photon?


title: How does gravity effects both time and light if they have no mass

text: I've been reading about how black holes can effect both time and light with gravity. So I was wondering, doesn't something have to have mass to be effected by gravity? And if so, does this mean both light and time have mass? And if not, how can gravity effect something that has no mass?


title: Is light affected by gravity? Why?

text: I would like to know if light is affected by gravity, also, I would like to know what is the correct definition of gravity: "A force that attracts bodies with mass" or "a force that attracts bodies with energy, such as light"? Is light massless after all?


title: What causes light to travel?

text: What is the force that causes it to move and why does it maintain the speed for so long? If it has no mass, why is it effected by mass?


title: Can electromagnetic radiation (i.e. photons) produce gravity?

text: I don't want to play with physical laws in a frivolous way. Assuming that the nature of matter and energy is the same, can a high density of highly energetic photons produce a gravity force? We do know that radiation is affected by space-time distortions, or in another way "feels gravity". Why do photons can (or cannot) produce a gravity field?


title: Why can light (photons) bends in a curve through space without mass?

text: I've heard that light can form a curve if they travel near high-mass stars or even a black hole with strong gravity. Which is according to this Newtonian formula $$\large F_{g}=\dfrac{Gm_1m_2}{r^2}.$$ But I've also heard that photons do not have (rest) mass! So it doesn't fit that equation anymore! But why can photons be pulled by gravities without (rest) mass? Could someone explain that?


title: Explain how (or if) a box full of photons would weigh more due to massless photons

text: I understand that mass-energy equivalence is often misinterpreted as saying that mass can be converted into energy and vice versa. The reality is that energy is always manifested as mass in some form, but I struggle with some cases: **Understood Nuclear Decay Example** In the case of a simple nuclear reaction, for instance, the total system mass remains the same since the mass deficit (in rest masses) is accounted for in the greater relativistic masses of the products per $E=\Delta m c^2$. When a neutron decays and you are left with a fast proton and a relativistic electron. If you could weigh those two without slowing them down, you would find it weighed as much as the original neutron. **Light in a Box** This becomes more difficult for me when moving to massless particles like photons. Photons can transmit energy from one heavy particle to another. When a photon is absorbed the relativistic mass (not the rest mass) of the (previously stationary) particle that absorbs it increases. But if my understanding is correct, the energy must still be manifested as mass _somehow_ while the photon is in-flight, in spite of the fact that the photon does not have mass. So let's consider a box with the interior entirely lined with perfect mirrors. I have the tare weight of the box with no photons in it. When photons are present the box has an additional quantifiable amount of energy (quantified below) due to the in-flight photons. Say there are $N$ photons... obviously assume $N$ is large. $$\Delta m = \frac{ E }{ c^2 } = \frac{ N h }{ \lambda c}$$ Interactions are limited to reflections with the wall, which manifest as a constant pressure on the walls. If I hold this box in a constant gravitational field (like the surface of Earth) then there will be a gradient in the pressure that pushes down slightly. Is this correct? Wouldn't there still technically be mass as the photons are in-flight, which would cause its own gravitational field just as all matter does? How is this all consistent with the assertion that photons are massless? Is it really correct to say that photons don't have mass? It seems to be a big stretch. Please offer a more complete and physically accurate account of this mirror- box.


title: If photons are deflected by a strong gravitational field, then how come photons do not have mass?

text: > **Possible Duplicate:**   >  Explain how (or if) a box full of photons would weigh more due to massless > photons It has been proved and showed through experiments that light can be bent by the Sun or any other body with considerable mass. Also light is nothing but photons. So can these photons be attracted by massive bodies if they have no mass?


title: What is the cause the light is affected by gravity?

text: I know that photons have no mass and that a photons exist only moving at the speed of light. So what is the cause that a massive astronomical object can bend a ray of light? I have two thoughts, but I am confused which of these, if any is correct:   1. A gravitational field of a massive astronomical object curves spacetime and affects the light traveling near this object **indirectly**. That is, for external viewers the light bends, but for the light itself it is still traveling in a straight line. So I mean, that the coordinates in that place are curved, but only for the external viewer.   2. Energy of a photon is equivalent to mass, so the gravity field interacts with photons **directly** the same way, as if they would have mass (equivalent to the energy).


title: Does non-matter energy curve spacetime?

text: I know that matter (mass) curves spacetime, but do other forms of energy do the same? I.e. is matter the only form of energy that curves spacetime?


title: What are the factors affecting the spacetime curvature?

text: Large masses in space as stars and planets cause a curvature in the spacetime fabric. What are the factors that affect this curvature? Is it only mass? And can we conclude these factors using Tensors?


