Anti-matter as matter going backwards in time? (requesting further clarification upon a previous post)

I understand this question has already been asked here, however, I don't have enough reputation points to place a comment (I suppose that's the reason) on a specific answer to request a reference. A user has answered (I'm not sure I can cite his name here, so I won't): > There is one technical inaccuracy in saying that antimatter moves back in > time (whatever it might mean). In quantum field theory we get positive > energy solutions (usual particles) and negative energy solutions. Negative > energy solutions behave in time as if they were propagating backward in > time. But they are not the antiparticles, they are just the "negative-energy > particles". Antiparticles are positive energy solutions, and they are > obtained by acting with charge conjugation operator on the negative-energy > solutions. So, antiparticles move forward in time, as usual particles. and, even though someone else comments > This is interpreting "going back in time" differently than anyone else does. I'd like a reference for (or further clarification on) that, since I have already heard that before (also informally). I understand this is a repost but if someone could at least give me some reference before closing it, I'd be thankful. PS: My first intention was to ask how to proceed on this case in "meta", but I don't have reputation points to place a question there. **EDIT:** To clarify further my question: I had originally the same question as the OP from the post I quoted. At first, I thought that an explanation of what are anti-particles and the interpretation as particles going back in time could be found in quantum mechanics alone, without more advanced quantum field theory. The argument I knew is that anti-particles are really related to the originally negative-energy solutions, that can be reinterpreted as positive- energy solutions, as long as one invert the sign of time in the unitary time- evolution operator. This would hence lead to the reinterpretation of such states as positive-energy anti-particles _going back in time_ or, at least, with _anomalous time dependency_. The particular answer I quoted, seems to contradict this interpretation, saying that it is a mistake. I'd like a further clarification here. Is this interpretation really wrong? From what is said on Wikipedia, looks like the _Feynman–Stueckelberg interpretation_ relies on the fact that anti-particles are to be understood as particles going backwards in time and then related to the latter by charge conjugation. Where does this interpretation primordially comes from? Where is it primordially needed if their association with the negative energy solutions of Dirac's equation (say) is a mistake? I understand that in quantum field theory one uses complex fields to describe both particles and anti-particles, relating them by charge conjugation. But isn't this done this way to accommodate Feynman–Stueckelberg's interpretation?


How much time has passed for Voyager I since it left the Earth, 34 years ago?

34 years have passed since Voyager I took off and it's just crossing the solar system, being approximately at 16.4 light-hours away. How much time have passed for itself, though?


Two masses with interacting forces and an external force

Two masses in 3d space attract each other with a potential relative to the distance between them. There is also an external force on each particle based on the distance from a origin. I want to find the equations of motion. I thought about reducing it to the 1 body problem or use Lagrangian formalism to find the equation of motion, however, the external force kind of ruins that approach. Any ideas on how to approach this?


Earth-Moon system

While learning about motion of centre of mass, I came across this statement online, > _It is also more accurate to say that the Earth and Moon together revolve > about their common center of mass._ I am not able to understand this. Can anyone please explain this statement?


clear picture about charge and flux

What is the difference between charge and electric flux?? According to my knowledge electric flux is nothing but charge enclosed by the closed surface.


Oscillation and wave

A box of mass $0.2 \,\text{kg}$ is attached to one end of a spring whose other end is fixed to a rigid support. When a mass of $0.8 \, \text{kg}$ is placed inside the box, the system performs four oscillations per second and the amplitude falls from $2 \, \text{cm}$ to $1\, \text{cm}$ in $30 \, \text{s}$ . Calculate the force constant, the relaxation time, and the Q-factor.


Do wide-angle videos make the first-person view seem slower than perceived in real life?

I considered posting this on other SE sites such as _Audio-Video Production_ and _Photography_ , but I didn't feel I'd get the definitive, fact-based (rather than experience-based) answer I'm seeking. **Background** I used a GoPro HD Hero (which has a 170-degree view) to record a track day (driving a car) and a snowboarding trip. Playing back the videos after each event, I sensed immediately that they seemed _slow_. However, since clocks and speech seemed to progress at a normal rate, I dismissed my senses, figuring adrenaline and wind must've made me perceive _real life_ faster than it actually was. And there's probably some truth to this adrenaline-and-wind explanation, but watching the videos again today (more than a year later), I sensed the slowness as I always did. I came up with a potential, partial explanation, but still lacking; I was hoping someone might be able to shed some light as to whether this "slow video" phenomena is merely psychological, or an expected effect of optics. **Partial Explanation** I think there _may_ be an effect, due to _depth perception_ being altered by wide-angle lenses. Suppose we were to record a skateboarder from the side, _i.e._ so the skateboarder would skate from left to right in our camera's field of view. In this case, because the axis of movement (the skateboarder's "line") would be perpendicular† to the axis of distortion (the camera's "line", or depth), there would be no distortion of perceived speed: the skateboarder moves relative to an undistorted frame of reference (the ground). However, if we were to mount the camera on the skateboarder's helmet, _i.e._ record in a first-person view, then the axes of movement and distortion would be one and the same. From here though, I'm not sure how to explain why this translates to slower-seeming playback. **Question** In layman's terms, my problem is, "The ground and the trees pass by me slower," when playing back a wide-angle video.   1. Is this psychological, or is there a physical explanation?   2. Is there a mathematical formula that can tell by how much one must speed up playback in order for "the ground and the trees" to pass by as they seemed to in real life? **Footnotes** † Technically, I guess "perpendicular" would be a curve, _i.e._ staying equidistant from the camera, but for simplicity, suppose the skateboarder is far enough from the lens that we may consider the curve a "line".


How does velocity cause time dilation?

I've been trying to learn about the speed of light and time dilation, but I'm at an impasse. The presented facts say that if I travel around the solar system at 50% the speed of light and then come back to earth I will have experienced less local passage of time than them. I will effectively have traveled to their future. I've also read that gravity causes time dilation too. I understand that space-time is a sort of unified thing and that affecting space affects time. It makes sense then that gravitational forces bending space will also bend time. But velocity? I can't wrap my head around it and I can't find a good explanation for it. People cite orbiting craft and planes as proof of time dilation, since their clocks will go slower than those in the surface (and vice-versa). This certainly explains gravitational time-dilation, but not necessarily velocity. Can't the change in local time passage be caused solely by the gravitational bending of space-time? Trying to find an answer, I came to a very recurring and frustrating example in texts that seek to explain time dilation. A man on a moving train throws a ball forward. Since he's moving with the train and the train is his point of reference, the ball to him only moves at the speed he threw it. But to a woman on the station the ball is moving at the speed of the train plus the speed it was thrown with. To some authors, this seems to open the mind to time-dilation understanding. To me it only explains the relative nature of motion. It says nothing of time. Another example I've found: if person A speeds away from person B very quickly, A's clock will seem to advance slower from B's point of view. How is this time dilation, though? The difference can be explained by the longer time it takes light to get to B, can't it? I realize I can't be right against the fine physicists out there, so I was hoping someone here could enlighten me. Where does the notion that velocity causes time-dilation come from?


