title: Time it takes two oppositely charged particles to collide

text: I think this is supposed to be a simple problem but I am having a hang up converting it to a one-body problem. It's one-dimensional. +q and -q a distance d apart, held stationary then let go at t=0. The potential is V(x)=kq^2/x. If I turn it into a one body problem, then m-->m/2, but how do i interpret the new x? Both particles are moving toward each other, so they travel a distance d/2 before colliding. I am guessing the relevant equation will be $t = {\sqrt{\frac{m}{2}}} \int \frac{dr}{\sqrt{E - V(x)}}$ What concepts am I lacking? I think this is supposed to be really easy, but it's not for me. edit, so x is now the relative distance between the two particles so it should be like one particle traveling the whole distance d ? I get a negative value, but is that acceptable? Something like $t=\frac{\sqrt{m}}{2} \int_d^0 \sqrt{\frac{d}{kq^2}} \frac{dx}{\sqrt{1-d/x}}$ And that isn't giving me a very good answer when I calculate it.


title: Two boxes that are connected pushed by force - what happens between two boxes?

text: So when two boxes are connected together, and force is applied, two boxes move with the same acceleration. (assuming force is constant.) My question is, how are forces between two boxes get cancelled out? When force is applied to the first box, it would exert force into the second box which pushes the first box with the same reaction force... And I am not sure afterawards. "By connected together" I mean that two boxes are stick together literally. There's no string or something like that. It's basically glued together.


title: Do photons age in a medium?

text: According to special relativity, time starts to slow down as we increase our speed and eventually stops once we get to the speed of light. By that logic, photons don't age in a vacuum state as, to us, the time stops for them. However, in a medium, their speed decreases, that means time is 'stationary'. Does that mean they start to age in a medium?


title: Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

text: Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, depending on the textbook S is either (up to a sign)   1. $\int \mathcal{L}dt$ where $\mathcal{L}$ is the interaction Lagrangian or   2. $\int \mathcal{H}dt$ where $\mathcal{H}$ is the interaction Hamiltonian. It is straightforward to prove that if you do not have time derivatives in the interaction terms these two expressions are equivalent. However these expressions are derived through different approaches and I can not explain from first principles why (and when) are they giving the same answer. Result 1 comes from the path-integral approach where we start with a Lagrangian and do perturbation with respect to the action which is the integral of the Lagrangian. Roughly, **the exponential is the probability amplitude of the trajectory**. Result 2 comes from the approach taught in QFT 101: Starting from the Schrödinger equation, we **guess** relativistic generalizations (Dirac and Klein-Gordon) and we **guess** the commutation relations to be used for second quantization. Then we proceed to the usual perturbation theory in the interaction picture. Roughly, **the exponential is the time evolution operator**. Why and when are the results the same? Why and when the probability amplitude from the path integral approach is roughly the same thing as the time evolution operator? Or restated one more time: Why the point of view where the exponential is a probability amplitude and the point of view where the exponential is the evolution operator give the same results?


title: Will Randall-Sundrum extra dimension scenario become defunct if not supported by LHC?

text: The Randall-Sundrum extra dimension scenario had been one of the most extensively studied class of theories. This offered a solution to the hierarchy problem. However, if this picture is not supported by the LHC, will it become completely defunct? What about theories like little higgs, composite higgs, technicolor, higgsless models (perhaps already practically abandoned?)?


title: Where to place a fan in order to cool a room the fastest

text: Imagine I have room that is very hot and it is colder outside. I want to cool it down using a fan. ## Scenario 1 There is only one door. I have one fan. Where should I put it and which way should it blow? ![Room with a door](http://i.stack.imgur.com/540TE.png) ## Scenario 2 I have two doors, one on each end. I still only have one fan. ![Room with two doors](http://i.stack.imgur.com/MtOg9.png) ## Scenario 3 Same as scenario 2, but I now have multiple fans. For convenience, it is fine to assume that the room is a cuboid and that the fan emits no heat.


title: Time evolution of a reduced density matrix

text: For a bipartite quantum system evolving under some master equation, is the time derivative of the reduced density matrix equal to the partial trace of the time derivative of the matrix? In other words, is the following true: $\dot{\rho}_{A} = Tr_B(\dot{\rho}_{AB})$ (Where $\rho_A = Tr_B(\rho_{AB})$) If not, is there some other simple method to find $\dot{\rho}_{A}$ from $\dot{\rho}_{AB}$?


title: What really is Spacetime?

text: What is a good definition on Space, Time and the most specific topic "Spacetime"? Because someone told me that spacetime is the foundation of the entire universe? And also, Is it possible to create spacetime?


title: On motivation for the definition of ADM mass

text: The **ADM mass** is expressed in terms of the initial data as a surface integral over a surface $S$ at spatial infinity: $$M:=-\frac{1}{8\pi}\lim_{r\to \infty}\int_S(k-k_0)\sqrt{\sigma}dS$$ where $\sigma_{ij}$ is the induced metric on $S$, $k=\sigma^{ij}k_{ij}$ is the trace of the extrinsic curvature of $S$ embedded in $\Sigma$ ($\Sigma$ is a hypersurface in spacetime containing $S$). and $k_0$ is the trace of extrinsic curvature of $S$ embedded in flat space. Can someone explain to me why ADM mass is defined so. Why is integral of difference of traces of extrinsic curvatures important?


title: Conservation law of energy and Big Bang?

text: Did the law of conservation of energy apply to the earliest moments of the Big Bang? If so, what theoretical physics supports this? I hear that Einstein's theory of relativity disputes the law of conservation of energy- so does that mean the law is false, or only some aspect of it?


title: How is dark energy consistent with conservation of mass and energy?

text: I initially thought that dark energy must in some way violate conservation of mass and energy since the component of the energy density of space that comes from dark energy is constant, and space is expanding. Therefore, as space expands the energy in the universe that comes from dark energy would increase. I presumed the source of this energy was not coming from the conversion of other types of energy to dark energy, so it must violate conservation. I decided to Google this and came upon this article: http://scienceblogs.com/startswithabang/2011/12/02/dark-energy-accelerated- expans/ It says that dark energy does NOT violate conservation and quotes Carroll, Press, and Turner (1992): "…the patch does negative work on its surroundings, because it has negative pressure. Assuming the patch expands adiabatically [i.e. without loss or gain of heat], one may equate this negative work to the increase of mass/energy of the patch. One thereby recovers the correct equation of state for dark energy: P = – ρ c2. So the mathematics is consistent." Is there a way to explain this in layman's terms? (The blog attempted to do this, but it was very unclear to me.) More specifically, can you explain where my initial train of thought described above fails when I erroneously concluded that dark energy violates conservation? Thank you.


title: Dark energy and conservation of energy

text: With accelerated expansion of universe which is same in all direction we know that dark energy increase with time because space between any two point in space time increases with time. So after some finite time we can not see nearby galaxy cluster which we can see now. So doesn't that violate conservation of energy which says energy neither can created nor can destroyed. Because with expanding universe energy in the form of dark energy increases with time so if we consider whole universe (visible + invisible) as isolated system then energy of whole universe increase means energy is created from nothing. Am I missing something over here?


title: Does the big bang violate the conservation of energy?

text: It is a fact that a thing is existing now because it had already been created. So why don't we take this to account to redefine law of conservation of energy.


title: Is false vacuum bubble nucleation possible in our universe?

text: Is it possible that a false vacuum bubble to nucleate into our universe rather than a true vacuum one ? If yes,it will expand at speed of light within our spacetime or what ?


title: Conservation laws in vacuum phase transition

text: Let's consider a bubble nucleation phase transition between different vacua via quantum tunnelling .For my understanding a particle must penetrate the potential barrier and find herself in an another energy state over the barrier. But this doesn't mean that the parent vacuum loses the particle ?This doesn't violate the conservation of energy laws?


title: Ampere's law on a long wire with varying current density

text: On a question from my book: > A long straight wire with a circular cross section of radius $R$ carries a > current $I$. Assume the current density is not constant over the cross > section of the wire, but rather varies as $J=\alpha r$ where $\alpha$ is a > constant. Given $I, R$ > > Find $\alpha$ > > Find the magnetic field as a function of r both inside and outside the wire I think it's just the calculus parts confuses me. My attempt: $$J=\alpha r' = \frac{dI}{dA}$$ $$dI = 2 \pi r'^2 dr' \alpha$$ $$I = 2 \pi \alpha \int_0^R r'^2 dr'$$ $$I = 2 \pi R^3 \alpha /3$$ $$\alpha = \frac{3I}{2 \pi R^3}$$ from here you just use ampere's and I believe there's no variance issues? $$\oint \vec{B} \cdot \vec{dl} = \mu_0 I_{in}$$ apply J=I/A $$B 2 \pi r = \mu_0 \alpha r A$$ $$B = \frac{\mu_0 \alpha A}{2 \pi}$$] $$B = \frac{3 \mu_0 I r^2}{2 R^3}$$ Is this right? The units seem to line up so I'm hopeful. Outside the wire is treated as the general uniform wire case I assume and am not too worried about that.


title: Thin Film Interference in Fluid Dynamics: Many are welcome to throw ideas in here, Please!

text: It was a few days ago, and there was a very heavy storm(currents were flowing beside the sidewalks as normal). I became intrigued by a thin film of gasoline that got caught in one of the currents. The visible colors of course changed by my angle of incidence, yet because there was a current, the thickness of the film also changed, as it spread out and moved along with the water. I would like to make something of this. How did an unchanged angle of observation result in so many different changes in which colors were the most visible at a certain time?


title: Satellite Collision Problem

text: I'm stuck on a problem of two satellites going towards each other. The mass of the first satellite is 400kg and the mass of the second satellite is 100kg. The altitude of the satellites is 1000km. I want to know if once the satellites collide, will they continue in orbit or will they crash and burn into the Earth? I know that you can get the velocity of each satellite by using this equation: $$v = \sqrt{GM/R}$$ Where $v$ is the velocity, $G$ is $6.67\cdot 10^{-11}$, $M$ is the mass of the planet, and $R$ is the radius.   I can get the velocities, but i don't know where to go from here. Also when I get $R$, do I also have to add the radius of the planet as well? so it would be 1000km + radius of planet? If you guys would help me with this, it would be amazing.