title: Light and Gravity - bending of light around a massive body

text: Well, as I have read, a massive body can cause light to bend around itself due to its gravitational attraction. What I don't understand is how, as the Newtonian formula for the force of gravitational attraction is $$F = \frac{Gm_1m_2}{r^2}.$$ As photons do not have (rest) mass, shouldn't there be no attraction between light and the celestial bodies, and therefore no bending? Surely I am missing a key point. Please point out my logical flaw?


title: Why does a photograph of a face look like it is always watching you?

text: Imagine a photo that is taken of someone looking straight at the camera. Why when we look at the photo now _from any direction_ it looks as if the person is looking straight at us?


title: Do the Maxwell equations yield the proper time of electromagnetic waves?

text: I apologize in advance for possible errors in my premises as I have no precise knowledge of Maxwell equations. Proposals for the correction or even abandon of my question are welcome. As Maxwell equations are a full description of electromagnetic waves I suppose that they also describe the **time** a wave takes from the place of its emission to the place of absorption. My question: As Maxwell equations are relativistically invariant (or at least compatible with special relativity), do they also yield the **proper time** (which is always zero for electromagnetic waves in vacuum)?


title: Why do I feel cold air in the shower?

text: Our house has a glass sliding door to the shower. The shower has the dimensions of about 2 feet wide, 5 feet long, and 6 feet high. Above the door (and shower head) there is about 1 foot of open space before the ceiling to let the heat escape. Sometimes I forget to slide the door all the way to the end of the other side, leaving about a 1 inch space in between the door and the wall. Cold air comes through so much that I _always_ known when there is a crack, the amount of cold air I feel does not seem proportional to the opening at all. I would think that the amount of heat being produced by the shower would be enough to where I don't feel the cold air, but that does not seem to be the case. What I'm thinking is that it is not that the heat is leaving through there, heat is escaping through the top since heat rises, but rather the cold air is coming through the opening due to a difference in pressure. Is this the case? How come it does not seem proportional to the opening? How come I can feel a fair amount of cold air coming from the small opening at the opposite end of the shower?


title: How to find the Green's Functions for time-dependent inhomogeneous Klein-Gordon equation?

text: I'm trying to find the Green's functions for time-dependent inhomogeneous Klein-Gordon equation which is : \begin{align*} \left[ - \nabla ^2 + \frac{1}{c^2} \dfrac{\partial ^2}{\partial t^2} + \kappa ^2 \right] \psi({\mathbf{r},t )} = \rho(\mathbf{r},t) \end{align*} It has been mentioned in the question that I can find the Green's functions : \begin{align*} &G_R(\mathbf{r} , t , \mathbf{r'} , t') = \dfrac{c}{8 \pi ^2 \mathbf{R} i }\dfrac{d}{d\mathbf{R}} \int_{- \infty}^{+\infty} \dfrac{e^{ i \frac{R}{c} \sqrt{q^2 - k^2 c^2}}}{\sqrt{q^2 - k^2 c^2}} e^{ - iq (t - t')} dq \\\ &G_A(\mathbf{r} , t , \mathbf{r'} , t') = -\dfrac{c}{8 \pi ^2 \mathbf{R} i }\dfrac{d}{d\mathbf{R}} \int_{- \infty}^{+\infty} \dfrac{e^{- i \frac{R}{c} \sqrt{q^2 - k^2 c^2}}}{\sqrt{q^2 - k^2 c^2}} e^{ - iq (t - t')} dq \end{align*} using the fourier transform, but when I use the fourier transform I don't gain the proper answer. The fourier transform which I use is the one which is generally given as : \begin{align*} f(r) = \dfrac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} e^{ik.r}\hat{f}(k) dk  \end{align*} but from this transform I cannot find $G_A$ and $G_R$. Is there another transform which I should use to find the Green's functions? **Edit** The Green's function which I wind up with is : \begin{align*} G_A(\mathbf{r} , t , \mathbf{r'} , t') = \dfrac{1}{(2\pi)^4} \int d^3\mathbf{k} dk' \frac{1}{k^2} e^{i\mathbf{k}.(\mathbf{r} - \mathbf{r'})}e^{ik'(t-t')} \end{align*} which is not even similar to the answer given here!


title: Speed of light, breaking the barrier

text: when the sound barrier is broken, a series of concentric waves of sound is produced.Does it mean when the speed of light barrier is broken, a ripple of photons are created in the space-time fabric?


title: How to replace $T$-product with retarded commutator in LSZ formula?