Are many-worlds and the multiverse really the same thing? Not too long ago, Susskind and Bousso uploaded the article "The Multiverse Interpretation of Quantum Mechanics" with the thesis that the many-worlds interpretation and the multiverse of eternal inflation are one and the same thing. The parallel worlds of one are exactly the same thing as the parallel worlds of the other. First, they claim decoherence can't happen over a complete description of the future light-cone of the measurement. Then, they apply that principle to eternal inflation. Without decoherence, superpositions of nucleating bubbles and metastable vacua can't decohere. According to the anthropic principle, most bubbles have no conscious observers, but an exponentially small minority do. Apply black hole complementarity to causal horizons. Then, somehow, in a way I can't follow, they combine causal diamond worlds into a global multiverse. Then they claim decoherence is reversible. My head is spinning. What are your opinions on this paper?


Is the total energy of the universe constant?

If total energy is conserved just transformed and never newly created, is there a sum of all energies that is constant? Why is it probably not that easy?


Would a gauss rifle based on generated magnetic fields have any kickback?

In the case of currently developing Gauss rifles, in which a slug is pulled down a line of electromagnets, facilitated by a micro-controller to achieve great speed in managing the switching of the magnets, does the weapon firing produce any recoil? If so, how would you go about calculating that recoil?


Moment of Inertia

Let $f(x) = \frac{1}{L}$ be a probability function, where $L$ is constant. Find the mean and variance. Discuss your results by making a connection to the moment of inertia definition.


Why isn't absolute $0 K$ temperature possible?

So $T$ is defined as $$T = \left(\frac{\partial E}{\partial S}\right)$$ and $S$ is defined as $$S = k_B \ln \Omega$$ where $\Omega$ is the number of accessible states of the system for a given $E$. I suppose you could write it as $\Omega(E)$. Hence, for $T$ to be zero, it must be possible to prepare a system that has an equivalent number of possible $|\psi\rangle_E$ for different $E$. It seems to me like you could find some weird system for which this applies. Why can't this be true?


How Does Dark Matter Form Lumps?

As far as we know, the particles of dark matter can interact with each other only by gravitation. No electromagnetics, no weak force, no strong force. So, let's suppose a local slight concentration of dark matter comes about by chance motions and begins to gravitate. The particles would fall "inward" towards the center of the concentration. However, with no interaction to dissipate angular momentum, they would just orbit the center of the concentration and fly right back out to the vicinity of where they started resulting in no increase in density. Random motions would eventually wipe out the slight local concentration and we are left with a uniform distribution again. > How does dark matter form lumps?


BCS theory, Richardson model and Superconductivity

I'm studying Richardson Model in second quantization. There are many initial points that I don't understand:   * We supposed that an attractive force between 2 electrons exists, due to electron-phonon interaction: an electron generates a phonon that is absorbed by the other electron, and we prove that the resulting potential is attractive. But is this really stronger than the Coulomb repulsive potential? How can one estimate this?   * Why does it neglect the attractive force between electrons with opposite spin? ($\pm$) They have an opposite magnetic momentum too...   * Why do we only consider the attractions between electrons with opposite spin?   * Why do we suppose $k_f ~R ~\ll ~1$ , where $k_f$ is (Fermi momentum)/$\hbar$, and $R$ the interaction distance?


If one is talking about antineutron production, what is the charge exchange?

In this question was given the antineutron production in the way $$ p + \bar{p} \rightarrow n + \bar{n}. $$ Internet search gives one source only where the charge exchange is mentioned. They talk about the "value for the $ \sigma (p + \bar{p} \rightarrow n + \bar{n}) $". So I couldn't understand how does charge exchange works. Where comes the missing energy from? What is the full equation? Can one give later primary sources?


Unsure how resistors are connected? (Series/Parallel)

![enter image description here](http://i.stack.imgur.com/vDRnq.jpg) I am currently attempting this practice question in my tutorial and while I am able to reduce the circuit below to ![enter image description here](http://i.stack.imgur.com/ZCFK4.png) I do not know whether the 6/6/6/6 pane is connected in series or parallel, it should not be parallal else R(Total) = 3+3+3 and Vo will be 2/3 * 12 = 4 The answer im expecting is 4/3V so my current attempt is wrong. I also doubt it is a Wye-Delta transformation. Can anyone explain how the resistors are connected and why? Much thanks in advance.


How can the Huygens-Fresnel principle be derived from the Maxwell equations?

The Huygens-Fresnel principle states that every point to which a luminous disturbance reaches becomes a source of a spherical wave. I have been trying to understand this considering a infinite screen with a microscopic hole $dS$ on which a plane wave is incident, but I see no obvious way to describe the resulting propagating wave. How can it be derived from the Maxwell equations?


Accelerometer on a wheel's rim

If I put a 3-axis accelerometer on a vertical wheel's rim which acceleration is going to give to me? Will the gravity interfere my results?


How are the finite speed of light and the atomic nature of mater related to the end of Moore's law?

In this article from 2007, Moore talks about the end of his Law. Can someone throw more light as to how the finite speed of light and the atomic nature of mater are related to the end of Moore's law?


Experiment with spring and two light sources emitting light in opposite directions

Two (almost massless) light sources connected with a (almost massless) spring emit light beams in opposite directions. What is the speed at which the spring extends?


What does the term 'hyperbolic model' mean?

I am reading this non-linear discrete dynamical system paper. The authors mention the term hyperbolic model. What does that mean?


How can I relate the energy of a gamma particle to electromagnetic radiation power?

I have a distinct feeling that dimensional analysis says these two (let's pick 1 MeV and +30dBm) can not be directly compared. What is the missing link between the two? I know one is energy and one is power. I suppose I'll have to get the energy of +30dBm radiated by an ideal +3dBm antenna at a distance of 2cm impacting an area of $10\times10\,mm$. $1.6\times10^{-13}$ joules for the 1MeV gamma [unknown] for the radio wave (for reference it will be in the low GHz frequency)


Evaluating the Einstein-Hilbert action

The Einstein-Hilbert action is given by, $$I = \frac{1}{16\pi G} \int_{M} \mathrm{d}^d x \, \sqrt{-g} \, R \, \, + \, \, \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{d-1}x \, \sqrt{-h} \, K$$ including the Gibbons-Hawking-York boundary term. A well-known derivation of the entropy of the Schwarzschild metric requires the evaluation of the boundary term. However, one must introduce a radial regulator $R$, and subtract off a counter-term which is the Gibbons-Hawking action of emtpy space with the same boundary. The final result is finite as $R\to\infty$. I am attempting to calculate the full action for a solution for which $R\neq0$, hence I need to also compute the pure Einstein-Hilbert piece. However, I must introduce a regulator, and the final action is not finite as I take it to infinity. My question: **is there an analogous procedure to "tame" the infinity of the pure Einstein-Hilbert piece, perhaps similar to the treatment of the Gibbons-Hawking term?** * * * I actually have to introduce two regulators. For my solution, the Ricci scalar is independent of a particular coordinate, $x_1$, so I get a factor of $x_1$ after integration evaluated at $\pm\infty$, so I introduced regulators, such that $L^{-} < x_1 < L^{+}$. As I take $L^{\pm}\to \pm \infty$, of course it is divergent.