title: Propagators, path Integrals, transition amplitudes, Green's functions etc

text: I'm trying to make a simple conceptual map regarding some of the things in the title as they pertain to quantum mechanics and or quantum field theory, and I'm finding that I'm a little perplexed about a couple of items. Let me summarize a few things I have as being true, and then state what I don't understand.   1. Generally the propagator $K$ or often $D(x-y)$ is a Green's function of the quantum operator sometimes the Schroedinger operator, or the Klein-Gordon operator or analogous items. In the K-G case we would have something like $(\partial_u\partial^u +m^2)D\propto \delta^4(x)$   2. The "transition amplitude" I would tend to think should quantitate a probability of a system in state progressing to another state in time, ie $\left \langle x'',t'' \right |\left. x',t'\right\rangle$   3. The path integral $\int \mathcal{D}e^{iS}$ should be interchangeble with a transition amplitude, at least according to a few of my texts. What I am struggling on could be precisely what is meant by "transition amplitude" in some cases. Take for example the propagator for the Klein-Gordon equation. $$D=\int d^4p{1\over 2\pi\hbar}^4{1\over E^2}e^{{i\over\hbar}p\cdot x}$$ As far as I can tell from it's form, the propagator for the k-g equation is NOT from it's form a (Dirac )delta function $\delta^4 (x)$ or even $\delta^3(x)$ I don't actually think that it should be given point number 1. However, I am failing to recognize its relation to "transition amplitude" in this case because I would usually equate a terminology such as "transition amplitude" with a probability. As the K-G propagator is NOT a normalized distribution, i.e. does not have the form of a delta function, what precisely is it supposed to quantify here? P.S: Update since original post (above)--I've since noted that the term 'propagator' may be used somewhat differently in different contexts. Specifially, going back to J.J Sakurai Modern Quantum Mech. --chapter 2.5(ish --dont have the book in front of me now) he uses the K to represent what is termed the propogator of the Schrodinger system. He then discussed this in it's equivalent to the Feynman Path integral approach to determine $<x'',t''|x',t'>$ The use of $D(x-y)$ ,also referred to as the propagator, in Quantum Field Theory approaches by contrast seems to have a different meaning. I now realize $D(x-y)$ is not equivalent to $\int\mathcal{D}e^{iS}$ but rather something different as is apparent to me now. So I think that straightens some significant things out in my head. If anyone has anything to add or correct me on, please do.


title: Fluid mechanics problem

text: I have a $2D$ fluid parcel with coordinates $(0.5,-0.5), (-0.5,-0.5), (0.5,0.5)$ and $(-0.5,0.5)$ and this parcel is deformed by a steady flow field of $u=ay$ and $v=0$, defined on the basis ${(1,0), (0,1)}$. I tried to calculate the velocity gradient tensor $\tau_{ij}$ given by the matrix $ \left( \begin{array}{ccc} 0 & a \\\ 0 & 0 \end{array} \right) $. I now need to decompose this into the symmetric strain rate tensor and the antisymmetric rotation tensor. However, this matrix isn't diagonalizable. Am I missing something here? * * * I need to use this part to "solve" the transformation equation of a point that is given by $$x_i(t+dt)=x_i+(u_i+du_i)dt$$where $du_i = \tau_{ij}dx_j$. Then, I must remove the rotation rate tensor from the velocity gradient tensor (which basically means I have the strain rate tensor left, if I am not mistaken) and use that to show that the transformation equation then becomes $$x_i(t+dt)=x_i+(u_i+\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})\partial x_j)dt$$ and finally determine the transformation equations for $x(t+dt), y(t+dt)$.


I've read and heard that quantum electrodynamics is more fundamental than maxwells equations. How do you go from quantum electrodynamics to Maxwell's equations?


title: Calculate signal strength received by an Ultrasonic Transducer

text: Below is an image showing transducer A transmitting a signal with an amplitude of 3000mV, I am trying to calculate the amplitude of the signal received by transducer B in mV, I understand that there are dB losses at each of the acoustic interference (1 to 5),   1. Which of the two formulae (given below) are relevant at each of the acoustic interference (1 to 5) and why?   2. How are these three parameters related - transmitted signal in mV (transducer A), dB loss and received signal in mV (transducer B)? I have looked at many articles online, but they haven't helped me. Can someone please suggest me or guide me to help answer the above questions. ![enter image description here](http://i.stack.imgur.com/tUYzk.png) **Formula 1:** The dB loss of energy on transmitting a signal from medium 1 into medium 2 is given by ![enter image description here](http://i.stack.imgur.com/wQb0M.jpg) **Formula 2:** The dB loss of energy of the echo signal in medium 1 reflecting from an interface boundary with medium 2 is given by: ![enter image description here](http://i.stack.imgur.com/rXs9P.jpg)


title: Doing theoretical physics after an engineering major?

text: Some background: I'm majoring in Electronics and Instrumentation engineering, but I want to pursue theoretical physics. I'm currently In my second year of undergrad engineering. I attend some physics classes in the weekends and I've learnt classical mechanics (from John R. Taylor's book predominantly, with occasional references to Goldstein and David Tong's lecture notes). My math background includes differential equations, multivariable calculus and a bit of linear algebra (working on this now). In the second year, we've started off with Griffiths electrodynamics and Griffiths quantum mechanics. Now my questions:   1. To supplement my lack of formal mathematical training (engg math is not rigorous) I've taken to books such as Artin Algebra, Boyce and diprima differential equations(I've completed this book), linear algebra by Sheldon Axler etc. Due to the sheer amount of math there is, I'm quite confused as to what to study. Which are the essential math topics I, as an aspiring theoretical physicist must know? Please suggest a textbook for them.    2. In my weekend classes, neither thermodynamics nor optics is covered. Must I study these in depth? If so, from which textbook?   3. Any other suggestions?


title: How can you calculate how fast a spinning ring/cylinder will accelerate a mass via gravitomagnetism?

text: Lets say you have a cylinder of length L, radius R, and mass M. How fast will it accelerate a mass of mass M2 that is entering the "throat" of the cylinder, considering the effects of gravitoelectromagnetism? By Gravitoelectromagnetism, I'm specifically talking about the effect where a spinning body will pull an object through the "throat" of its spin. Also, I'm wondering what the practical limit of this is: considering the strongest known materials, how fast could a cylinder be spun?


title: The order of seeing event in different spacetimes

text: Assume this question: > Three events A, B, C are seen by observer O to occur in the order ABC. > Another observer O$^\prime$ sees the events to occur in the order CBA. Is it > possible that a third observer sees the events in the order ACB? I could draw spacetime diagram for three arbitrary-ordered events for O and O$^{\prime}$. But I couldn't draw spacetime diagram for third observer. Could someone help me figure this out?


title: How do I get the total acceleration from 3 axes?

text: For a project I'm working on I'm using an accelerometer which measures acceleration in 3 directions, x, y and z. My question is: How can I calculate the total acceleration in a certain direction from these 3 values? Considering this simple graph layout: ![enter image description here](http://i.stack.imgur.com/UX9S8.png) My initial idea is:   * Take the sqrt of (x^2 + z^2) to calculate the resulting value in the zx plane.   * Take this value, square it and add y^2, take the square root of that   * Final equation: Sqrt(y^2 + Sqrt(x^2 + z^2)) Is this correct? On some sites I see x^2 + y^2 + z^2 being used, but I don't know if that's right and why it's right. **EDIT:** I just figured that taking the Sqrt of (x^2 + z^2) and squaring it just results back in x^2 + z^2, so that's why I can use x^2 + y^2 + z^2. Another thing: Do I have to normalize for gravity? I think I do, but how would I go about this? Do I need to know the exact position and tilt of my device as it will be in the end?


title: Time-Dependent Potentials in Quantum Mechanics

text: A potential that depends on time is usually solved using the time dependent perturbation theory in standard undergraduate textbooks in quantum mechanics. The reason usually mentioned is that time dependent potentials cannot be solved using separation of variables. Is this the only reason why it has to be solved using time dependent perturbation theory (Just because we cannot use methods of separation of variables)?


title: Discretization of Hamiltonian using finite difference always justified?

text: I have this continuum version $$ H_{R}=\int dx\psi^{\dagger}(x)(\frac{p^{2}}{2}+V)\psi(x) $$ with $V$ as constant potential. Is it always justified to go from this to $$ \sum_{i}c_{i}^{ \dagger }\left[c_{i+1}+c_{i-1}-2c_{i}\right] +V \sum_{i}c_{i}^{ \dagger }c_{i} $$ using the finite difference form of the one-dimensional second derivative? Ignore the factor $-1/2 $ and assume lattice constant $a= 1 $ and $\hbar =1 $. Actually I am thinking whether it is justified to use this even when the derivative of eigenfunction is discontinuous at some of the points in real space like for the delta-function barrier. Will that affect the second derivative of the field operators ?


title: Theoretical considerations on the conservation of energy and the conservation of linear momentum

text: I report to you an interesting excerpt from my Physics book. It is an Italian version, so I apologize in advance, as I'm sure I won't give proper justice to its beauty in the translation as the authors would have done. Talking about > There is a profound theoretical connection between quantities that are > preserved and symmetries of nature. So the principle of conservation of the > linear momentum is related to the _spatial_ symmetry of nature, which > implies that an experiment done in a place give identical results to an > equal experiment performed in any other place. [...] The principle of > conservation of energy is related to the _temporal_ symmetry: the result of > an experiment today will be equal to the result of the same experiment did > yesterday. Physics I - Resnick, Halliday, Krane After this words, the authors start talking about the conservations of the linear momentum in the canonical way. However, I was captured by this fly away and I want to find out more. How can I do that? In synthesis:   1. _Spatial_ symmetry of nature $~\Rightarrow~\ldots?? \ldots~\Rightarrow~$ Conservation of the linear momentum    2. _Temporal_ symmetry of nature $~\Rightarrow~\ldots?? \ldots~\Rightarrow~$ Conservation of energy  What do you think is in between?


title: Can relativistic energy transformation be explained by time dilation and E=h f?

text: Can one explain the relativistic energy transformation formula: $$E = \gamma\ E',$$ where the primed frame has a velocity $v$ relative to the unprimed frame, in terms of relativistic time dilation and the quantum relation $E=h\nu$? I imagine a pair of observers, A and B, initially at rest, each with an identical quantum system with oscillation period $T$. Now A stays at rest whereas B is boosted to velocity $v$. Just as in the "twin paradox" the two observers are no longer identical: B has experienced a boost whereas A has not. Both observers should agree on the fact that B has more energy than A. From A's perspective B has extra kinetic energy by virtue of his velocity $v$. Relativistically A should use the energy transformation formula above. But we should also be able to argue that B has more energy from B's perspective as well. From B's perspective he is stationary and A has velocity $-v$. Therefore, due to relativistic time dilation, B sees A's oscillation period $T$ increased to $\gamma\ T$. Thus B finds that his quantum oscillator will perform a factor of $\gamma\ T/T=\gamma$ more oscillations in the same period as A's quantum system. Thus B sees that the frequency of his quantum system has increased by a factor of $\gamma$ over the frequency of A's system. As we have the quantum relation, $E=h\nu$, this implies that B observes that the energy of his quantum system is a factor of $\gamma$ larger than the energy of A's stationary system. Thus observer B too, using his frame of reference, can confirm that his system has more energy than observer A's system. Is this reasoning correct?


title: Formula for Rabi frequency

text: I have calculated the dipole transition elements of electronic states $\langle a|D_1^m|b \rangle$ following the book of Cohen-Tannoudji (Complement $E_X$) and tried then to calculate from that the Rabi frequency $$\Omega_{a\to b}=\frac{1}{2\hbar}E_m \langle a|D_1^m|b \rangle.$$ But what is the direction of the E-field vector $E_m$ for the $\pi$, $\sigma^+$, $\sigma^-$ transitions?