text: I am reading Itzykson and Zuber's Quantum Field Theory book, and am unable to understand a step that is made on page 246: Here, they consider the elastic scattering of particle $A$ off particle $B$: $$A(q_1) + B(p_1) ~\rightarrow~ A(q_2) + B(p_2)$$ and proceed to write down the $S$-matrix element using the LSZ formula, with the $A$ particles reduced: $$S_{fi}=-\int d^4x\, d^4y e^{i(q_2.y-q_1.x)}(\square_y+m_a^2)(\square_x+m_a^2)\langle p_2|T \varphi^\dagger(y) \varphi(x)|p_1 \rangle \tag{5-169}$$ Then they say that because $q_1$ and $q_2$ are in the forward light cone, the time-ordered product can be replaced by a retarded commutator: $$T \varphi^\dagger(y) \varphi(x) ~\rightarrow~ \theta(y^0-x^0)[\varphi^\dagger(y),\,\varphi(x)]\,.$$ This justification for this replacement completely eludes me. What is the mathematical reason for this?


title: Extension of an unfixed spring

text: Suppose a spring with spring constant 1 N m^-1 is held horizontally. If a pull of 1 N is applied to its left end and a pull of 2 N is applied to its right end, how much longer would the spring be as compared to its unstretched length? I came up with this question because the spring related qurstions I've seen all have one end fixed some where. I am totally stuck on this one and I really have no idea where to start. All I can figure out is that the spring will accelerate due to a net force. It seems inappropriate to be taking net forces, since pulling both ends with a larger force makes the spring more elongated. P.S. this is not homework


title: Electric Force is to Magnetic Force as Gravitational Force is to ...?

text: One can no nothing about the magnetic force and yet arrive at it by taking the relativistic effects of a current and a moving charge system into account. I ask whether there exists such an inherent force in case of gravity.


title: Long time deviations from exponential decay in radioactivity

text: Are there any examples of common substances whose decay is not exponential? We're used to thinking about radioactivity in terms of half-lives. This is a concept that makes sense only for a decay that is exponential. However, there are plenty of physics articles on the subject of non exponential decay. It seems to be theoretically ubiquitous. For example: > The decay of unstable quantum states is an ubiquitous process in virtually > all fields of physics and energy ranges, from particle and nuclear physics > to condensed matter, or atomic and molecular science. **_The exponential > decay, by far the most common type, is surrounded by deviations at short and > long times$^{1,2}$._** The short-time deviations have been much discussed, > in particular in connection with the Zeno effect$^{3,4,5}$ and the anti-Zeno > effect$^{6,7,8,9}$. Experimental observations of short$^{10,11}$ and > long$^{12}$ time deviations are very recent. A difficulty for the > experimental verification of long-time deviations has been the weakness of > the decaying signal$^{13}$, but also the measurement itself may be > responsible, because of the suppression of the initial state > reconstruction$^{2,14}$. > > 1) L. A. Khalfin, Zurn. Eksp. Teor. Fiz. 33, 1371 (1957), English > translation: Sov. Phys. JETP 6 1053 (1958).   >  2) L. Fonda and G. C. Ghirardi, Il Nuovo Cimento 7A, 180 (1972). 10.1103/PhysRevA.74.062102, F. Delgado, J. G. Muga, G. Garcia-Calderon   _Suppression of Zeno effect for distant detectors_ So are there any examples of deviations from long time decay? If not, then why not? Is the theory wrong or simply impractical? And is there a simple, intuitive explanation for why long decays should not be exponential?


title: Physics of tsunami: the relationship between wavelength, sea depth and the height of the water

text: If I understand correctly, when an earthquake occurs, energy will be transferred to the water, resulting in water waves. As the waves reach seashore, because the sea depth is getting shallower and wavelength is getting shorter, the height of the wave gets push up, resulting in tsunami. In other words in deep sea, water won't get pushed up as high as the water in shallow seashore. Is my understanding correct? Is there a quantitative way to express the physics behind all this?


title: Is it mathematically possible for a black hole to be connected to a white hole or a worm hole?

text: I know that there's no evidence at this point for "white holes" however would it even be mathematically possible for a black hole to be connected to a white hole (total opposite so everything would be expelled, after some really extreme physical conditions)? Maybe with a wormhole connecting them? If this was even possible (if the black hole or connection could actually be created and be stable enough), would that matter be expelled into a different universe, etc? Maybe even a different region of spacetime? Just curious, as it would be a cool idea.


title: Can the earth's magnetic field be generated by heat from a natural nuclear reactor?