Gravity and the Standard Model

Gravity is ignored in the SM. The proton rest mass is ~0.938 GeV/$c^2$. LHC protons will move with 7 TeV energy, presumably with a relativistic mass about 7,450 times rest mass. A cosmic ray with the highest energy was detected at about $6\times 10^{21}$ eV. If it was a proton, its relativistic mass would have been about $6.4\times 10^{12}$ times rest mass. My question is, at what energy levels would it be necessary to include gravity in the SM? I recognize this isn't an issue for forseeable particle accelerators, but astronomical events like supernova are capable of generating extremely high energy particles.


Fate of light ray

![enter image description here](http://i.stack.imgur.com/VQWpx.png) This question has been on my mind since I first learned optics. Where will the light ray go? First, I thought of it taking a U turn. But, I don't think that will happen. Now I have studied a bit about wave optics and I think that the light will go... everywhere? Can someone please explain it in realm of ray optics? I wouldn't mind a bit about wave optics. Just don't go deep down there. Thanks.


Can Black Holes be the Dark Matter?

Seems to fit the definition: interacts with gravity, doesn’t radiate energy (except Hawking Radiation) and could create gravity lensing without absorbing very much of the light. Could 80% of the original matter created in the Big Bang have ended up in lots of black holes thereby giving us the amount of mass to balance the universes gravity? Kent


Complex Representation of a gauge group and a Chiral Gauge Theory

In this John Preskill et al paper, a statement is made in page 1: > We will refer to a gauge theory with fermions transforming as a **complex > representation of the gauge group** as a **chiral gauge theory** , because > the gauged symmetry is a chiral symmetry, rather than a vector-like symmetry > (such as QCD). But my question is: why does a **Complex Representation of gauge group** imply a **Chiral Gauge Theory**? If fundamental representation of SU(3) is a complex representation (with complex conjugate anti-fundamental Rep), then isn't QCD with fundamental representation of SU(3) a perfect counter example where the gauge symmetry is vector-like, instead of chiral??? ps. See this page, or learn that from Wiki: > In physics, a complex representation is a group representation of a group > (or Lie algebra) on a complex vector space that is neither real nor > pseudoreal. In other words, the group elements are expressed as complex > matrices, and the complex conjugate of a complex representation is a > different, non-equivalent representation. For compact groups, the Frobenius- > Schur indicator can be used to tell whether a representation is real, > complex, or pseudo-real. > > For example, the N-dimensional fundamental representation of SU(N) for N > greater than two is a complex representation whose complex conjugate is > often called the antifundamental representation.


Centripetal issue when considering gravity

Forgive me if my question seems silly, but I am quite baffled. Suppose you have a satellite orbiting a horizontal swing planted into the ground and we want to find the velocity with which the satellite must be moving in order to succeed in taking a fully circular path around the swing (no failure followed by parabolic fall). Indeed if we solve for F = m(v^2)/R where F can be equated to g, we can also solve for the velocity; all nice and good, and the nature of this method makes some intuitive sense but there is a problem (at least for me): the force that the satellite experiences is an inward force towards the center, and therefore the force described in the aforementioned equation is the centripetal force. However, this conflicts with the method of solving the question, since at the top of the path, the centripetal Force and the gravitational force will be both pointing downwards. Therefore shouldn't the satellite actually be "doubly" compelled to accelerate to the ground? But clearly when we for for v with g, we assume that they cancel each other out - hence they point in opposite directions. The idea that the cancel also "explains" the intuition that at the top of the circular path, the satellite will be feeling the most "weightless" it will throughout its revolution. Additionally from a different reference frame / non centrifugal one, it makes sense that at the top, the most force will be on the satellite to turn its path downward as it is after that instant that the satellite sees its downward acceleration. I am, essentially confused with how to handle centripetal/centrifugal "fictitious" forces in this worked example.


Two-Person Problem

The following is a thought experiment I've been stuck on. (No, this isn't homework. I made it up.) Here it is:   **Two-Person Problem**   Let's say we have a superposed particle going through a tube. There are two people on each side of the tube, as well as two detectors, one 1 yard away from the start of the tube, and the other 2 yards away from the start of the tube, each on different sides. Also they are two people on each side. Now, when the particle reaches the first detector, it is collapsed to person A, but still superposed to person B. Let's say that the result was $|0 \rangle$. My question is, is it possible to be collapsed and superposed at the same time? Even worse, when the particle reaches the second detector, it is possible it could have a result of $|1 \rangle$? And no, in the experiment they are not allowed to communicate with each other.      What's happening here!


Relation between classical and quantum information

It is known that for a classical system the amount of information needed to store its state is the same as the amount of information that can be stored in that system. This amount is equal to Shannon's entropy or its classical generalizations. On the other hand the amount of information needed to describe the state of a quantum system is greater than the amount of information that can be stored in such quantum system. My impression is that to describe the state of a quantum system of $N$ qubits one needs $2^N$ bits of information, on the other hand one can only store $N$ bits of information inside such system, which is in my impression equal to von Neumann's entropy of the system. Thus I have two possible explanations of such discrepancy. a) The true entropy of a quantum system is exactly von Neumann's entropy. The difference is only due the fact that a classical storage can only independently operate with digits not less than one bit, thus to store, say 4 variables of 1/4 bit each a classical system needs 4 bits (at least one bit for each variable) while a quantum system can use parts of a digit independently and thus store all 4 variables in one binary digit. b) The true entropy of a quantum system of $N$ qubits is exactly $2^N$ bits but the amount of **extractable** information is only $N$ bits. Which explanation do you consider more correct or are there possible other explanations?


Superposition and particle mediated interactions

Suppose particle A interacts with particle B by means of an exchange particle E. Presumably, since this is quantum physics, the exchange particle emission and receipt are just one of many possible paths by which the two particles can interact. Does that mean that A is in a continual state of superposition of having emitted an exchange particle in every direction? And that B then is in a continual state of superposition of having absorbed it (and thus moved if there was a force involved) and not? Somehow that seems like the most sensible option, because the alternative is that A knows where B is and sends it E at some frequency determined by the strength of the fields. The process above would also entangle A and B as they cannot disagree over whether they exchanged E.


The potentiality of the electric field

Could you please explain using just words why electric the field is potentially? I know the proof using integral: $$A = \int_{12}q\vec{E}\cdot{d}\vec{r} = qQ\int_{12}\frac{\vec{r}\cdot{d}\vec{r}}{r^3} = qQ\int_{r_1}^{r_2}\frac{dr}{r^2} = qQ\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$$ (so the work depends only on the coordinates of the beginning and the end). I can hardly understand it and I want to know how to prove it using only words (maybe with simple school math).


Optical depth in astrophysics

How can be the exponential decay of the intensity $I$ via the optical depth $\tau$ be derived? $$I(\tau)=I(0)e^{-\tau} $$


Representations of the Poincare group

Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?


What if $F\neq \frac{dp}{dt}$?

I was thinking of this idea that maybe there are esoteric cases where the force is not given in classical mechanics as $F=dp/dt$ but as some function of $F=F(p,q,\dot{p},\dot{q})$ E.g, something like: $k\cdot \frac{dp}{dq}$ with a suitable constant $k$, or any other sort of function of p,q and its time derivatives. Are there any toy models that theorists suggest on this idea?


Does the Earth's core have tides?