title: Intuitively Re-Deriving Equations of Mathematical Physics

text: Using the intuitive interpretation of the Laplacian $\vec{\nabla}^2$ as the difference between the average value of a field in the neighbourhood of a point & the value of the field at that point, one can pretty easily & quickly derive the form of the heat equation, Poisson's equation & the wave equation (as is done in that link if anyone's interested: Davis - Fourier Series & Orthogonal Functions P196). I quite honestly cannot remember those equations, I re-derive them using the intuition that the Laplacian affords me mixed with physical reasoning drawn from the situation (drawn from the field we're using whether it's temperature, concentration, electric potential or displacement). I'm wondering if similar intuition can be used to derive the form of the Helmholtz equation, the Schrodinger equation, the Dirac equation, & really any other nice equation from mathematical physics that people have nice intuition for in their head & wouldn't mind sharing. It doesn't have to be in any way rigorous or even necessarily completely logical so long as you get the right result, though it should really be quick & to the point, thanks.


title: How does a SCFT avoid the Haag-Lopuszanski-Sohnius theorem?

text: According to the Haag-Lopuszanski-Sohnius theorem the most general symmetry that a consistent 4 dimensional field theory can enjoy is supersymmery, seen as an extension of Poincarè symmetry, in direct product with the internal gauge symmetry. But we know that conformal theories, having as a symmetry group the conformal group (which is indeed an extension of the Poincarè group) in direct product with the internal gauge group exist. Also there exist superconformal theories, which enjoy both conformal symmetry, supersymmetry and gauge internal symmetry. All this theories are consistent, from a theoretical point of view, and well definite in $d=4$. Therefore I ask, how does superconformal field theories avoid the Haag- Lopuszanski-Sohnius theorem?


title: How do magnets work in classical electromagnetism?

text: A charged particle moving in a magnetic field experiences a Lorentz force $F=qv\times B$. A charged particle in a electric field experiences a force given by Coulomb's inverse square law. But for a magnet we have what seems to be adhoc ways to calculate how the magnet will rotate. In the above case we speak concretely of a charged particle being influenced by a magnetic field or an electric field. But what is the analogous "thing" or "particle" causing the magnet to rotate? It feels like classical electromagnetism is Maxwell's equations plus laws for how a charged particle behaves in a magnetic field and electric field, but notably missing corresponding Coulomb and Lorentz force laws for "magnets".


title: What exactly are super WIMPs?

text: I recently got confused (and slightly annoyed by the lack of technical details) when reading a popular article (authored by Jonathan Feng and Mark Trodden) introducing the concept of super WIMPs. The article characterized super WIMPs (without giving more detailed explanations) as follows:   * WIMPs could probably decay to so-called super WIMPs, which would only gravitationally interact with visible matter    * different kinds of super WIMP particles could interact via additional newly postulated weak "dark forces" ( = gauge bosons ?) with each other   * this kind of dark matter particles can probably interact with dark energy ( how? What is dark energy in this particular scenarios suposed to be? )   * the authors vaguely stated the super WIMP models are some kind of extensions of supersymmetric models that lead to the "ordinary" WIMPs From this characterization I really dont get what super WIMPs are suposed to be so my question is: What are the underlying theoretical ideas behind these phenomenological models? Are they derived in some "top down" approach from high energy theories or is some "buttom up" extension of something like the MSSM for example applied ? And I would appreciate a technically more accurate description of the super WIMP particles and their interactions.


title: How were non-Euclidean manifolds applied to physics before Einstein?

text: In the letter of introduction to Einstein's 1916 paper on General Relativity, he writes, "The mathematical tool sthat are necessary for general relativity were readily available in the 'absolute different calculus,' which is based upon the research on non-Euclidean manifolds by Gauss, Riemann, and Christoffel, and which has been systematized by Ricci and Levi-Civita and has already been applied to problems of theoretical physics." In what way had it already been applied to problems of theoretical physics? See http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf for the quote.


title: Is the conclusion that the cosmic burst observed in the Draco constellation from gamma rays that are streaming towards a massive black hole correct?

text: Referencing this news article: http://www.smh.com.au/technology/sci- tech/cosmic-burst-in-far-away-galaxy-puzzles-nasa-20110408-1d6kz.html It also references an event id:   (GRB) 110328A The article seems to imply that the reason for the unusually long period of gamma bursts being visible is because the remnants of the exploding star is near the centre of that galaxies nucleaus, and hence probably near the supermassive black hole. It goes on to say that the observable event is a stream of this energy towards the sm-bh.   Is this the most logical hypothesis for this event?   What other possibilities could explain such a lengthy burst of gamma radiation being visible? Edit:   If it is even speculation that it is a star being ripped apart, then what other possibilites could explain the explosion that lead to the unusually long bursts?


title: Rotation matrix is always a unitary operator

text: Can someone explain why the rotation matrix is a unitary, specifically orthogonal, operator?


title: Who invented the term "Coulomb logarithm"?

text: Who is the author of the term "Coulomb logarithm"? In fact, Coulomb logarithm was computed by Langmuir in his paper of 1928 where the term "plasma" was introduced into physics, but the term "Coulomb logarithm" seemed to appear later. The earliest reference I found is the paper Relativistic kinetic equation  by Belyaev & Budker (1956). UPDATE: Using Google Labs I found a reference dated by 1937 in very rare Georgian journal in Russian.


title: What is a "classical Schrodinger field", really?

text: I have read through the wikipedia page and several lecture notes/arxiv papers from my google search (and several related P.SE questions), but I'm still hopelessly confused.   1. Consider a 'classical Schrodinger field in a box' problem. Since the field evolves just like a normal wavefunction, I could still extract the x-projection of any operators I want from it, in addition to the field momentum $\frac{\partial\mathcal{L}}{\partial\dot{\phi}}$ or any other field variables.   2. From my understanding, there are two ways to arrive at QFT:     * from QM/RQM: we change our basis into particle-number basis (fock space).     * from CFT: we "2nd quantize" the field and field-momentum. Looks like the "particle creation" interpretation of the field in QFT can be deduced only from the first approach. Which means, the interpretation of the latter approach can't be completely independent from the former. But they don't give me any hint on how to interpret the classical field. Or am I missing something? Or should I just give it up because the quantum field is not an observable? Random comment: I could still see $\hbar$ lurking inside CFT (some people might argue that it's there to keep the dimension consistent, but I could apply the same argument to Schrodinger equation, too).


title: Can spin liquids without spin-rotation and time-reversal symmetries possess nonzero Spin Density Wave (SDW) order parameters?

text: For those spin liquids with _SU(2) spin-rotation symmetry_ **or** _time- reversal(TR) symmetry_ , the Spin Density Wave (SDW) order parameters are always zero, say $\left \langle \mathbf{S}_i \right \rangle=\mathbf{0}$, due to the spin-rotation symmetry or TR symmetry. But if a spin-liquid state is _neither_ spin-rotation symmetric _nor_ TR symmetric, _e.g._ the exact _chiral spin liquid_ ground state of the generalized Kitaev model on the _decorated_ honeycomb lattice, is there any possibility that the SDW order parameters $\left \langle \mathbf{S}_i \right \rangle\neq \mathbf{0}$ ? Therefore, if a spin-liquid state has _nonzero_ order parameters $\left \langle \mathbf{S}_i \right \rangle$, why we still call it a "spin liquid" rather than a "SDW phase"? **Remarks:** The _spin-rotation_ mentioned here should be understood as the **continuous $SU(2)$ or $SO(3)$ one** , say **all** the spin-rotation transformations. Although the Kitaev model(on the _decorated_ honeycomb lattice) and its ground state **break** this _continuous spin-rotation symmetry_ , they still **possess** the _$\pi$ spin-rotation symmetry_ about $S_x,S_y$ and $S_z$ spin-axes, and this is _enough_ to ensure $\left \langle \mathbf{S}_i \right \rangle=\mathbf{0}$. Thanks in advance.


title: What are specific arguments against the ensemble interpretation (as promoted by L. Ballentine)?

text: Leslie Ballentine develops in QM: A Modern Development an interpretation based on the ensemble interpretation, and responds to most criticisms. My question: what criticisms still exist against this interpretation such that it is not recognized as the standard interpretation within the physics community? What problems still exist? http://en.wikipedia.org/wiki/Ensemble_interpretation


title: Explain $\rho_{0}\dot{e} - \bf{P}^{T} : \bf{\dot{F}}+\nabla_{0} \cdot \bf{q} -\rho_{0}S = 0$

text: I am trying to understand the balance of energy -law from continuum mechanics, fourth law here. Could someone break this a bit to help me understand it? From chemistry, I can recall $$dU = \partial Q + \partial W$$ where $U$ is the internal energy, $Q$ is heat and $W$ is the work. How is the fourth law of conservation in CM: $$\rho_{0}\dot{e} - \bf{P}^{T} : \bf{\dot{F}}+\nabla_{0} \cdot \bf{q} -\rho_{0}S = 0$$ related to that? **Terms**   * $e(\bar{x}, t) = \text{internal energy per mass}$   * $q(\bar{x}, t) = \text{heat flux vector}$   * $\rho(\bar{x}, t) = \text{mass density}$ **Operations**   * $: \text{ -operation} = \text{Frobenius inner product?}$ (related)   * $\dot{\text{v}} = \text{derivative of vector } v$   * $\dot{\text{M}} = \text{transpose of matrix } M$


title: Are there books on Regularization and Renormalization in QFT at an Introductory level?

text: Are there books on Regularization and Renormalization, in the context of quantum field theory at an Introductory level? Could you suggest one? _Added_ : I posted at math.SE the question _Reference request: Introduction to mathematical theory of Regularization_ and accepted this answer by Willie Wong.


title: Good reference for renormalization

text: > **Possible Duplicate:**   >  Are there books on Regularization at an Introductory level? I am looking for a good introduction to renormalization (group). I have read several books about it but never really understand the idea. I know that the rough idea is to introduce the bare mass etc to absorb infinite integrals in perturbation theory. But my understanding stops at this point. I would really appreciate your help.


title: Schrödinger evolution for a Klein-Gordon equation

text: I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with the time evolution operator, which is crucial on deriving the perturbative expansion $-$ the main tool in QFT I believe. c So here's what I have a problem with: when we make the leap from Schrödinger equation to a Klein-Gordon equation, we get a second order time derivative, and hence loose the simple concepts from nonrelativistic QM like: the Hamiltonian, time evolution operator etc. But for a scalar quantum field we can make a Lagrangian density: $$ \mathcal{L}(x) = \hbar^2 c^2 g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi^* - m^2c^4 \phi \phi^* $$ and perform the "second quantization", from which we get a Hamiltonian, canonical commutation relations and the ability to use pictures (Schrödinger's, Heisenberg's...). So how does this work? Before there was no Hamiltonian in principle, and now there is. Is this the Hamiltonian we pluck into the perturbative expansions' formulas? What changed, when compared to the single solution wave equation in the beginning?