text: I've come upon Dr. J. Marvin Herndon's theory that the earth's magnetic field is generated by a hot nuclear reactor operating in the center of the earth. This is backed by various papers, some of them peer reviewed: http://www.nuclearplanet.com/Herndon%27s%20Nuclear%20Georeactor.html His theory purports to explain various anomalies such as the high level of He3 in basalt in Hawaii and their dependency with time. A recent Herndon arXiv paper is: _Uniqueness of Herndon's Georeactor: Energy Source and Production Mechanism for Earth's Magnetic Field_ http://arxiv.com/abs/0901.4509 His papers suggest that the natural reactors sometimes seen in uranium deposits must also occur at the center of the earth. He says that the conventional explanation for the earth's magnetic field fails because the Rayleigh number for the core is inappropriate as far as determining whether convection exists. Since this is not accepted geology, what are the problems with the theory?


title: What is changed when proton has finite radius?

text: How the field and interactions are changed when we assume that proton has finite radius in atom for example? What gives the finite size effect? Is it the higher moments of multipole expansion?


title: Planck's Law in terms of wavelength

text: I am drawing a blank when it comes to equation transformation. Wikipedia gives two equations for the spectral radiance of black body:   * First as a function of frequency $\nu$: $$I(\nu, T) = \frac{2 h \nu^3}{c^2}\cdot\frac{1}{e^\frac{h \nu}{k T} - 1}$$   * Then as a function of wavelength $\lambda$ : $$I'(\lambda, T) = \frac{2hc^2}{\lambda^5}\cdot\frac{1}{e^\frac{h c}{\lambda k T}-1}$$ And I don't see how they get $\lambda^5$ term. I'm assuming that the transformation is just $\nu \rightarrow c/\lambda$, but that gives $$ \frac{2 h \nu^3}{c^2} \Rightarrow \frac{2 hc}{\lambda^3} \neq \frac{2hc^2}{\lambda^5} $$ Similar transformation happens at other parts in the article also. I'm obviously missing something, likely completely trivial.


title: Wien's displacement law in frequency domain

text: When I tried to derive the Wien's displacement law I used Planck's law for blackbody radiation: $I_\nu = \frac{8 \pi \nu^2}{c^3} \frac{h \nu}{e^{h\nu/k_bT}-1}$ Asking for maximum: $\frac{dI_\nu}{d \nu}=0:~0= \frac{\partial}{\partial\nu}(\frac{\nu^3}{e^{h \nu/k_bT}-1}) = \frac{3\nu^2(e^{h \nu/k_bT}-1) - \nu ^3h/k_bT \cdot e^{h\nu/k_bT}}{(e^{h \nu/k_bT}-1)^2}$ It follows that numerator has to be $0$ and looking for $\nu>0$: $3(e^{h \nu/k_bT}-1) - h \nu/k_bT \cdot e^{h\nu/k_bT}=0$ Solving for $\gamma=h\nu/k_bT$: $3 (e^\gamma-1) - \gamma e^\gamma=0 \rightarrow \gamma=2.824$ Now I look at the wavelength domain: $\lambda = c/\nu:~ \lambda =\frac{h c}{\gamma k_b} \frac{1}{T}$ but from Wien's law $\lambda T = b$ I expect that $hc/\gamma k_b$ is equal to $b$ which is not: $\frac{h c}{\gamma k_b}= 0.005099$, where $b = 0.002897$ Why the derivation from frequency domain does not correspond the maximum in wavelength domain? I tried to justify it with chain rule: $\frac{dI}{d\lambda} = \frac{dI}{d\nu} \frac{d \nu}{d \lambda} = \frac{c}{\nu^2} \frac{dI}{d \nu}$ where I see that $c/\nu^2$ does not influence where $dI_\lambda/d \lambda$ is zero.


title: Different versions of Planck's law

text: For a presentation in physics I am going to talk about black body radiation since our book just mentions Planck's law, Wien's displacement law and Stefan- Boltzmann's law. I want to derive Wien's law and Stefan-Boltzmann's law from Planck's law. I have managed to derive Stefan-Boltzmann's law from: $$u(\nu)=\frac{2\pi h \nu^3}{c^2}\frac{1}{e^\frac{h\nu}{kT}-1}$$ by integrating over all $\nu$ from $0$ to $\infty$. However, my textbook says: $$u(\lambda,T)=\frac{2\pi hc^2 }{\lambda^5}\frac{1}{e^\frac{hc}{\lambda kT}-1}$$ If I can get from the second formula to the first, I will be happy alltough, a simple and informal derivation of Plack's law would be very satisfying. I also managed to derive Wien's law by derivation and setting equal to zero by using: $$u_\lambda=\frac{8\pi hc }{\lambda^5}\frac{1}{e^\frac{hc}{\lambda kT}-1}$$ What are the differences? And please, since I am Norwegian and do not know very advanced physics, do not use any advanced expression without explaining them. Thanks!


title: Magnitude refers to number or number with units?