Does the moon produce a measurable tidal-effect on the Earth's (liquid) core? If so, how strong is it? Would it play a factor in other geological effects like earthquakes, volcanoes, etc?


Cross-section of a wave packet

In text books, wave packets are one-dimensional drawings. But we live in a three-dimensional world. Suppose a wave packet from a HI-cloud (frequency 1420 MHz) is approaching the earth, distance about 10 million miles and it will arrive here in a few seconds. How long is this packet (FWHM)? How large is the cross-section? Is the diameter zero? Or 1/10 of a wavelength or three time the wavelength? Or even bigger? Any good ideas? (This question is partly inspired by this Phys.SE answer.)


Partially polarized light in laboratory

Does anybody know a way to create (in laboratory, but with no super- sophisticated equipement) a beam of light with partial linear polarization whose degree of polarization can be adjusted (at least over the range 0-0.5)?


Ground states of the Hamiltonian of a two spin system

For the spin system shown in this graph (http://i.stack.imgur.com/3lg1R.png), the Hamiltonian is $$S^{(1)}_z\cdot S^{(1)}_z=\frac{1}{4}\begin{pmatrix} 1 & 0 &0 &0 \\\ 0&-1 &0 &0 \\\ 0 &0 &-1 &0 \\\ 0 &0 &0 &1 \end{pmatrix}.$$ The eigenvalues of this Hamiltonian are $-\frac{1}{4}$ when the eigenstate is $$\begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix}$$ or $$\begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$ and $\frac{1}{4}$ when the eigenstate is $$\begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix}$$ or $$\begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}.$$ Moreover, the expectation value of the Hamiltonian for $$\begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$ is $-\frac{1}{4}$ and for $$\begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}$$ is $\frac{1}{4}$. Based on these information can we say that $$\begin{pmatrix} 0 \\\ 1 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 1 \\\ 0 \end{pmatrix}$$ are the ground states of the Hamiltonian while $$\begin{pmatrix} 1 \\\ 0 \\\ 0 \\\ 0 \end{pmatrix}$$ and $$\begin{pmatrix} 0 \\\ 0 \\\ 0 \\\ 1 \end{pmatrix}$$ are the first excited states?


Without apparatus can we say that the system is measured(decohered) by the environment?

"Einselection" and "tridecompositional uniqueness theorem" seem to resolve the preferred basis problem. But the premise is that there are three parts in discussion.(system, apparatus, environment) However, it seems that in many situations we don't have the role of apparatus, and thus there are just system and environment. For instance, we usually say that the system is monitored by its environment and thus in a state with determinate classical physical value. In these situations, in which there are just two parts(system and environment) and the Schmidt form of the total system is not unique, can we say that the system is measured(decohered) by the environment?


Proof of the Hydrostatic weighing equation?

I've been trying to derive this equation for about two hours: ![enter image description here](http://i.stack.imgur.com/VoM8x.png) But I can't seem to get it. In my class, we are using it to accurately measure the volume of irregulary-shaped objects, but I completely missed the derivation. I'm not looking to leech off of somebody's proof, but any help with this would be greatly appreciated. Thank you in advance! * * * Just in case anyone would like to see my implementation of Mark's answer: $$ w_{dry} = m_{dry}g $$ $$ m_{dry}g = \rho_{o}Vg $$ $$ \rho_{o} = \frac{m_{dry}}{V} $$ The forces acting on the same object, but submerged fully in a fluid are: $$ w_{wet} = m_{dry}g - B $$ $$ = m_{dry}g - \rho_{f}gV $$ $$ m_{dry} - m_{wet} = \rho_{f}V $$ $$ \rho_{f} = \frac{m_{dry} - m_{wet}}{V}$$ The specific gravity of the object in relation to the fluid it is submerged in is therefore: $$ S_{g} = \frac{\rho_{o}}{\rho_{f}} $$ $$ = \frac{\frac{m_{dry}}{V}}{\frac{m_{dry} - m_{wet}}{V}} $$ $$ = \frac{m_{dry}}{m_{dry} - m_{wet}} $$


Induced EMF of a spinning metal rod

Hey everyone, I'm currently taking an Eletricity and Magnetism course and I'm having dificulty answering this problem. I've gone to my tutor, but sadly, he hasn't taken E+M for a while now and couldn't recall off the top of his head how to go about solving this. We went through my book together, but couldn't find any examples pertaining to this type of problem. (We did find examples of induced EMF's obviously, but making the leap to a spinning rod seems to be too great a task) Anyways, here is the problem I'm working on: A metal rod is spinning on an axis that is located in the middle of the rod, and perpendicular to the rod's length. If the rod is in an external magnetic field that is parallel to the axis the rod is spinning on, how would I go about finding the induced emf? The rod's thickness is negligible but has length L. It's spinning at 1500 rev/s and the magnetic field is 7.5T Any pointers would be appreciated!


Why do bar magnets have least attraction in its center?

Why do bar magnets have the least attraction in its center?


How is parity relevant to determining angular momentum?

# Question: > Particle A, whose spin $\mathbf{J}$ is less than 2, decays into two > identical spin-1/2 particles of type B. > > What are the allowed values of the orbital angular momentum $\mathbf{L}$, > the combined spin $\mathbf{S} = \mathbf{s}_1+\mathbf{s}_2$ (where > $\mathbf{s}_1$ and $\mathbf{s}_2$ are the spin vectors of the B particles), > and the total angular momentum $\mathbf{J}$ of the decay products? Give the > results from two cases: first if particle A has odd parity and then if A has > even parity. # My thoughts: Particle A can be spin-1/2, spin-1, or spin-3/2. Since $\mathbf{J}<2$, we see that there are four possibilities for A: $$ \begin{align*} &(1): \;\;\mathbf{S}_A = 1/2 \quad\quad \mathbf{L}_A = 0 \quad\Rightarrow\quad \mathbf{J} = 1/2 \\\ &(2):\;\;\mathbf{S}_A = 1/2 \quad\quad \mathbf{L}_A = 1 \quad\Rightarrow\quad \mathbf{J} = 3/2 \\\ &(3):\;\;\mathbf{S}_A = 1 \quad\quad\quad \mathbf{L}_A = 0 \quad\Rightarrow\quad \mathbf{J} = 1 \\\ &(4):\;\;\mathbf{S}_A = 3/2 \quad\quad \mathbf{L}_A = 0 \quad\Rightarrow\quad \mathbf{J} = 3/2 \\\ \end{align*} $$ The total spin of the B particles can be either $1$ or $0$, and each particle can individually have an orbital angular momentum, along with the angular momentum of the particles as a system. With this thought, cases 1,2, and 4 are impossible because the orbital angular momentum of the B particles is an integer, as is their total spin (and therefore their total angular momentum too). Thus we find that only case 3 is allowed, so the total angular momentum of the B particles is $1$ and their orbital angular momentum is $0$ (so $\mathbf{J}=1$). I have a strong feeling that this is incorrect, because the question asks for the cases when A has odd parity and even parity (what does that even mean?!) so I suspect there should be more than one possible answer. Where did I go wrong?


How to understand topological order at finite temperature?