title: Integral in different coordinate systems

text: In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r}'-\mathbf{r}|}\mathrm{d}\tau'.$$ This is, of course, justified by the fact that each cartesian component of $\mathbf{A}$ obeys Poisson's equation, according to the first equation. But then he went on to say that to evaluate the integral you are restricted to use cartesian coordinates because that was our assumption in deriving the second equation from the first. (4th edition, page 244, footnote 19). This seems wrong to me. As far as I can imagine, the value of the integral is independent of the system of coordinates you use.


title: What are the most important discoveries/breakthroughs in physics recently?

text: Can you provide a list of the most important discoveries/breakthroughs in physics recently? By recent, I mean the past decade or so. All branches of physics are welcome. Basically, I am interested in major physics breakthroughs/discoveries which haven't become well-known yet outside their narrow specialties. Most breakthroughs in the 90s like string dualities and the accelerating universe have already become common knowledge.


title: Seeing one's back on the event horizon

text: If we would hypothetically be exactly on the event horizon, we should see our own back, because of the circular motion of photons on the event horizon, right? But what would be the image size, or $-$ asking differently $-$ how far away, would our back seem to be? Would it be magnified or minified, when compared to the image of a person $2 \pi R_{Schwarzschild}$ away?


title: Proving Operator identities (Quantum Physics)

text: How would I go about showing: $$\hat{A}^{\dagger} + \hat{B}^{\dagger} = \left( \hat{A} + \hat{B} \right) ^{\dagger}$$


title: Finding $\psi(x)$ from Fourier modes

text: In quantum physics we've defined: $$ \psi (x) = \sqrt{ \dfrac{1}{2 \pi \hbar} } \int^{ \infty }_{-\infty } \phi (p) \exp \left( i \dfrac{px}{ \hbar} \right) dp $$ Now, $$a(k) \equiv \sqrt{ \hbar } \phi (p)\quad {\rm and}\quad k = \dfrac{p}{ \hbar } $$ Where, $$ a(k) = \left\\{ \begin{array}{cccc} 0 & k < \- \dfrac{ \epsilon }{2} \\\ \sigma + \dfrac{2 \sigma }{ \epsilon } k & \- \dfrac{ \epsilon }{2} < k < 0 \\\ \sigma - \dfrac{2 \sigma }{ \epsilon } k & 0 < k < \dfrac{ \epsilon }{2} \\\ 0 & k > \dfrac{ \epsilon }{2} \\\ \end{array} \right.$$ Normalizing $a(k)$ I get $\sigma$ to be: $$ \sigma = \sqrt{ \dfrac{3}{ \epsilon } } $$ But I can't get anything reasonable from the Fourier Integral.


title: Voltmeter Question

text: I understand that a voltmeter is used for measuring potential electrical difference, but how exactly should said voltmeter be connected with the resistor in circuit?


title: Mechanism of Hawking's radiation and entropy of the black hole

text: During Hawking's radiation, a virtual particle with negative energy and mass (from pair of particle and antiparticle) fall into black hole and its real partner having positive energy escape from vicinity of black hole. And it _appears_ to have been emitted from black hole. _Since these pair of particles were present outside the black hole_ , flow of negative energy particles reduces its mass. As the black hole loses its mass the area of event horizon gets smaller. And so, it must decrease the entropy of the black hole. So, how the entropy of the black hole always increase? What I am missing here. Please explain.


title: Magnetic field of a wire with over 1000 amperes?

text: Does a large thick wire that has over 1000 amperes of current flowing through it generate powerful magnetic fields? What formula is best here to predict $B$?


title: Dispersion relation of silicon

text: In the case of dispersion relation of silicon having crystal plane orientation 111; what is the Sellmeier's equation for refractive index $n$ of silicon orientation 111 & what it's extinction coefficient $K$?


title: What exactly is resistance?

text: I am trying to teach myself about electricity and magnetism and I have a few questions about resistance. How does resistance of say a long wire compare to that of a a short wire? what about thick and thin?


title: Calculating the power of a lightbulb

text: How do I calculate the power of a lightbulb? I have values but I don't know the equation to use.


title: Mirror reflection

text: **Question** ![http://i.imgur.com/ALUdyoB.png](http://i.stack.imgur.com/9JcHo.png) Create a set-up, as shown in the Figure, consisting of two mirror tiles (A and B) and a bird face, facing to the right . Make sure the two mirrors are perpendicular to the table and parallel to each other. The situation shown is a front view of the situation. Question A: If look at a mirror from the same direction and angle as one of the two arrows that are drawn, you will see the bird in the mirror. Draw the two mirror images of the bird, making sure the bird mirror is facing to the correct side! **My Try** I tried to draw a mirror reflection. ![\[IMG\]http://i.imgur.com/9U67i12.png\[/IMG\]](http://i.stack.imgur.com/32wfh.png) But Can the light ray (in yellow) simply go through the bird?


title: How does a car turn without any skidding?

text: The rear wheels of a car always face in the direction the car is moving. The front wheels are able to turn left or right and thus can point in the direction the car is moving towards. What I don't understand is how a car can turn with all four wheels rotating (not skidding). That is, how is it possible that the front two tires can face in one direction, the rear two tires in another direction, with the four tires all connected by rigid rods and with all four tires rotating without skidding? I'm trying to visualize this assuming the car is moving very slowly, but even then the situation just seems impossible to me. Is it in fact that the rear tires are skidding in just very small micro-steps so that we don't actually observe it happening?


title: Fermion Field of Standard Model

text: Why fermion field is treated as anti-commuting and boson field as truly classical in standard model?


title: $p\ dq$ is the "tautological" one-form?

text: The one-form $$\theta=\sum_i p_i\, \text dq^i$$ is a central object in hamiltonian mechanics. It has a bunch of applications: $\omega=\text d\theta$ is the symplectic structure on phase space, $S=\int\theta$ is the classical action, and so on and so forth. It is associated with the names Liouville one- form, Poincaré one-form, canonical one-form, and symplectic potential, none of which surprises me, but its Wikipedia entry informs me that the preferred[by whom?] name for it is actually "tautological" one-form, on the grounds that 'canonical' (which would be my natural choice) is 'already heavily loaded', and because of the risk of confusion with some algebraic thingammy. This name completely mystifies me. **Why was the name "tautological" chosen for this object?** When, where, and by whom? Or was this name chosen because that's its name?


title: What's the physical intuition for symplectic structures?

text: I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector field. To simplify things, let's consider the configuration space $TQ$, we know that $T^{*}Q$ always have a symplectic structure by putting $\omega = d\theta$ where $\theta = \sum p_i dq_i$ is the Liouville one-form, then the hamiltonian vector field is defined by $\omega(X_H, Y) = dH(Y)$ and I can change from the Lagrangian $L : TQ \longrightarrow \mathbb{R}$ to the hamiltonian $H :T^{*}Q \longrightarrow \mathbb{R}$ by the mass (1, 1)-tensor $M=M_i^j$. So what's the physical intuition for $\omega$, $X_H$ and $\theta$? Why do people uses a symplectic structure in mechanics (if it's to define $X_H$, what's the utility of $X_H$?)? Furthermore is the unique utility in changing the lagrangian to the hamiltonian the existence of a symplectic form in $T^{*}Q$?


title: Velocity after inelastic collision between bullet and block

text: The following is the problem that I am working on. > A bullet of $.01\: \mathrm{kg}$ is shot into a block of mass $.89\: > \mathrm{kg}$ that is hanging from the ceiling. After the bullet has been > caught in the block, it swings and rises $.40\: \mathrm{m}$ from its initial > height. Find the initial velocity of the bullet. I though that $\frac{1}{2}m{v_0}^2 = (m+M)gh$ would give me the solution $25.6\: \mathrm{m/s}$, but the answer is supposedly $2.5 \times 10^2\: \mathrm{m/s}$. What am I missing ?


title: Why does the electric field dominate in light?

text: I read a book on the wave property of light where the author mentioned that the electric field, instead of magnetic field, dominates the light property. I don't understand why. In Maxwell's theory, a light field has an electric and magnetic field at the same time and they are perpendicular. Also, in some books, where they consider the polarization, they only use the electric field as example. For example, if the vibration of the electric field is up and down, it cannot go through a polarizer which orients 90 degree to the vibration direction of the field, so no light goes through the polarizer. But what happened to the magnetic field? The magnetic field is perpendicular to the electric field, so in this case, the magnetic field should pass the polarizer, and we should have outgoing light -- but we don't. Why is this so?


title: The requirements for superconductivity

text: Which properties are sufficient evidence for a material to be not superconducting? I am looking for a set of statements like > If the material is semiconducting, it is not superconducting Edit: I am not looking for a definition of superconductivity, or for introductional literature like the famous W. Buckel. I am looking for properties, that would forbid superconductivity. If you have a source for it i would be very glad. As far I remember magnetic atoms will forbid superconductivity too, but i could not find a source yet.


title: Can high charges (like $1\times 10^{-3}$ coulomb) be acheived?

text: How can we charge a metal electrode with large charges ? When i saw a video about measuring a charge using a visiostat on a balloon , the charge was 0.6 nano-coulomb. Is the charge of 1 coulomb unachievable?


title: Why can we see the dust particles in a narrow beam of light (and not in an all lighted area)?

text: Let us say that I am sitting in a room with all the drapes open. Bright sunlight is coming through the window. The whole room is brilliantly lighted. I will not be able to see the dust particles suspended in air. Now, if I draw the drapes close, keeping a small slit open, allowing only a beam of sunlight to come in, I will readily see the suspended dust particles in that beam. The same thing will happen in a dark night with the beam of light from a handheld battery torch. What will be the scientific explanation for this? I can not see the dust particles when I have more light. But when I actually reduce the light and there is only one narrow beam present, I can see those minuscule particles. How does a narrow beam of light enable me to see those fine elements?


title: Why does light of high frequency appear violet?

text: When people are asked to match monchromatic violet light with an additive mix of basic colours, they (paradoxically) mix in red. In fact, the CIE 1931 color space chromaticity diagram shows this effect begins at about 510nm (greenish- cyan), where people mix in no red. From that point on, the higher the frequency of the light source, the more red they mix in. This effect is reflected by the red curve of the CIE standard observer color matching functions, which has an additional bump in the area of blue light. However, that curve does not match the actual spectral sensitivity of red cones. So where does this additional perception of red at higher frequencies come from?


title: Is the color wheel just an optical illusion?

text: As a kid, I was taught that that blue and yellow make green, yellow and red make orange, and red and blue make purple - forming the subtractive color wheel. As an adolescent I was taught that blue and green make cyan, green and red make yellow, and red and blue make magenta - forming the additive color wheel. Somewhere in that time I was also taught that the colors of the rainbow are ROY G. BIV. That last one is the only one that makes _much_ sense to me in physics because perceivable light is not somehow a cycle, but rather a tiny segment of the EM spectrum. Why do the colors at the top and bottom of the spectrum seem to be mixable? Is there anything in nature that indicates this should be the case? Is this just a psychological or physiological phenomenon?