text: This question is about terminology for physical quantities. When we talk about magnitude (while talking about scalars and vectors) do we refer to just number or Number along with units? example: If a person weighs 120 pounds, then "120" is the numerical value and "pound" is the unit. Which is magnitude? 120? or 120 pounds? **EDIT:** In the book I'm using its written as > The number indicates the magnitude of the scalar quantity and is inversely > proportional to the unit chosen. This statement is wrong. Right? Its not the number alone. Its along with units.


title: Applications of the particle in a box and the finite square well

text: What are some "real" world applications of the particle in a box (PIB) and the finite square well (FSW) which are discussed in an intro quantum mechanics class? For instance, I know that the PIB can applied to quantum dots and the FSW to the Ramsauer-Townsend effect. How about other applications?


title: Why is there this asymmetry between the two foci of an orbital ellipse?

text: Why does the Earth revolve with the Sun at one of its foci? Does the other focus do nothing? Why is there this asymmetry in our solar system?


title: What exactly means "is a singlet under $SU(N)$"

text: I don't get a grip of what that exactly means. What IS an abstract singlet, doublet,... under $SU(N)$ or other groups?


title: What does this notation mean?

text: Terminology question. Reading this, in the middle of the second page, when it says > Left-handed quarks form 3 (3; 2; + 1 6 ) multiplets Qn (n = 1; 2; 3); What does this (3;2;+1/6) mean?


title: Notation for Standard Model Charges?

text: Does anybody know what these following numbers describing an electron $(1, 1, -1)$ represent in $SU(3) \times SU(2) \times U(1)$? Or, these numbers that describe an up quark: $(3, 1, 2/3)$? I'm really confused!


title: Assuming collision , are there fundamental forces associated with absorbtion?

text: I just learned that strong and weak nuclear forces relate to decay/emission. I know absorbtion depends on Energy levels(QM) and heat(thermodynamics , kinetic energy , entropy) and E = gamma mc^2 ( special relativity). Assuming collision , are there fundamental forces associated with absorbtion ? I add that I am not an expert in QM. Edit : I noticed my question is not so good , but that was because of my ignorance. My apologies.


title: What happens to the Lagrangian of the Dirac theory under charge conjugation?

text: Consider a charge conjugation operator which acts on the Dirac field($\psi$) as   $$\psi_{C} \equiv \mathcal{C}\psi\mathcal{C}^{-1} = C\gamma_{0}^{T}\psi^{*}$$ Just as we can operate the parity operator on the Lagrangian, and we say that a theory has a symmetry if $$\mathcal{P}\mathcal{L}(t,x^{i})\mathcal{P}^{-1} = \mathcal{L}(t,-x^{i})$$ Suppose we operate $\mathcal{C}$ on the Dirac Lagrangian what should we get? $$\mathcal{C}\mathcal{L}_{Dirac}(x^{\mu})\mathcal{C}^{-1} = \mathcal{L}^{*}_{Dirac}(x^{\mu}) ?$$ in analogy to the transformation of the scalar field $\phi$ under charge conjugation. On a same note one can ask what equation should $\psi_{C}$ satisfy? Should it satisfy the conjugated Dirac equation as $$(i\gamma^{\mu}\partial_{\mu} + m)\psi_{C} = 0 ?$$ If so can someone give me the physical interpretation for it. I am asking this question as I want to explicitly use $\psi_{C}$ and check whether it keeps the Dirac Lagrangian invariant. I have done a calculation by substituting $\psi_{C}$ in the Dirac equation and have found it is not satisfying as shown below. $$(i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} = i(\gamma_{\mu}C\gamma_{0}^{T})\partial^{\mu}\psi^{*} - (C\gamma_{0}^{T})m\psi^{*}$$ We will use $C^{-1}\gamma_{\mu}C = - \gamma_{\mu}^{T}$ and $\\{\gamma_{\mu}^{T},\gamma_{\nu}^{T}\\} = 2g_{\mu\nu}$.   Consider \begin{align} \gamma_{\mu}C\gamma_{0}^{T} &= CC^{-1}\gamma_{\mu}C\gamma_{0}^{T} \\\ &= -C\gamma_{\mu}^{T}\gamma_{0}^{T} \\\ &= C\gamma_{0}^{T}\gamma_{\mu}^{T} \end{align} Hence substituting back we will get   \begin{align} (i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} &= C\gamma_{0}^{T}(i\gamma_{\mu}^{T}\partial^{\mu}\psi^* - m\psi^*) \\\ &= C\gamma_{0}^{T}[(i\gamma_{\mu}^{T}\partial^{\mu}\psi^* - m\psi^*)^{T}]^{T} \\\ &= C\gamma_{0}^{T}(i\psi^{\dagger}\gamma_{\mu}\partial^{\mu} - m\psi^{\dagger})^{T} \\\ &= C\gamma_{0}^{T}[(i\bar{\psi}\gamma_{0}\gamma_{\mu}\partial^{\mu} - m\bar{\psi}\gamma_{0})]^{T} \\\ \end{align} Now \begin{align} \gamma_{0}\gamma_{\mu}\partial^{\mu} &= (\gamma_{0}\partial^{t} + \gamma_{i}\partial^{i})\gamma_{0} \end{align} If we substitute back we will get \begin{align} (i\gamma_{\mu}\partial_{\mu} - m)\psi_{C} &= C\gamma_{0}^{T}[\\{i\bar{\psi}(\gamma_{0}\partial^{t} + \gamma_{i}\partial^{i}) - m\bar{\psi}\\}\gamma_{0}]^T \\\ &\neq 0 \end{align}