I have heard that in 2+1D, there are no topological order in finite temperature. Topological entanglement entropy $\gamma$ is zero except in zero temperature. However, we still observe some features of topological order in fractional quantum Hall effect, such as fractional statistics, symmetry factionalization. So what is the meaning when people say "there is no topological order at finite temperature in 2+1D"? And what is the situation in 3+1D?


Why do or don't neutrinos have antiparticles?

This was inspired by this question. According to Wikipedia, a Majorana neutrino must be its own antiparticle, while a Dirac neutrino cannot be its own antiparticle. Why is this true?


Interaction speed between electric charges and magnetic materials

Einstein said that the speed of a matter in universe cannot exceed the speed of light.   1. Is it correct for electric force transmission speed from one electric charge to other one?    2. What is transmission speed between each other?    3. Is the reaction delay of one of them speed of light time when we move another one?   4. Also same question for magnetic force interaction speed with each other.   5. And is there any particule theory about the subject as gravitation can be created by gravitons? Thanks a lot for answers


Is it possible to "add cold" or to "add heat" to systems?

> Amanda just poured herself a cup of hot coffee to get her day started. She > took her first sip and nearly burned her tongue. Since she didn't have much > time to sit and wait for it to cool down, she put an ice cube in her coffee > and stirred it with a metal spoon. After a moment, she felt the spoon warm > up, but when she took another sip, the coffee was cooler. She was pretty > sure, the ice added cold to her coffee, and the coffee added heat to her > spoon. Would you agree?


It would take an elephant, balanced on a pencil, to break through a sheet of graphene the thickness of Cling Film

I'm currently doing some work on a presentation about graphene, and have come across numerous articles which claim something along the lines of > It would take an elephant, balanced on a pencil, to break through a sheet of > graphene the thickness of Saran Wrap / Cling Film. My question is, is there any proof / calculations to back up this claim? Every article I come across seems to be very similar and I cannot find the original source which might contain proof, or a link the relevant paper/study **Sources**   * http://www.dailymail.co.uk/sciencetech/article-2045825/Graphene-strong-sheet-clingfilm-support-elephant.html   * http://www.scientificamerican.com/article.cfm?id=balancing-act


Thermodynamics shell-and-tube heat exchanger

A shell-and-tube heat exchanger is used to cool a flow of steam using a stream of cooling water cooling water. The steam enters the heat exchanger at $0.1 ~bar$ and a quality of $91$% , and exits as a saturated liquid at the same pressure. The mass flow rate of the steam is $2.6 ⋅ 10^5 \frac{kg}{hour}$. The cooling water enters the heat exchanger at $5 ~MPa$ and $15^ \circ{C}$ and exits at $38^ \circ{C}$at the same pressure. Neglect stray heat transfer and neglect kinetic and potential energy effects. As well, assume the system is operating under steady-state conditions. Draw a schematic of the system and locate the state points on a T-v diagram. Determine the mass flow rate of cooling water in kg/s and determine the rate of heat transfer between the steam and the cooling water in MW. ..... I know by neglecting PE and KE and rearranging we have, $$m_3 = m_1 ⋅ \frac{h_1-h_2}{h_4-h_3}$$ which i solved for using some table values for enthalpy to get $m_3 = 1634.44 \dfrac{kg}{s}$ now how do I determine the rate of heat transfer between the steam and the cooling water? and if someone could show me the T-v diagram, would be awesome!


1. Why are some materials diamagnetic, others paramagnetic, and others ferromagnetic?   2. Or, put another way, which of their atomic properties determines which of the three forms of magnetism (if at all) the compound will take?   3. Is paramagnetism to ferromagnetism a continuous spectrum, or is there no grey zone in between?


How to define the mirror symmetry operator for Kane-Mele model?

Let us take the famous Kane-Mele(KM) model as our starting point. Due to the time-reversal(TR), 2-fold rotational(or 2D space inversion), 3-fold rotational and mirror symmetries of the honeycomb lattice system, we can derive the intrinsic spin-orbit(SO) term. Further more, if we apply a spatially uniform electric field perpendicular to the 2D lattice(now the mirror symmetry is broken), a (extra) Rashba-type SO term will emerge. To present my question more clearly, I will first give a more detailed description of the above symmetry operations in both first- and second- quantization formalism. In the follows, a 3D Cartesian coordinate has been set up where the 2D lattice lies in the $xoy$ plane. **First-quantization language:** (1) TR symmetry operator $\Theta:$ $\Theta\phi(x,y,z)\equiv \phi^*(x,y,z)$, hereafter, $\phi(x,y,z)$ represents an **arbitrary wave function** for single electron. (2) 2-fold rotational operator $R_2:$ $R_2\phi(x,y,z)\equiv \phi(-x,-y,z)$, where we choose the middle point of the nearest-neighbour bond as the origin point $o$ of the coordinate. (3) 3-fold rotational operator $R_3:$ $R_3\phi(\vec{r} )\equiv \phi(A\vec{r})$, where $A=\begin{pmatrix} \cos\frac{2\pi}{3}& -\sin\frac{2\pi}{3}& 0\\\ \sin\frac{2\pi}{3}& \cos\frac{2\pi}{3} & 0\\\ 0& 0 & 1 \end{pmatrix}$ $\vec{r}=(x,y,z)$ and we choose the lattice site as the origin point $o$ of the coordinate. (4) **Mirror symmetry operator $\Pi:$ $\Pi\phi(x,y,z)\equiv \phi(x,y,-z)$.** **Second-quantization language:** (1) TR symmetry operator $T:$ $TC_{i\uparrow}T^{-1}=C_{i\downarrow}, TC_{i\downarrow}T^{-1}=-C_{i\uparrow}$, where $C=a,b$ are the annihilation operators referred to the two sublattices of graphene. (2) 2-fold rotational operator $P_2:$ $P_2a(x,y)P_2^{-1}\equiv b(-x,-y), P_2b(x,y)P_2^{-1}\equiv a(-x,-y)$, $P_2$ is unitary and we choose the middle point of the nearest-neighbour bond as the origin point $o$ of the coordinate. (3) 3-fold rotational operator $P_3:$ $P_3C(\vec{x})P_3^{-1}\equiv C(A\vec{x}), \vec{x}=(x,y),C=a,b$, where $A=\begin{pmatrix} \cos\frac{2\pi}{3}& -\sin\frac{2\pi}{3}\\\ \sin\frac{2\pi}{3}& \cos\frac{2\pi}{3} \\\ \end{pmatrix}$ and $P_3$ is unitary, we choose the lattice site as the origin point $o$ of the coordinate. (4) **Mirror symmetry operator $M:$ ????** as you see, that's what I want to ask: how to define the mirror symmetry operator $M$ **in terms of second-quantization language** for this 2D lattice system? Or maybe there is no well defined $M$ for this model? Thanks in advance. **Remarks:** (1) A direct way to verify your definition of $M$ being correct or not is as follows: The intrinsic SO term $i\lambda\sum_{\ll ij \gg }v_{ij}C_i^\dagger\sigma_zC_j$ should be invariant under $M$ while the Rashba term $i\lambda_R\sum_{<ij>}C_i^\dagger \left ( \mathbf{\sigma}\times\mathbf{p}_{ij}\right )_zC_j$ will not be. (2)Here **mirror** operation is just reflection in one of the three spatial axes (i.e. $(x,y,z)\rightarrow (x,y,-z)$), not the “parity” operation in the context of "CPT symmetry" in field theory.