title: Why does the visible light spectrum appear to be circular

text: I'm not sure if this is an acceptable place to ask this question, as it may have more to do with the biological workings of human eyes than with the physical properties of light, but I'd rather hear a physical explanation than a biological one (if one exists). Essentially, we often describe the visible spectrum with a color wheel. In this wheel, Red appears next to Purple (violet), but Red and Violet are at opposite ends of the visible light spectrum. What accounts for this? That is, why do we perceive the Visible Light Spectrum as circular, instead of linear?


title: Is an entropic force an actual force that can be explained as a fundamental interaction?

text: Fundamental interactions, such as electromagnetism, the strong force, the weak force, and possibly gravitation, all have something in common: They can be described in terms of relativistic quantum fields, and are clearly the results of interactions between two different kinds of fields. For example, with an electron interacting via the electromagnetic force, one can describe the electromagnetic field using quantum electrodynamics, and an equation of motion can be obtained from the QED lagrangian: $$\mathcal{L}_{QED} = \bar{\psi} (i \gamma^{\mu} D_{\mu} - m)\psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}$$ And, the interaction picture here is clear: The electron field ($\psi$) interacts with the electromagnetic field, resulting in its change in motion. However, with an entropic force (such as the elasticity of a polymer), can the same be done? Can one construct a Lagrangian for the elastic force of a polymer, and is there some sort of "entropy force field" that can act as the relativistic quantum field with which polymer molecules interact? Or is the entropic force not a force at all, just a consequence of the universe tending towards maximum entropy? Or, alternatively, is it possible that every entropic force that exists in this universe is actually a manifestation of one of the fundamental forces (electromagnetism, strong force, weak force, gravity), meaning that the elastic force in a polymer is really something like electromagnetic interaction?


title: Magnet alignment within solenoid

text: Imagine I have a solenoid connected to a power supply. Solenoid produces an electromagnetic field. Now I take a permanent magnet and place it inside the solenoid. How will the magnet align itself (assuming there is no gravity) inside the solenoid and will it rotate itself around the central axis "aligning" itself? From these two cases cases, which one is correct behaviour and where would the "north" pole point (up, down, left or right): One: ![enter image description here](http://i.stack.imgur.com/8rI1t.png) Two: ![enter image description here](http://i.stack.imgur.com/uVdWC.png) P.S. What if the magnet is shaped differently, for example like a horseshoe, will it change its behaviour?


title: How do universal battery chargers work

text: How does a universal battery charger, such as this one work, i.e. how does it know what voltage, current and polarity is appropriate for the battery inserted? Do batteries have some form of protocol to tell the charger these parameters?


title: Local symmetry and General Relativity

text: First I want to consider an example of 1D motion. Lagrange equation: $$ \frac{d}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0 $$ If we transform $ L \rightarrow L+a $ with $a$ a is constant, the equation of motion remains unchanged. This is global symmetry. To obtain local symmetry we want when transforming $L \rightarrow L+a(x) $ we still have the same equation. To obtain that we introduce the "total derivative": $$\frac{Df}{dt} = \frac{df}{dt} + \frac{\partial a}{\partial x}$$ Then the equantion of motion would be unchanged under any local transformation: $$\frac{D}{dt} \frac{\partial L}{\partial \dot x} - \frac{\partial L}{\partial x} = 0$$ $\frac{\partial a}{\partial x}$ is considered related to the geometry of 1D space. I see $\frac{\partial a}{\partial x}$ is similar to the Christoffel symbols in general relativity. Is there anyway to construct General Relativity by demanding local symmetry like this?


title: Quantum Fourier Transform and Entropy

text: QFT is a nonlocal unitary transformation and so can generate entanglement in a system. It means a separable pure state can be converted into an entangled pure state. Now since the presence of entanglement can be witnessed via an increase in the entropy of the subsystems. Since all the subsystems witness a positive entropy change ,does the entropy of the complete system also increase (it seems to increase since entropy is additive) ? Now if it does increase , It seems to violate reversible nature of Quantum algorithms. I am very confused.


title: How to calculate the electrostatic force between two dielectric rings of charge?

text: Let's say there is a dielectric ring of charge with radius $a$ and charge $q_1$. I could calculate the electrostatic force applied on a point charge with charge $q_2$ and which is collinear with the centre of the ring with these equations(the distance between ring and point charge is $r$ and the distance between the centre of the ring and point charge is $d$): $$ dE= (k.dq)/r^2 $$ $$ dE_x=(k.dq)/r^2\cos(\theta) $$ $$ dE_x=(k.d.dq)/(a^2+d^2)^{(3⁄2)} $$ $$ E_x=∫(k.d.dq)/(a^2+d^2)^{(3⁄2)} =(k.d)/(a^2+d^2)^{(3⁄2)}∫dq=kdq/(a^2+d^2)^{(3⁄2)} $$ $$ F=kdq_1q_2/(a^2+d^2)^{(3⁄2)} $$ Here is the figure of the ring: ![enter image description here](http://i.stack.imgur.com/0Ha0o.png) However, I couldn't find a proper answer to how to calculate the electrostatic force between two dielectric rings of charge with radius $a_1,a_2$ charge $q_1,q_2$ respectively. I would be grateful if someone could help me on this subject. Also please show me which formula did you use rather than just a solution. Feel free to ask me anything about this question.


title: related to projectile motion

text: I was given this worksheet from my teacher and well im finding it really tough so far .. the question below is just crushing my skull .. so i would appreciate it if you guys would guide me through it > A particle projected from ground takes time $t_1$ to reach $\frac{15}{16} H$ > and time $t_2$ to cover the rest of distance to the ground. If $H$ is the > maximum height attained. find the ratio $\frac{t_1}{t_2}$ well my options were many ... they were $\frac{1}{3}$, $\frac{3}{1}$, $\frac{5}{3}$ and $\frac{3}{5}$ and well using my knowledge of atleast what i know is:   1. the object has **almost** reached the max height so the ratio wont be that big    2. i tried using the equations of motions and to no avail i still didnt get anywhere   3. $u \sin(\theta) t_1-\frac{1}{2}g (t_1)^2= \frac{15}{16} H$    4. and $\frac{\left(u \sin\theta\right)^2} {2g} =H$ keeping in mind that $t_2$ has to obviously be greater than $t_1$ ... the second and the third options are eliminated ... well it surely cant be $\frac{1}{3}$(maybe) .. because that is a really big ratio ... is there some other extra formula or something other than the common formulae (like max height and range and splitting of vectors)i am supposed to know to solve this question am i missing out anything?


title: If light rays obey to the wave equation, why can they be thought as straight lines?

text: I'm a newbie with physics but I'm wondering how a ray of light can essentially be represented. I have always known that a ray of light proceeds in a straight line until it encounters another object (or material) that refracts/reflects it. But a light ray should be part of an electro-magnetic wave, is this correct? If it is so, it should obey to the wave equation and this doesn't seem to me to describe a straight line ray. I'm having problems visualizing how light is emitted and how it relates with the wave equation. Can someone with a clear understanding of the problem explain it to me in simple terms?


title: Quantum description of Raman effect

text: In the classical description of Raman effect the object of study is the electric polarizability of the system. Since I'm interested in learning the quantum description of the Raman effect and in Bernath's "Spectra of atoms and molecules" is said that we are interested in studying the transition dipole moment like: \begin{equation} M_{10}=\langle \Psi_1\mid\mu\mid\Psi_0\rangle \end{equation} Where $\mid\Psi_i\rangle$ are the $i-$level state of the system. How this two description are related?


title: Is it time or duration?

text: Taking this post: "Is there a proof of existence of time?", as a starting point. Therein was mentioned that there is confusion between: " _time_ " and " _flow of time_ ". There was a comment (of mine) that the confusion is not between _time_ and _flow of time_ (which are equivalent), but between _time_ and _duration_ of which one is a dimension (i.e _duration_ ). Given the importance of the problem of _time_ in General Relativity and Quantum Gravity. Having made this disctinction is an important step, since duration can easily be considered as a dimension (with the proper $c$ factor) along with other space dimensions, than actual _time_ (or _flow of time_ ). > Can we say that _time_ parameter/dimension in SR/GR actually represents not > _event time_ but _duration_ (i.e time-interval)? _By the way this would elucidate the wick-rotation method, as transforming from "duration" to "frequency" representation._ (Not to mention that one can have as many duration dimensions in a manifold as one wants with **no** conceptual or definition problems like when one attempts that with extra _time_ dimensions, per some theoretical proposals)


title: Error limits when we have 4 lengths adding up and then finding its mean?

text: I have the following readings for length of a wire: 10.2 ± 0.1 cm 10.3 ± 0.1 cm 10.1 ± 0.1 cm 10.2 ± 0.1 cm Now, when I find out the mean value, I get: (10.2 ± 0.1 + 10.3 ± 0.1 + 10.1 ± 0.1 + 10.2 ± 0.1) cm / 4 So, will I get the answer as (10.2 ± 0.1) cm or (10.2 ± 0.4) cm? Why?


title: How do you measure numerically the central charge of a system?

text: Let's say that you are doing some Monte-Carlo simulations of a statistical system on a lattice and you observe scale invariance, meaning that you are at a conformal point. Can you get a numerical appreciation of the central charge? I know how the central charge is related on the free energy (on a cylinder for example) or to the stress-energy tensor but these are not direct observable in a Monte-Carlo. Is there a systematic method for that? Has it already been done?


title: Why and how maximum force is $\frac{dF}{dx}=0$

text: In an certain question my teacher asked to find the maximum force. She said that the maximum force in electrostatics means $\frac{dF}{dx}=0$. Why is it like that?


title: Difference between the increase in optical path due to refraction and that due to reflection

text: If we place two glass plates of refractive index n and each having thickness t,on the way of a light ray the increase in optical path becomes ![enter image description here](http://i.stack.imgur.com/7p0h5.jpg) (S2P-S1P)=2(n-1)t due to refraction through them,and the path difference(or extra path traversed by light) due to reflection at the second surface of one glass plate is ![enter image description here](http://i.stack.imgur.com/NNxzf.gif) 2ntcos(alpha).Without seeing the expression if we look at the phenomenon directly,aren't they basically same? as both are going through a glass media of same length!!


title: Is spacetime discrete or continuous?

text: Is the spacetime continuous or discrete? Or better, is the 4-dimensional spacetime of general-relativity discrete or continuous? What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous? Are there experimental evidences of continuity/discreteness? When particles move inside space do they occupy spacetime by little chunks? What would imply if spacetime is discrete on continuous theories? I've found little information on the web and books. Probably my question is ill-posed and I apologize for this.


title: Quantized spacetime structure?