title: Electromagnetic Field as a Connection in a Vector Bundle

text: I would like to know more about Ehresmann connections in vector bundles and how they relate to the electromagnetic field and the electron in quantum mechanics. **Background:** The Schrödinger equation for a free electron is $$ \frac{(-i\hbar\nabla)^2}{2m} \psi = i\hbar\partial_t \psi $$ Now, to write down the Schrödinger equation for an electron in an electromagnetic field given by the vector potential $A=(c\phi,\mathbf{A})$, we simply replace the momentum and time operator with the following operators $$\begin{array}{rcl} -i\hbar\nabla &\mapsto& D_i = -i\hbar\nabla + e\mathbf{A} \\\ i\hbar\partial_t &\mapsto& D_0 = i\hbar\partial_t - e\phi \end{array}$$ I have heard that this represents a "covariant derivative", and I would like to know more about that. **My questions** :   1. (Delegated to Notation for Sections of Vector Bundles.)   2. I have heard that a connection is a "Lie-algebra-valued one-form". How can I visualize that? Why does it take values in the Lie-algebra of $U(1)$?   3. Since a connection is a one-form, how can I apply it to a section $\psi$? I mean, a one-form eats vectors, but I have a section here? What is $D_\mu \psi(x^\mu)$, is it a section, too? I apologize for my apparent confusion, which is of course the reason for my questions.


title: Confused by renormalization

text: > **Possible Duplicate:**   >  Suggested reading for renormalization (not only in QFT) I'm trying to learn QFT. I don't quite understand why renormalization works. If you are calculating a Feynman diagram and you get an infinity as an answer, you can identify the infinite terms and set the value of some Lagrangian parameters such that the infinities cancel and those Lagrangian parameters become infinite. I don't understand why does this ad-hoc procedure give the correct answers. Can some-one point me to a more rigorous but easy treatment of renormalization?


title: Angular momentum representation

text: It is well know that, using position representation $$\langle r\lvert L\rvert \psi\rangle =r \times (-i\hbar\nabla\langle r|\psi\rangle )=r \times (-i\hbar\nabla\psi(r)).$$ However, I read from some books that if $L$ is acting on some position ket directly, then $$L|r\rangle ~=~ r \times (i\hbar\nabla|r\rangle).$$ Can anyone explain the latter equation regarding the missing "-" sign?


title: String tension in vertical circular motion

text: Suppose that I have a point mass attached to a massless string and I am rotating it vertically. That means The mass is in uniform circular motion and the path of its motion is vertical circle. How does the tension change with respect to the position of the mass. More specifically is the tension in the string is only due to circular motion ($mv^2/R$) or gravity plays a part in it ($mv^2/R$ + something due to weight)?


title: Impulse Response and Linearity

text: Lately in my physics class (2nd-year undergraduate) we've been learning about the impulse response of systems and using Green's Functions to model the response of a system to more complicated input forces. Since it's a physics class, not a math class, some of the arguments for the math have been a bit handwavy. I've been going back through the work, then, to try to convince myself of all of the steps. I'm on board with everything up until the final point: To get the response of a system to a complicated input force, you can find the response of the system to a series of impulses, then add together the resulting responses to get the total response. This requires the assumption that the motion of a system as a function of the given force is linear, i.e.: $$y(F_1(t))+y(F_2(t))=y((F_1+F_2)(t))$$ Where $y$ is the system's position at some given time, and $F_1(t)$ and $F_2(t)$ are different time-dependent forces applied to the system. More visually, you need the assumption that, in the following picture, it's legitimate to get the actual motion of the system by adding together the two dashed red lines, each of which is the response to an impulse at a different time. ![Image](http://i.imgur.com/U3aDbs3.png) Now, this is obviously correct. The trouble is, I can't see _why_ , mathematically speaking, it should be. My question, then, is how we actually prove that this is the case. I'm sure it's a simple observation that I'm just missing, but I can't for the life of me figure out right now.