Lightning Power?

What is the power source of a lightning? It doesn't consume fuel, nor an atomic fission/fusion, but it discharges a large amount of energy. Is it feasible to create an artificial lightning using the same method of the natural lightning? If it is, can we create a power plant from the same source?


Taylor expansion in classical 1D harmonic chain (classical field theory)

I'm struggling with something apparently really easy but not entirely straightforward. In "Condensed Matter Field Theory" of Altland and Simons the classical 1D harmonic chain is treated as introductory example to classical field theory. Consider an harmonic chain of N particles on a periodic 1D lattice (N+1=1). The position of the I-th particle is $R_I=Ia$. The Lagrangian reads $L=\sum_{I=1}^N[\frac{m}{2}\dot{R_{I}^2}-\frac{k_s}{2}(R_{I+1}-R_{I}-a)^2]$ Now he parametrizes $R(t)=R_I+\phi(t)$ where the deviation from equilibrium is small, so that $L=\sum_{I=1}^N[\frac{m}{2}\dot{\phi_{I}^2}-\frac{k_s}{2}(\phi_{I+1}-\phi_{I})^2]$ Now he goes to the continuum limit, that is, he introduces continuum degrees of freedom $\phi(x)$ and applies a Taylor expansion such that: $\phi_I\rightarrow a^{1/2}\phi(x)\left.\right|_{x=Ia} \hspace{30pt} \phi_{I+1}-\phi_I\rightarrow a^{3/2}\partial_x\phi(x)\left.\right|_{x=Ia} $ I can't figure out how this substitution and Taylor expansion works. If you look to the first, you should get for the second one: $\phi_{I+1} - \phi_I\rightarrow a^{1/2}\left[\phi(x+a)-\phi(x)\right]\left.\right|_{x=Ia}$ How is this now equal to the derivative, should i take a limit for $a\rightarrow 0$? And where does the prefactor $a^{3/2}$ come from? Cheers


Neutrinos: how can they carry information about universe?

I know that neutrinos are particles with a very small mass and no electric charge. They infrequently interact with matter and so they can give us information about the "old" universe. But how can they do it?


How to calculate force generated from a magnet coil to a magnet?

I go to a website http://easycalculation.com/engineering/electrical/solenoid- force.php They have the calculator for the force between a solenoid coil and another piece of ferromagnetic material. What if I have a strong magnet (max pull force is 50lb) instead of ferromagnetic material? How does the force change?


Energy conservation in nuclear reactions and radiactive decay

Reading "Fundamentals of Nuclear Physics" by Atam P. Arya, I understand that in a nuclear reaction, let say $x+X \to y+Y$ meaning that "when a particle $x$ strikes a target nucleus $X$, the outcome of the nuclear reaction is a recoil nucleus $Y$ and a particle $y$. In many cases more than one type of particle may be given out." Now, Arya applies the energy conservation to the nuclear reaction, writing $$E_i=E_f$$ where $$E_i=K_x+m_xc^2+K_X+M_Xc^2$$ and $$E_f=K_Y+M_Yc^2+K_y+m_yc^2$$ being $K_x, K_X, K_y, K_y$ the kinetic energies of the particle $x$, the target nucleus (parent), the recoil nucleus (daughter) and the particle $y$, respectively. And $M$ & $m$ its respective masses. Rewriting energy conservation, he defines a quantity $Q$ (the disintegration energy): $$Q:=(K_Y+K_y)-(K_X+K_x)=(M_X+m_x)c^2-(M_Y+m_y)c^2 \qquad (1)$$ Now, my **question** is whether or not the equation (1) (and its interpretation) can be applied to alpha and beta decay (where there are no particle colliding with the nucleus).


Why is my speed imaginary?

A particle is released from a dense atmosphere free planet of radius r at a distance R (R >> r ). find the speed on impact. $$F = \frac{GmM}{(r + R)^2} = m \frac{dv}{dt} = mv\frac{dv}{dR}$$ $$GM\int^{R_f}_{R_0}(r + R)^{-2} dr = \int^{V_f}_{V_o}vdv$$ But $R_f$ = 0 when the object strikes the planet $$-GM[r^{-1} - (r +R_0)^{-1}] = 1/2[v_f^2 - v_0^2]$$ $$-2GM[\frac{R_0}{r(r + R_0)}] + V_0^2 = V_f^2$$ why is it imaginary?


Stroboscope-and-telegraph problem

**Narrative:** Consider, in a suitably flat region, two straight lines which don't necessarily intersect. Let vector $\mathbf{x}$ point along one line, and vector $\mathbf{y}$ point along the other. Let $X$ and $Y$ be the two points closest to each other on these two lines, or otherwise let $P$ be their intersection point. The real number $\phi := \text{ArcCos}\bigl( \frac{\mathbf{x} \, \cdot \, \mathbf{y}}{|\mathbf{x}| \, |\mathbf{y}|}\bigr)$ is supposed to be known as part of the description of how the two lines are related to each other. Now consider a stroboscope "$B_x$" moving uniformely along one of these lines (in direction of vector $\mathbf{x}$, passing point $X$), and a telegraph "$A_y$" moving uniformely along one of these lines (in direction of vector $\mathbf{y}$, passing point $Y$). The case of intersecting lines may be dealt with later on. Stroboscope $B_x$ is supposed to _blink_ ceaselessly (therefore its name: "$B_x$") with equal durations $\tau_B$ from any one " _blink_ " ($b_j$) to the next one ($b_{j+1}$); and telegraph $A_y$ is supposed to re-shuffle its " _arms_ " all the time (... cmp. "$A_y$") with equal durations $\tau_A$ from settling into any one signal state ($a_k$) to settling to the next ($a_{k+1}$). The real "ratio" number $r := \tau_B / \tau_A$ is supposed to be known, too. Let $a_0$ be the state of telegraph $A_y$ while passing point $Y$, and $b_0$ be the state of stroboscope $B_x$ while passing point $X$. Let integer $p_{BA}$ be given as the number of " _blinks_ " stated by stroboscope $B_x$ after having passed point $X$ (i.e. after having been in state $b_0$) and before observing telegraph $A_y$ passing point $Y$ (i.e. before observing that telegraph $A_y$ had been in state $a_0$). If the described order of observations is inverted (i.e. if stroboscope $B_x$ first observed that telegraph $A_y$ had passed point $Y$ and only later passed point $X$) then integer $p_{BA}$ is obtained as negative value (or as value $0$). Finally suppose that stroboscope $B_x$ is (partially) surrounded by a shiny case such that telegraph $A_y$ can see its (own) reflection and recognize its states in the mirror image; and suppose that telegraph $A_y$ is (perhaps rather diffusely) reflective as well such that stroboscope $B_x$ would be able to see reflections of its " _blinks_ ". Now: For any particular signal state $a_k$ telegraph $A_y$ can count how many subsequent times it went on to re-shuffle its arms before recognizing the reflection (from $B_x$) corresponding to state $a_k$; lets call this number $\alpha_{Ak}$. Along with that, again for any particular signal state $a_k$, telegraph $A_y$ can count how many subsequent times it saw stroboscope $B_x$ _blink_ subsequently before recognizing the reflection (from $B_x$) corresponding to state $a_k$; lets call this number $\alpha_{Bk}$. Correspondingly, stroboscope $B_x$ may determine the numbers $\beta_{Bj}$ and $\beta_{Aj}$ ...   (No!, too bad: stroboscope $B_x$ may not readily determine these numbers because it cannot easily tell the reflection of any one _blink_ from that of any other. But this may lead to yet another question ... In order to fix the narrative let's here just assume that "stroboscope $B_x$" is in fact a "telegraph" as well, with distinct states $b_j$ whose reflections from $A_y$ can be individually recognized, too.) **Question (1):** Do the numbers $\alpha_{Ak}$ have a minimum, $\alpha_{\text{min}}$ ?,   in particular, if the two lines don't intersect.   (And similarly: Do the numbers $\beta_{Bj}$ have a minimum, $\beta_{\text{min}}$ ?) And if so: **Question (2):** Can the "ratio" number $r$ be calculated from the (given) numbers $\phi$, $p$, $\alpha_{Ak}$, $\alpha_{Bk} $, $\beta_{Bj}$ and $\beta_{Aj}$ ?, or perhaps at least approximated for sufficiently large numbers $\alpha_{\text{min}}$ and/or $\beta_{\text{min}}$ ?