text: Is quantized spacetime more like a chessboard or a matrix of lights? I have heard two analogies, the light bulb one appears to allow for movement whilst the chess piece one seems to violate zenos paradox. Whilst is a better example? http://plato.stanford.edu/entries/paradox-zeno/#Sta


title: The heat equation in relation to smoothing algorithm

text: Recently I learned about a technique in image processing, which has its roots in something called the 'heat equation' from physics. The original creators of this technique were inspired by the physics of how heat diffuses through an object. The objective of course is to 'smooth out' the image, for general noise removal. This was done by taking a Gaussian kernel, and convolving it with the image. However, the professor says that the heat-equation is actually a generalization of this process, and in fact, we can get much better techniques using the heat-equation framework. Essentially, if the original image we have is $I$, then the heat equation framework says that: $$ I(t) = \nabla \cdot (\ D(x,y) \ \nabla I) $$ where the $\nabla$ means spatial derivative, (I think). The $I(t)$ indicates the image new image at some point in time as it evolves - as heat flows - as this algorithm is run. Finally, the $D(x,y)$ is the "diffusion co-efficient", and if $D=1$, (or any constant), then the above simply collapses to a Gaussian kernel convolving the image $I$. Now, what I am hoping for is the following: I am hoping someone here can add some intuitive insight into how/why this is working, vis-a-vis a physical analogy to 'heat flow' in the image. The way I currently understand it, is that we have an image. The bigger the amplitude of certain pixels, the 'hotter' those pixels are. In fact every image pixel is as hot as its amplitude. Now, we also know from physics and entropy, that the heat will try to dissipate, so that eventually, the "temperature" across the image becomes equal. (This I take it, is what is happening when a gaussian convolves the image - this is the 'smearing' we are seeking for removal of noise...). Now, with the diffusion co-efficient being a constant or 1, this 'heat flow' occurs everywhere. However, if the diffusion co-efficient is, say, a binary function of the spatial image, the the co-ordinates of where $D(x,y)$ equal 0, are where no heat can flow through, and so those are pixels that are spared from the heat flow... Is my understanding of this physically inspired algorithmic technique correct? Are we really doing nothing but 'simulating' heat flow, except in the frame work of an image? Thank you so much.


title: How "earthing" electricity work?

text: I made a simple bulb-battery circuit and then I cut one of the wires and attached both ends to cemented floor, the bulb didn't glow, this means **`cemented floor`** is a poor conductor of electricity. Then how does **earthing** work ? This idea of activity came from when I got a shock being barefoot but got no shock from same source with slippers on.   So, `How can electrons pass through insulator like cemented floor during earthing ?`


title: Solving circuits using kirchhoff's laws and elimination method ( not any matrix method)

text: I need to solve this problem as part of my review in college physics. The task is to find the value of 5 branch currents using Kirchhoff's laws and elimination method (or maybe called elimination by substitution BUT not using any matrix method). The circuit is: ![enter image description here](http://i.stack.imgur.com/DNVkI.png) I got the KCL equations: a: $I_1 - I_2 - I_3 = 0$ b: $I_3- I_4- I_5 = 0$ and the KVL equations: $\text{Loop 1}: 15 - 4I_1 - 6I_2 = 0$ $\text{Loop 2}: 6I_2 - I_3- 3I_4 = 0$ $\text{Loop 3}: 3I_4 - 2I_5 - 10 =0$ My problem is I can't get the correct answer by elimination method. Please help me to provide the solution if the given answers are: > $I_1 = 1.89$   >  $I_2 = 1.24$   >  $I_3 = 0.65$   >  $I_4 = 2.26$   >  $I_5 = -1.61$ Please show the solution and give tips on how to solve this type of problems using elimination method so other students can benefit and learn here.


title: Decay rate of overpressure for a nuclear blast (shockwave) over an urban area?

text: I have come across a very new idea for me concerning nuclear weapons effects, check this please. In brief, that person states that blast overpressure attenuates/decays by causing damage for each individual building it crosses in a radial line, which makes some sense. But, the numbers this guy gave were really shocking, for example, he stated (calculated from data given by W.Penney) that 1% of the blast energy is lost for each wood-frame house the blast crosses, and about 5% for each concrete or brick building. If that is true, then by correcting the blast overpressure rings driven from the usual scaling laws we find that for example : for a 1 MT, the ideal 20 psi ground range is about 2.6 km, then by applying the decay/attenuation rate over say 50 concrete buildings in a modern city (just a rough estimate for buildings in 2.6 km radial line), we get 0.95^50 = 7.5% of the value of the overpressure expected for an ideal surface at 2.6 km, which is about 1.5 psi, so that makes nuclear weapons much much weaker than we ever imagined. So now my back to my question. Is there a way to calculate how the buildings will affect the attenuation of the blast wave in a common urban area ? Also, do the numbers of 1% and 5% makes sense for you ? I am sorry if I can't express what I want in a better way, English is not my first language so I did my best and I hope that was enough to explain what I want to know.


title: Ultra-Relativistic Fermi Gas - Chemical Potential and Energy

text: I tried deriving the expression for chemical potential of a Relativistic Fermi Gas using asymptotic expansion (for large z) in : $$ N = \frac{V}{h^3}4\pi (KT)^3g_s \int \frac{p^2 dp}{e^{p-\nu}+1} $$ where $ z = e^\nu $ and $ \nu =\frac{\mu}{KT} $ After evaluating the integral by asymptotic expansion, I obtained the value (upto first order in T) : $$ \mu(T) = \mu(0)\bigg[1-\frac{2\pi^2}{3} \frac{KT}{\mu} \bigg] $$ I am also able to obtain the energy for the system using this and expression for energy to be : $$ E(T) = \frac{3}{4}N\mu(0)\bigg[1-\bigg(1-\frac{2\pi^2}{3}\bigg) \frac{4KT}{\mu} \bigg]$$ I am unable to check the correctness of these expressions. Although energy gives the right answer for the one at T=0K


title: The 6-j symbol and intersecting Wilson loops, redux

text: This is a quite specific question continuing the problems I have with computing the expectation value of intersecting Wilson loops I laid out here. Using the tools from the answer there, I quite quickly arrive at the following expression for the local factor associated to a vertex, where two Wilson loops with reps $\alpha_1$ and $\alpha_2$ meet, and where the four surrounding regions have reps $\beta_1$ to $\beta_4$: $$ G(\alpha_1,\alpha_2,\beta_{1,2,3,4})_{\mu\nu}^{\sigma\rho} := \epsilon_\mu^{ijk}(\alpha_1,\beta_1,\beta_4)\epsilon_\nu^{lmn}(\alpha_2,\beta_1,\beta_2){\epsilon^*}^\sigma_{ijk}(\alpha_1,\beta_2,\beta_3){\epsilon^*}^\rho_{lmn}(\alpha_2,\beta_3,\beta_4)$$ The Greek indices are the indices incurred from decomposing tensor products as $a^i \otimes b^j \otimes b^k = \epsilon_\mu^{ijk}e^\mu$, leading to integral results like $$\int \alpha_i(V_b)^i_{i'} \beta_c(V_b)^j_{j'} \beta_{c'}(V_b)^k_{k'} \mathrm{d} V_b = {\epsilon^*}^\mu_{i'j'k'}\epsilon^{ijk}_\mu$$ (see previous answer). Since the $\epsilon^*$ that has the $\mu$ this is summed with lives on the opposite end of the (part of) the Wilson line, the Greek indices must necessarily remain open at the vertices. I am totally fine with this being the result of the computation, but I am still puzzled why the relation to the 6j symbol is so casually tossed about. Let me first remark that the above equation is already suspiciously similar to the very first equation in the definition of $6j$ symbols, but the free indices are irritating me. If $\epsilon_\mu^{ijk}(\alpha_l,\beta_m,\beta_n)$ is the $3jm$ symbol (with $i,j,k$ playing the role of the $m$ and the reps corresponding to the $j$), what is the additional index $\mu$ doing here? If it is not the $3jm$ symbol (which I am currently thinking), then why would the $G$ defined above be the $6j$ symbol (and why has it free indices)? (If these are neither $3jm$ nor $6j$ symbols, then why do Witten, Ramgoolam, Moore, etc. insist they are?) Note that the $6j$ symbol cannot arise after summing the Greek indices, since the $G$s the second index belongs to are, in general, at other vertices, and so have not exactly the same 6 reps as arguments. Furthermore, the $3jm$ symbols are, if I understand them correctly, essentially the Clebsch-Gordan coefficients for expanding a tensor product of _two_ irreducible reps in a third, and the $\epsilon$ above expand the tensor product of _three_ irreducible reps in all possible fourths (which are then summed over in form of the Greek indices). Something does not add up here, and I heavily suspect it is only in my understanding of the symbols, so I would really appreciate someone clearing up my confusion. **EDIT** : Ok, I think I have found something, but I am still a far shot from solving this riddle, and it requires to think more carefully about the coefficients $\epsilon^{ijk}_\mu$: Let $\alpha,\beta,\gamma$ be reps with basis elements $a^i,b^j,c^k$as before. Then, we can decompose the tensor product stepwise instead of at once as: $$ a^i \otimes b^j \otimes c^k = \sum_{\rho \subset \alpha \otimes \beta} C(\alpha,\beta,\rho)^{ij}_\zeta e(\rho)^\zeta \otimes c^k = \sum_{\rho \subset \alpha \otimes \beta} \sum_{\sigma \subset \rho \otimes \gamma} C(\alpha,\beta,\rho)^{ij}_\zeta C(\rho,\gamma,\sigma)^{\zeta k}_\mu e(\sigma)^\mu$$ (I apologize for the abundance of symbols, but it really becomes clearer what's happening that way.) Here, the $C(j_1,j_2,j_3)$ are now manifestly Clebsch-Gordan coefficients for $j_1,j_2$ in $j_3$, and thus essentially $3jm$ symbols, and the notation $\rho \subset \alpha \otimes \beta$ means that the irreducible rep $\rho$ occurs as a subrep in $\alpha \otimes \beta$. Since Clebsch-Gordan coefficients for reps not appearing in a given tensor product are zero, we can drop the constraint on the sums and sum over all irreducible reps. Thus, $\epsilon^{ijk}_\mu = \sum_\rho C(\alpha,\beta,\rho)^{ij}_\zeta C(\rho,\gamma,\sigma)^{\zeta k}_\mu$. Now, in the integral result above, by the Peter-Weyl theorem (see also previous answer), the $\epsilon$ are only summed over the $\mu$ belonging to trivial subreps of $\alpha \otimes \beta \otimes \gamma$, i.e $\sigma = 0$, if we denote the trivial rep by $0$ in analogy to $j = 0$ in the spin case. Therefore, we have that the result of the integral is $$I := \int \alpha(g)^i_{i'}\beta(g)^j_{j'}\gamma(g)^k_{k'} = \left(\sum_\rho C(\alpha,\beta,\rho)^{ij}_\zeta C(\rho,\gamma,0)^{\zeta k}_\mu C^*(\alpha,\beta,\rho)_{i'j'}^\eta C^*(\rho,\gamma,0)_{\eta k'}^\mu\right) $$ But the trivial irreducible rep has only one dimension, so the sum over the $\mu$ is just the multiplicity $n(\rho,\gamma,0)$ of $0$ in $\rho \otimes \gamma = \bigoplus_\sigma n(\rho,\gamma,\sigma)\sigma$, i.e. $$ I = \sum_\rho n(\rho,\gamma,0)C(\alpha,\beta,\rho)^{ij}_\zeta C(\rho,\gamma,0)^{\zeta k} C^*(\alpha,\beta,\rho)_{i'j'}^\eta C^*(\rho,\gamma,0)_{\eta k'}$$ This gets rid of the annoying $\mu$, would lead to the $G$ from the beginning of the question to be comprised of the sum over a product of 8 $3jm$ symbols (the $C(\alpha,\beta,\gamma)$), of which 4 each are summed over their $m$ indices, yielding the product of two $6j$ symbols summed over one of their $j$s (the $\rho$). Also, the index structure in $e^{ijk}e^*_{ijk}$ in $G$ would translate to an index structure of the $C$ exactly matching that of the $3jm$ in a $6j$ symbol. But before I work that out, can anybody tell me if this is the right track or if I butchered something along the way (I am not comfortable enough with my skills yet to fully trust my reasoning when it leads me to an answer whose shape I already know)? Or should I perhaps take this to the mathematicians, since the answer seems to be purely group theoretic so far?