title: What is many body localization?

text: Is there any good definition of many body localization? It is the property of one state or it is the property of a Hamiltonian? Why does disorder play an important role in many body localization? What is the relation between Anderson localization and many body localization?


title: What is our location relative to the Big Bang?

text: Given what we know about space, time and the movement of galaxies, have we or can we determine what our position is in relation to the projected location of the Big Bang? I've read some introductory papers on the superstructure and galaxy cluster movements, but none of them specifically mentioned space in terms of relative or absolute positions relating to the original position of the Big Bang. So my question is, does our current understanding of the structure and workings of the Universe give us a good enough estimate to determine our location relative to the Big Bang or can we never guess at it, since every viewpoint in our universe looks the same in every direction?


title: Explaining UV radiation to a 6 year old

text: My (just completed) PhD involved a considerable amount of research involved with the detection of solar UV radiation. This generated quite a bit of interest, especially when I was conducting my experiments outside. A friend's 6 year old was most fascinated, but could not grasp the concept of UV radiation, primarily as she could not 'see' it. She understood the importance of protecting oneself from too much UV (this is a big thing in Australian schools). On reflection, it occurred to me that a lot of people don't truly grasp what UV radiation actually is, despite knowing of the risks involved with overexposure. Conversely, I know and understand the technical side of it, but struggle to put it in simpler terms. What is a simple and meaningful way of explaining UV radiation?


title: Entanglement of qubits circuit- Bell states

text: I know that the quantum circuit $\text{CNOT}\; (H \otimes I)$, where $\text{CNOT}$ is the controlled-not gate and $H$ the Hadamard gate, takes the computational basis of two qubits $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ to the Bell states, which are maximally entangled. Would $\text{CNOT}\;(H \otimes I)$ give us an entangled state even if the qubits are initially not in a standard basis state, _i.e._  $|0\rangle$ or $|1\rangle$? I was interested if, when you say that some qubit A is maximally entangled with another qubit B, is there a similar circuit that **(a)**  produces that state, and **(b)**  from which you can easily show that the qubits are maximally entangled?


title: How scientists estimated that our universe is 4% normal matter, 21% dark matter, and 75% dark energy?

text: Is there a simple way to understand how scientists estimated/calculated the following percentages? ![enter image description here](http://i.stack.imgur.com/0GktV.jpg)


title: Generalized functions in physics

text: Prior to the Dirac delta function, what other distributions functions where physicists using? I find it hard to motivate the theory of generalized functions with just the delta function alone.


title: Approximation of skeleton diagrams

text: I'm studying the diagrammatics for a Bose system (in the superfluid phase) developed by Gavoret and Nozieres (Annals of Physics 28 349 (1964)). In this paper, they show how to solve the problem using skeleton diagrams. In other words, they give equations for the two-point, three-point and four-point functions, involving the full Green's functions and irreducible diagrams (that they don't discuss, but I think they mean 2-PI irreducible here). In particular, they discuss in great length how these different functions are linked together, and what kind of Ward identities they have to fulfill in order to respect conservation laws. Of course, this approach is useless unless one does approximations in order to compute correlation functions. What I don't get is how to approximate this skeleton diagrams in a consistent way to recover (for example) standard perturbation theory, which in this context is Bogoliubov theory, and at the same time be sure that the conservation laws are consistently recovered. I can't find a nice reference that would tell me how to start from these diagrams and what I should do to them. All the textbooks I've looked at only have few pages on skeleton diagrams, and just show how to express the self energies and vertices with them, without discussing anything more.


title: Angular momentum conservation in pion decay?

text: I have seen the charged pion decay $$\pi^{-}~\to~ \bar{\nu}_{\ell} +\ell^{-}$$ represented with diagrams containing a $W^-$ in the $s$-channel. The $\pi^-$ and $W^-$ have angular momentum $0$ and $1$ respectively, though. How does this process conserve angular momentum? I see that this question has been asked before, but I haven't found an answers.


title: How to calculate total distance and time a puck will move?