Experimental observation of elementary particles?

I posted a similar yet totally unrelated question recently, and got really satisfying responses to it. Thus, on the same theme... **How have we come to realize the existence of elementary particles in general? What evidence have we accumulated over the years that proves their existence?** Once again, please note that I'm not _doubting_ the existence of elementary particles, but am just curious as to how we have found out about them.


Does diffraction occur before interference?

In the double slit experiment, light passes through two slits and the waves interfere and form an interference pattern. A single slit is required for diffraction. So, I was thinking about whether the light in the interference setup undergoes diffraction first at the two slits? I think that's the case, because light waves spread due to diffraction at the two slits. The two waves interfere to form the interference pattern. Is my thinking correct?


What exactly is regularization in QFT?

**The question.** Does there exist a mathematicaly precise, commonly accepted definition of the term "regularization procedure" in perturbative quantum field theory? If so, what is it? **Motivation and background.** As pointed out by user drake in his nice answer to my previous question Regulator-scheme-independence in QFT , it is often said that in a renormalizable quantum field theory, results for physical quantities (such as scattering amplitudes) when written in terms of other physical quantities (like physical masses, physical couplings, etc.) _do not depend on the regularization procedure_ one chooses to use. In fact, user drake takes this desirable property as part of the definition of the term "renormalizable." In my mind, in order for such a statement or definition to be meaningful and useful, it helps to have a precise, mathematically explicit notion of what constitutes a regularization procedure. That way, when one wants to compute something physical, one can use any procedure one wishes provided it satisfies some general properties. **My current understanding in a (small) nutshell.** When we perturbatively compute, say, correlation functions for some theory pre-regularization/renormalization, we obtain formal power series' in the bare parameters that characterize the theory. Such power series' contain expressions for loop integrals that generically diverge, often due to high- momentum (UV) effects, so we "regularize", namely we implement some procedure by which these integrals are made to depend on some parameter, call it $\Lambda$, and are rendered finite provided $\Lambda$ doesn't take on a certain limiting value $\Lambda_*$ (that could be $\infty$) corresponding to the physical regime (like the UV) that led to the original divergence. We then renormalize and find (in renormalizable theories) that $\Lambda$ drops out of physical results. **What sort of an answer am I looking for?** I am looking for something like this. > A _regularization procedure_ is a prescription by which all divergent > integrals encountered in perturbation theory are made to depend on a > parameter $\Lambda$ and which satisfies the following properties: (1) All > divergent integrals are rendered finite for all but a certain value of > $\Lambda$. (2)... I know that there are other properties, but I don't know what constitutes a sufficiently complete list of such properties such that if you were to show me some procedure, I could say "oh yes, that counts as a valid regularization procedure, good job!" Surely, for example, the putative regularization procedure cannot be too destructive; I cannot, for instance, simply replace every loop integral with $3\Lambda$ and call it a day because that would completely destroy all information about how many loops the corresponding diagrams contained. How much of the "formal structure" of the integrals does the procedure need to preserve? As far as I can tell, there is no discussion of this in any of the standard QFT texts who simply adopt tried-and-true procedures like a hard UV cutoff, dim reg, Pauli-Villars, etc. without commenting on general conditions sufficient to guarantee that these particular procedures count as "good" ones. There is, of course, a lot of discussion of whether certain regulators preserve certain symmetries, but that's a distinct issue. **Edit. (January 8, 2014)** Discussions with fellow graduate students have led me to believe that the appropriate definition proceeds by appealing to the idea of effective field theory. In particular, if we view our theory as an effective low-energy description of some more complete theory that works at higher energy scales, then imposing a high momentum cutoff has a conceptually privileged position among regulators; it is the natural way of encoding the idea that the putative theory only works below a certain scale. This can then be used to define a regularization procedure that, in some sense, reproduces the same structure encoded in regularizing with a cutoff. Unfortunately, I'm still not entirely sure if this is the correct way to think about this, and I'm also not sure how to formalize the notion of preserving the structure the comes out of cutoff regularization. My inclination is that the most important structure to preserve is the singular behavior of regularized integrals as the cutoff $\Lambda$ is taken to infinity.


What do the components of light velocity look like in polar coordinates?

The Schwarzschild solution makes use of polar coordinates, and I'm wondering how the different components of velocity of light change with the position. Might I get some examples of light velocity vectors (in units of $c$ such that $\langle 1,0,0,0\rangle$ is the velocity vector for a light ray moving radially away from the black hole at the speed of light)?


Bremsstrahlung: why is electron slowed/stopped by the positive nucleus?

I can't understand why the electron is slowed/stopped by the nucleus. The electron is a negative charge and the nucleus is positive... they should attract each other...


What is mathematical definition of a strong gravity?

Mathematical definition of a weak gravity is simple $g=\frac{GM}{r^2}$ but what is mathematical definition of a strong gravity? (blackhole-like or close to a blackhole-like object)


How do wind turbines cause lightning?

I saw this paper about Lightning discharges produced by wind turbines More lightning strikes wind turbines than comparable non-moving structures. What would be the physical cause of this? Does it depend upon the motion of the wind turbine blades or is it because these are connected to the electric power grid?


How to get Gell-Mann matrices?

How to get Gell-Mann matrices $f_{i}$ (more or less strictly)? What are the requirements for getting them, excluding $||f_{i}|| = 1$, commutational law $[f_{i}, f_{j}] = if_{ijk}f_{k}$ and hermitian nature? Is that enough?


How an accelerated object sees another accelerated body in special relativity?

Assume two objects are moving with a constant acceleration $a_1$ and $a_2$, which are the measured accelerations by respective object (or constant force being applied to each of the objects). My problem is how one object sees the motion of the other one. I know that the accelerations are transformed according to the relation (Why proper acceleration is $du/dt$ and not $du/d\tau$?), $$ a = \frac{a'}{\gamma^3} $$ where $a'$ is the measured acceleration and $a$ is the acceleration in a inertial frame.