title: Null geodesics and rotation in stationary axisymmetric space-times

text: It is well known that in Schwarzschild space-time, a torque-free gyroscope in circular orbit at _any_ permissible angular velocity at the photon radius will, if initially tangent to the circle, remain tangent to the circle everywhere along its world-line. This is usually explained as follows. Say I'm described by an arbitrary worldline in an arbitrary space-time and I orient a gyroscope and a photon gun along some direction at an initial event; there is a swarm of mirrors surrounding me in my infinitesimal neighborhood and at each instant of my proper time, I shoot out a photon in the direction of the gyroscope axis at that instant and reorient the gyroscope axis in the direction of the reflected photon when it arrives back to me, with the direction being relative to my worldline of course. In doing this, the gyroscope axis gets Fermi-Walker transported along my worldline; see section 2 of https://www.zarm.uni- bremen.de/uploads/tx_sibibtex/2001LaemmerzahlNeugebauer.pdf and problem 5 on p.161 of Geroch's general relativity notes http://home.uchicago.edu/~geroch/Links_to_Notes.html and p.164 of said notes for the solution. Coming back to the photon radius in Schwarzschild space-time, it is clear from the above prescription that Fermi-transport of the gyroscope implies the gyroscope always points tangent to the circle because a photon emitted from my photon gun will travel along this circle, and so will the reflected photon, irrespective of my angular velocity. I have two questions with regards to the above: (1) I don't quite understand what constraints there are on the worldline of the mirror in the above operational procedure, which is basically Pirani's "bouncing photon" method. Certainly it cannot follow an arbitrary worldline in my infinitesimal neighborhood, so what kind of restrictions are there on said mirror's worldline? Consider for example an observer and a mirror at rest on a rigidly rotating disk in flat space-time that are separated by an infinitesimal radial amount. Then there exist both future and past directed radial null geodesics given by $\frac{dr}{dt} = \pm (1 - \omega^2 r^2)^{1/2}$ from my worldline to that of the mirror's so a photon emitted by myself towards the mirror will come back to me in the same radial direction. Will not then the gyroscope remain fixed in the radial direction? But this does not constitute Fermi-Walker transport as the polar axes of the local frame fixed to the disk rotate relative to the momentarily comoving local inertial frame due to Thomas precession. So where in the above is my incorrect application of Pirani's "bouncing photon" method? (2) Let $(M,g_{\mu\nu})$ be a static axisymmetric space-time having a time- like and hypersurface orthogonal Killing field $\xi^{\mu}$ and an axial Killing field $\psi^{\mu}$; it is easy to see that $M$ being static is equivalent to $\psi_{\mu}\xi^{\mu} = 0$. Now consider a congruence of circular orbits in the rest space of $\xi^{\mu}$ given covariantly by the time-like Killing field $\eta^{\mu} = \xi^{\mu} + \omega \psi^{\mu}$ where $\nabla_{\mu}\omega = 0$ with $\omega$ being the angular velocity relative to spatial infinity. Then, given the above conditions, it can be shown that $\eta_{[\alpha}\nabla_{\beta}\eta_{\gamma]} = 0$ for _all_ permissible values of $\omega$ on a time-like integral 2-manifold of $\eta^{\mu}$ if and only if the null vector field $k^{\mu} = \xi^{\mu} + \Omega \psi^{\mu}$ is a geodesic on this 2-manifold, where $\Omega^2 = -\xi_{\mu}\xi^{\mu}/\psi_{\nu}\psi^{\nu}$. This is a generalization of the situation at the photon radius in Schwarzschild space-time due to $\eta_{[\alpha}\nabla_{\beta}\eta_{\gamma]} = 0$ being equivalent to Fermi- Walker transport. However intuitively I do not understand why the space-time must be static as opposed to just stationary. Consider a freely falling circular photon orbit in some stationary (but not static) axisymmetric space-time. If an observer were to be in circular orbit at this photon radius then why couldn't he just use Pirani's "bouncing photon" procedure from above to conclude that Fermi-Walker transport of a comoving gyroscope implies that its axis remains tangent to the circle? Intuitively speaking, under what circumstances would this fail in a stationary but non-static space-time, and why? Thanks in advance!


title: Does a rotating magnetic monopole have electric and magnetic moment in classical view?

text: Would a rotating sphere of magnetic monopole charge have electric moment ? In a duality transformation E->B.c etc. how is the magnetic moment translated m = I.S -> ? Mel = d/dt(-Qmag/c).S ? A more general question would a sphere charged with monopoles have other moments that the monopole in the multipole expansion ? Like electron is predicted to have electric dipole moment also.


title: How does electricity propagate in a conductor?

text: On a systems level, I understand that as electrons are pushed into a wire, there is a net field and a net electron velocity. And I've read that the net electron drift is slow. But electricity travels through the wire, essentially at c, and I want to understand that mechanism. My apologies if my question is poorly stated, I know bare bones undergraduate quantum theory circa 1990s but it doesn't explain the motion of electricity in detail. Here's my conception, I'm hoping someone will fill in the holes, ha ha. An electron moves into the wire. It's got a kinetic energy. After travelling a short distance, it spontaneously emits a photon, which hits another electron in a valence shell. That electron then presumably does the same. If this conception is simply wrong, please enlighten me. The questions that arise:   1. Presumably at this level, electrons are acting more like waves and less like particles, but is there any classical component in the picture, ie are electrons coming in imparting other electrons with kinetic energy through repulsion, or does it not work that way?   2. If electrons momentarily have energy, then pass it on by a photon, what determines when that photon is emitted, and what frequency it will be? I assume that electrons in this cloud are not limited by any kind of exclusion principal, and that any frequencies are possible?   3. Why should a photon emitted by an electron be in the direction of travel? Conservation of momentum tells me that if an electron is moving, the photon should be emitted in that direction, slowing the electron, but could an electron emit a photon in the opposite direction? If it did, I assume it would somehow have had to absorb energy from elsewhere? That sounds possible by analogy with quantum tunneling.   4. What is the mechanism by which electrons propagating increase the temperature of the material? Are they transmitting energy to the electrons in the valence shell, which tug at the nucleus, do some photons hit the nuclei directly, or is there some other way?    5. Presumably, electricity travels slower than light, because there is some time in each exchange, and some time when electrons are moving at sublight speeds before emitting a photon. By how much is this slower than light, and what is the speed of each interaction?


title: Conduction and propagation

text: What is the difference between conduction of electric wave in conductor and propagation of electromagnetic wave in dielectric? Why propagation term is used for dielectric and conduction for conductor?. Somehow why propagation of electromagnetic wave (is it energy wave) is not possible in conductor, but in dielectrics, and conduction (power signal) not possible in dielectric.


title: what exactly do electric waves transfer from one point of the wire to other..?

text: > **Possible Duplicate:**   >  how does electricity propagate in a conductor? so basically i was considering the speed of the charges inside wire defined by the drift velocity and speed of electric waves that's equal to speed of light..so my point is if electric waves don't carry any charge (as mostly light that we encounter from sun's radiation is em and since they don't carry any charge we don't get shock)so what do they actually carry..i was trying to figure out its resemblance to mechanical waves ..as mechanical wave don't itself carry any matter with it they only transfer energy..then so do the electric waves they can't carry charge but they must transfer energy in the form of you know electric waves..or could i say now that these electric waves are source of electric field..then thing that's confusing me is that if electric field propagates at speed of light(speed of electric waves)..then do the voltage that we define as eL or voltage or potential at a point is=electric field intensity * L(distance) then should the voltage drop or potential also varies accordingly or i could say that the potential also following the electric field intensity at the speed of light..but this is contradictory to our general observation in which we define particular time period and phasors which define variation of voltage that..so not at the speed of light..


1. Why non integer spins obey Fermi statistics?   2. Why is fractional statistics and non-Abelian common for fractional charges?


title: Just a contraction of indices

text: I came across a contraction which is not giving the desired result. This is a toy problem in how to get a supergravity theory in low energy limit of a superstring theory using the vanishing of beta function in the $\sigma$ model, however there is no use of this context. What I want to ask is contracting d and m in the following expression where T is torsion two-form and R is the curvature two-form. The expression is $T^{[c}_{[pq}\delta^{d]}_{m]} +2a\partial_{[p}R^{cd}_{{qm}]}=0$ There is one extra condition $R^{mc}_{mq}=0$. It is clear that second term would look like $\frac{2a}{3}\partial_m R^{mc}_{pq}$, apart from some sign probably. The first term however after writting out explicitly gives $\frac{1}{6}T^c_{pq}$ ( one can eliminate terms like $T^c_{cm}$ which can shown to be cancel ), which gives something like $T^c_{pq}+4a\partial_m R^{mc}_{pq}$=0 ( May be there is a minus sign before $4a\partial_m R^{mc}_{pq}$ but that is not the problem). The problem however is that the result should be following one $T^c_{pq}+\frac{4a}{9}\partial_m R^{mc}_{pq}$=0 and I don't know how he is getting the factor of 9 there. For reference one can look at Castellani ' Supergravity and superstrings: a geometric perspective vol. 3' around page 1850.


title: Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?

text: Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\rangle$ and the first excited state $|1\rangle$. $\langle 0|\hat p|0 \rangle$ is zero right? Since the particle can either be travelling to the left or right, where $\hat p$ is the momentum operator. Similarly, I think $\langle 1|\hat p|1 \rangle = 0$ But, $\langle 0 | \hat p | 1 \rangle$ is non-zero, right? Since they are different states. Also, since $\hat p$ is Hermitian, $\langle 0 | \hat p | 1 \rangle = \langle 1 | \hat p | 0 \rangle $, right?