text: To simulate a puck's movement, I use the following model (please restrain from discussing the model, it makes the simulation look the way I want it to) and would like to know how to calculate the total way the puck will move and how long it will take to do so. My setup is: Per second, the puck looses 20% ( **=1-k** ) of its previous speed, so after a time **t** the speed will be **v(t) = v(0) * k^t**.   * If I look now at time **t(n)** , how far has to puck traveled?   * How long will it travel until its speed falls below a defined **v(min)**?   * How far will it go until its speed drops below **v(min)**? I have tried all kinds of formulas I remember from my physics lessons but I'm really stuck :-(


title: The ideal trampoline

text: Suppose we have a mass attached to the top of an ideal (linear and massless) spring oriented vertically in a uniform gravitational field, and on top of that mass there is another mass resting on it. The two masses are not attached at all, so they will lose contact with each other as the normal force is about to become negative. Also suppose that once the two masses separate and collide again, they undergo perfectly elastic collisions. First of all, is there a name for systems like this? It seems like an "ideal trampoline" to me but searching for that doesn't yield much. Has anyone ever discussed it in a book? Second of all, is this system chaotic? For sufficiently small oscillations, of course, the masses remain in contact the whole time and you get simple harmonic oscillation, but above some threshold the free mass will keep bouncing off the spring-attached mass and it's quite nontrivial to figure out what eventually happens. Do you get interesting things like period doubling?


title: Learning physics online?

text: I'm thinking of following some kind of education in physics online. I have a master degree in Computer Science and have reasonable good knowledge in physics. I would like a program of 1-2 years and I'm more interested in particle physics. Is there any good online program that offer something similar?


What is the definition of a charge-neutral operator? I guess it means something like: it is invariant under charge conjugation. It that correct?


title: How to modify the bullet trajectory based on the ballistic coefficient?

text: I am new to the physics surrounding bullet trajectory and how it is calculated. I am a software developer and I am working on a ballistics calculator for rifles. I am using wiki for the trajectory calculation I am currently using the equation under the "Angle θ required to hit coordinate (x,y)" section. This is all well and good, but it doesnt take into account the drag of the bullet(ballistic coefficient). I have searched all over trying to figure out how to apply the coefficient to this equation. I am really at a loss and and would be very thankful for any direction in this matter. Maybe I have a gap in my understanding, but I have found plenty of other calculators and other documentation on trajectory and the coefficient but nothing that marries to the two together.


title: How would a newtons cradle act in zero gravity?

text: I imagine that the first ball would strike the rest as normal, but what would the last ball do, without gravity to swing it back?


title: Does Nantenna (nano antenna) violates 2nd Law of Thermodynamics?

text: **Does Nantenna (nano antenna) violates 2nd Law of Thermodynamics ?** Nantennas absorb infrared heat and convert it in direct current. Quote from Wikipedia: > He did not discuss whether or not this would violate the second law of > thermodynamics. Some people said that that Nantennas and Solar Cells are not thermodynamics. I do not understand that thesis. That is why I have re-phrased question: **Are Nantennas a perpetual motion machine of Second Kind ?** As I know: 2nd law XOR perpetual motion of second kind **PS: I can't agree that answers are prooving that that second law is not violated by nantennas and MIM diods. If someone has to say anything about the subject then I would be happy to read that. Thank You all for responses!**


title: What is the RC time constant in a superconductor?

text: In conventional conductors, the RC time constant is the time required to charge or discharge a capacitor through a resistor by ≈ 63.2 percent of the difference between the initial value and final value: $$\tau = R \cdot C $$ However, in a superconductor, the resistance is exactly zero. $$\tau = 0 \cdot C = 0$$ Which would require infinite current because the capacitor is charged in 0 time. So the RC time constant equation above must not be valid for superconductors. How should the RC time constant be defined in a superconductor such that it does not require infinite current? That is, what is the expression for the time constant of charging a superconducting capacitor by ≈ 63.2 percent? **Edit: I re-framed the question because we were getting bogged down about signal propagation time. Now instead of using special relativity to point out the flaw in the equation, I note that it would require infinite current, which would not be physically realizable.**


title: About the Large Hadron Collider

text: I think Physics is fascinating , especially those really clever experiments one can do that demonstrate some important principle and probably don't cost a lot of money to make. Also the Thought Experiments of Galileo and Einstein and Maxwell (I think) and many others ; thought experiments that cost nothing and point out interesting ideas. It's great when some scientist figures out a relatively inexpensive way to test some complicated idea. I know there is a great 'need' to analyse the fundamental structures of matter but could this be done without spending a million dollars a day as they do with the Large Hadron Collider( If that monetary value is accurate). Can expensive experiments be redesigned so they are inexpensive yet still effective?


What would be the requirement to learn matrix mechanics? More specifically, what math do I need? Can anyone recommend me a book that covers all maths needed for matrix mechanics?


title: How does Higgs Boson get the rest mass?

text: Higgs Boson detected at LHC is massive. It has high relativistic mass means it has non-zero rest mass. Higgs Boson gives other things rest mass. But, how does it get rest mass by itself?