title: Simplifying a seemingly simple gamma matrix identity

text: When studying from the book by Wise and Manohar, _Heavy Quark Physics_ (pg 102), I came across a seemingly simple identity that I am not able to prove. It's likely an easy problem but I can't for the life of me figure out why. First they define the transverse covariant derivative, $$D_\perp \equiv D^\mu - D \cdot v v ^\mu$$ where $v^\mu$ is some four-vector that squares to 1 and then they write down the identity, $$ \require{cancel} \cancel{D_{\perp}}\cancel{D_{\perp}} = D_\perp^2 +\frac{1}{2}[\gamma_\mu,\gamma_\nu] D_\perp^\mu D_\perp^\nu$$ I would have naively thought that we would still have the usual, $D_\perp^2= \cancel{D_{\perp}}\cancel{D_{\perp}} $. Why is this not the case?


title: Surface Tension - Lung Alveoli

text: So, the way I understand this is as follows : The alveoli (pretend they're bubbles) have diameters of the order of microns implying a massive pressure required to inflate them by the Young-Laplace equation. $p_{in}-p_{out}=\frac{2\gamma}{r}$ However, the presence of pulmonary surfactant molecules (lets just pretend they're like detergents molecules in washing liquid) can effectively reduce the surface tension at the unexpanded alveoli and hence allow easy inflation. Now this bit I don't understand : As the alveoli expand the **distance between the individual surfactant molecules on the alveoli increases and hence the surface tension rises again** therefore decreasing the rate of expansion. What is the mathematical connection between surface tension and separation between surfactant molecules ? How can I rationalise the statement in bold ?


title: Electric field between electrodes of different shapes

text: How to calculate electric field between two electrodes - square and circular?


title: Are there any substances that allow sound to travel better then air?

text: The question is the same as the one I put in the title but here it is in more words. I was wondering if there were any substances that allowed sound to travel more efficiently. For example water is obviously a much worse way to transport sound then air.


title: Why are the bounds to the permittivity $\epsilon$ a circular arc in the complex plane?

text: I'm reading this paper which is essentially about connecting the complex permittivity $\epsilon$ with the microstructure of a thin film. They talk about how you can place limits on the possible values of $\epsilon$ depending on how much information you have on the microstructure, and say: > We use the fact that all bounds are circular arcs in the complex $\epsilon$ > plane to give... Why is this the case? My intuition tells me it has something to do with the Kramers-Kronig relations or energy conservation, but I don't know why.


title: Temperature on the surface of the sun calculated with the Stefan-Boltzmann-rule

text: In a German Wikipedia page, the following calculation for the temperature on the surface of the Sun is made: $\sigma=5.67*10^{-8}\frac{W}{m^2K^4}$ (Stefan-Boltzmann-constant) $S = 1367\frac{W}{m^2}$ (solar constant) $D = 1.496*10^{11} m$ average distance earth-sun $R = 6.963*10^8 m$ (radius of ths sun) $T = (\frac{P}{\sigma A})^\frac{1}{4} = (\frac{S4 \pi D^2}{\sigma 4\pi R^2})^\frac{1}{4}=(\frac{SD^2}{\sigma R^2})^\frac{1}{4} = 5775,8$ (Wikipedia gives 5777K because the radius was rounded to $6.96*10^8m$) This calculation is perfectly clear. But in Gerthsen Kneser Vogel there is an exercise where Sherlock Holmes estimated the temperature of the sun only knowing the root of the fraction of D and R. Lets say, he estimated this fraction to 225, so the square root is about 15, how does he come to 6000 K ? The value $(\frac{S}{\sigma})^\frac{1}{4}$ has about the value 400. It cannot be the approximate average temperature on earth, which is about 300K. What do I miss ?


title: Justification of ignoring large set of entanglements

text: If we can think about the universe as a wave function then many particles should be entangled with many other particles in the universe. The obvious question arises why we don't see those entanglements in everyday circumstances. One standard explanation given is those entanglements average out and cancel so we can ignore those. However, hardly any mathematical justification is given for them to cancel. My question is how much trust one should have on that particular assertion? Is there any mathematical arguments already put forward by anyone?


title: Terminal velocity of a body in a vacuum

text: What velocity would a small body moving towards a large body reach under gravity alone? More importantly, is there a 'terminal velocity' for the small body due to the increasing energy required to accelerate the body as its velocity increases? Assumptions: No other forces, gravity acts over infinite distances. Parameters: Large mass=$10^{24}$kg Small mass=$100$kg, Distance between them=$10^{12}$m.


title: Is holomorphy the real reason for nonrenormalization in supersymmetry?

text: Seiberg traced the nonrenormalization of supersymmetric theories to holomorphy of the superpotential in chiral superspace. However, this overlooks the fact that with a different number of supersymmetric generators, supersymmetry can be real or symplectic, instead of complex. But yet, even for those cases, we still have nonrenormalization. If holomorphy isn't the reason, what is? If the reason for nonrenormalizability is totally different for different dimensions and number of supersymmetry generators and choice of superfields, is it then _ad hoc_ that nonrenormalizability always holds in each case?


title: Why are some of the biggest stars known blue?

text: My question refers to an overview of the biggest stars we know: http://farm5.static.flickr.com/4138/4820647230_faba1c9f3b_o.jpg Why are some of those blue?


What does it mean that the Higgs has a nonzero vacuum expectation value? Are there any other important field with nonzero VEV? How is this related to the 100-orders magnitude wrong prediction?


title: How to place a mirror parallel to a wall?

text: For one of my experimental setup I need to place a mirror perfectly parallel to a wall. It can be placed at any distance from the wall. I would like to use any method other than direct measurement. I am free to use the following: a webcam a secondry mirror Edit: It's not necessary to use all or either of them.


title: Hail stones in summer?

text: This has always puzzled me. Yesterday (in London) it started hailing despite it being about $20^oC$. A couple of years ago I experienced hail in Sicily when it was about $35^oC$ in the shade!. How is this possible?


title: Question on energy mass conversion

text: I have a question regarding the energy-mass conversion. Well, when a particle starts moving with a speed comparable to that of light, its (relativistic) mass increases that means some matter is created and that too of the same particle...energy being converted to mass is ok but how does energy perceive what atoms it has to form? Say I take a stone to a high speed, then constituents of stone is formed. And if I perform same thing with another substance, its constituents are formed..How? Energy can be converted to mass but a mass of what? Does that mean we can create matter of any desirable substance?


title: Continuity domain for momentum operator

text: I know this is essentially a mathematic question, but I received no answer on math SE. Moreover it has a direct application in physics, so I thought to ask this here too. The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. It isn't continuous in fact if I consider the sequence $$ g_n(x)=\frac{e^{inx}}{\sqrt{2\pi n}} $$ it's a Cauchy sequence but $\\{Pg_n\\}_n$ does not converge. I am searching a domain where $P$ is a continuous functional. My professor gave me the example $$D_P=\\{\varphi \in L_2(0,2\pi):\varphi(0)=\varphi(2\pi) \\}$$ but I'm not convinced because if I consider the function $\psi$ \begin{equation} \begin{cases} -\frac{x}{\sqrt{\pi}}+\sqrt{\pi},\hspace{0.5cm} 0\leq x<\pi\\\\\sqrt{x-\pi},\hspace{0.5cm} \pi\leq x \leq 2\pi \end{cases} \end{equation} it belongs to $D_P$, but if I apply $P$ on it I obtain \begin{equation} \begin{cases} \frac{i}{\sqrt{\pi}},\hspace{0.5cm} 0\leq x<\pi\\\ -\frac{i}{2\sqrt{x-\pi}},\hspace{0.5cm} \pi\leq x \leq 2\pi \end{cases} \end{equation} that is not square-integrable on $(0,2\pi)$. ![](http://i.stack.imgur.com/Mv1d7.png) Am I wrong? If not, which is a correct continuity domain for $P$?


title: How do I find the unitary operator for a combined system?

text: I want to describe the interaction between a target and a probe and know that this is described by some unitary operator that acts on both systems. My operator needs to implement a $\pi/q$ rotation in the equatorial plane. I need to perform a Von Neumann measurement on the probe which is described by a projection onto one of the probe states. In my case this is the spin either up or down at and angle $\phi$ to the x-axis. I have a formula for the unitary operator that acts in the tensor product space which is $$U=\sum{u_{nk,n\prime k\prime}|n>|S_{k}><n\prime|<S_{k\prime}|}$$ where {|n>} are the probe states, $|S_{k}>$ are a set of basis states for the target, and $u_{nk,n\prime k\prime}$ are the matrix elements of U Could somebody give an example explicitly of finding such a unitary matrix for a combined system or some hints for my problem.


title: If light was able to pass through a wall, would the wall be invisible to the human eye?

text: In addition, to get light to other side of the wall, could it be converted to radio waves and then back to light waves? Edit: My idea was if there was a special material that was painted on both sides of a wall that converted radio waves to light waves and light waves to radio waves would the wall appear invisible? (My thought being that the electromagnetic radiation would travel through the wall since radio waves can travel through walls) Also, thanks for all the responses!


title: Is it possible to determine the state or values of something without measuring it

text: To give context to this question, I am currently looking into non-locality / hidden variables / Bell's Theorem, EPR / etc. I've noticed the assertion that the values / state of something when measured aren't what the values / state would be if the measurement were not made. How is this known? Or is it an assumption? I have no science background so please explain in layman's terms, if possible.


title: How important is mathematical proof in physics?

text: How important are proofs in physics? If something is mathematically proven to follow from something we know is true, does it still require experimental verification? Are there examples of things that have been mathematically proven to some reasonable degree of rigor (eg satisfy a mathematician) that turned out to be false based on experiment?


title: Perspective and changes in kinetic energy

text: Say you have two planes flying next to each other at the same speed and one decides to pick up speed by burning a tank of rocket fuel. If someone on the ground wanted to know that plane's new speed after burning the fuel and knew the plane's mass, initial speed, and energy stored in the fuel, he could calculate it using conservation of energy. If the pilot of the other plane wanted to calculate the same thing but didn't know his own speed, he could calculate the speed of the first plane relative to himself using conservation of energy--the initial relative speed is 0 and all the work done by the rocket fuel is converted into the plane's new kinetic energy. He could then relay this relative speed to someone on the ground who knew his plane's speed, but adding the speed of the second plane to the calculated speed of the first relative to the second would give a different value for the first plane's speed than the first guy on the ground would have calculated. I see why this doesn't work mathematically (a^2+b^2 isn't (a+b)^2), but why can't the second pilot calculate the new speed of the first plane relative to himself by converting the stored fuel energy into kinetic energy? Why when on the ground do we not consider our motion relative to, say, the moon when calculating the first plane's speed relative to us, but the second pilot must consider his speed relative to the ground before calculating the first plane's speed relative to himself? Or maybe to put it better, why is having no movement relative to the Earth instead of relative to the plane's initial speed (the second plane) or anything else the correct way to approach this problem?