Modeling magnetic field shielding

I am wondering what is the best way to model magnetic and electric field shielding from a magnetic dipole in the near field? For example, let's say you have a coil of current carrying wire in the x-y plane which creates an AC magnetic field along the z-axis. I am interested in a model in which I could put in material parameters (such as $\sigma$, $\epsilon$, $\mu$) for a shielding plate located distance $r$ from the source. I tried using section 5.4 of an online electromagnetic book which talks about reflection, and transmission of of EM waves. I would estimate wave impedance as $Z=\omega\mu_0r$ which is the equation for near field magnetic source. I determine the intrinsic impedance of the shielding material as $\eta=\sqrt {\frac {j\omega\mu} {\sigma + j\omega\epsilon}}$. I want to try good conductors as well as ferrite so I'm not making any assumptions such as good conductor or loss-less medium. I implemented this but the result was not what I expected. I would expect highest energy reflection with the ferrite because it has low reluctance to magnetic fields and would guide the flux through it and return to source. However, a good conductor (e.g aluminum) had almost 100% energy reflection, but I was expecting the shielding mechanism of conductors would be all energy lost as absorption from eddy currents. Let me know your thoughts of whether this is a legit model; I can give more info about parameters/equations used if that's helpful.


Newton's First Law of Motion; Empirical Aspects

Newton's first law states that in an inertial frame, a body at rest continues to be at rest, and a body in constant rectilinear motion continues its motion, unless an external force is applied upon the bodies. My question is how far have these assertions been tested experimentally? More precisely, is there any need to test this statement, or is it taken as a "definition" of sorts? I assume the body at rest is a trivial case. On the other hand the case of the body in constant motion would be considerably harder to test, since no external force on a body is an idealization. But I am still curious whether there have been any attempts to test the case of the moving body.


Why does elastic energy only depend on first derivatives?

Say we have an elastic material that is deformed with displacement function $u : \mathbb{R}^n \rightarrow \mathbb{R}^n$. It is reasonable to assume that the energy required for such a displacement is an integral over local properties of $u$: $$E = \int_S F(u_i, u_{i,j}, u_{i,jk}, \cdots)$$ Where $u_{i,jk\cdots z} = \frac{\partial^m u_i}{\partial x_j \partial x_k\cdots \partial x_z}$. It is clear that $F$ cannot depend on $u_i$, because the energy is translation invariant. As it turns out, for small deformations $F$ depends on all $u_{i,j} = \frac{\partial u_i}{\partial x_j}$ and does not depend on higher derivatives. Furthermore, $F$ is a quadratic in $u_{i,j}$, namely $F = \sum T_{ijkl} u_{i,j} u_{k,l}$ where $T_{ijkl}$ depends on the material. Is there a reason for this, or is this just an empirical fact? Is there a set of intuitive assumptions that lead to this form of $F$?


What's the mathematical background to the representation for Gaussian beams?

# Background A general optical system (not necessarily having an axis of rotational symmetry) can be represented, for small deviations from a base ray, by the matrix transfer equation, $$ \left[\begin{array}{c} n'L'\\\ n'M'\\\ x'\\\ y' \end{array}\right] = \left[\begin{array}{cc} B & -A\\\ -D & C \end{array}\right] \left[\begin{array}{c} nL\\\ nM\\\ x\\\ y \end{array}\right] = D(g) \left[\begin{array}{c} nL\\\ nM\\\ x\\\ y \end{array}\right] $$ where $L,M,N$ are the direction cosines of a ray, $n$ is refractive index, $x,y$ are coords on a reference plane, $A,B,C,D$ are $2\times 2$ matrices and a prime (') denotes the image space and unprimed quantities are in the object space. The sign convention for the matrix sub- blocks is from ``The Ray and Wave Theory of Lenses'' by A. Walther. The $4\times 4$ transfer matrices $D(g)$ are the defining represention of the symplectic group $Sp(4,R)$ carried on a 4-d vector space $V_{4}$ . They obey, $$ D(g)^{T} \left[\begin{array}{cc} 0 & 1\\\ -1 & 0 \end{array}\right] D(g) = \left[\begin{array}{cc} 0 & 1\\\ -1 & 0 \end{array}\right] $$ where the $0$ is $2\times 2$ and $1$ is the unit $2\times 2$ matrix. Thus one can say that geometric optics is the study of a finite dimensional representation of $Sp(4,R)$ and each optical system is a $g\in Sp(4,R)$. Furthermore, two optical systems $g$ followed by $h$ are equivalent to the single optical system $hg$. It turns out (see Guillemin and Sternberg ``Symplectic Techniques in Physics''), provided the optical system is lossless, that diffractive optics in the Fresnel approximation is the study of an infinite-dimensional projective representation of $Sp(4,R)$ carried on a Hilbert space. In this case, the complex amplitude $u(x,y)$ on a reference plane is a vector $|u\rangle$ and the transfer equation between object and images reference planes is now, $$ |u'\rangle=K(g)|u\rangle $$ where $K(g)$ is a kernel which is actually a projective representation of $Sp(4,R)$ carried on the Hilbert space. Note that, in a book on diffraction, like Walther's, the transfer equation would be appear as, $$ u'(x',y')=\int K(x',y'|x,y)u(x,y)dxdy $$ and the kernel $K(x'y'|x,y)$ depends on the matrices $A,B,C,D$ via a function called the eikonal. The details are not important for this question. # Question If one studies the diffraction of a Gaussian beam, $$ u(x,y)=\exp{\left\\{\frac{i}{2} \left[\begin{array}{cc} x & y \end{array}\right]\left(\frac{q}{n}\right)^{-1} \left[ \begin{array}{c} x \\\ y \end{array} \right]\right\\}} $$ by an optical system in the Fresnel approximation, then the output beam is also Gaussian and the relation between the $2\times 2$ input and output complex beam parameter matrices is: $$ \frac{q'}{n'}=\left(C\frac{q}{n}-D\right)\left(B-A\frac{q}{n}\right)^{-1} $$ This is another representation (realization) of $Sp(4,R)$ carried on the space of symmetric $2\times 2$ matrices like $q/n$. It's some sort of conformal transformation because, for cylindrical lenses, $q/n$ is a complex number and $A,B,C,D$ are real numbers, so the above equation is a conformal transformation. The existence of this representation was a complete surprise for me; it's clearly related to the infinite-dimensional projective representation of $Sp(4,R)$ because the derivation proceeds via Fresnel theory, but it's carried on the 3-d space of symmetric $2\times 2$ matrices whilst the projective representation has no irreducible subspaces other that the even and odd parity irreps. My question is, ``What is the mathematical background to the representation for Gaussian beams?''


What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?

I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry. One alternative way of formulating it: I am looking for an equivalent of Weaire–Phelan structure that would be able to fill in a sphere, while stacked in layers around the central core. Another alternative way of formulating it: I am looking for a crystalline structure that would result in growth of spherical or (almost) spherical crystals. In case it is impossible, what would be the closest quasicristaline structure for this? Any pointers towards textbooks or publications or just intuitive explanations would be more then welcome.


How do I solve these integrals of wave function and operator?

1. First integral $$\int \Psi^*({\bf r},t)\hat {\bf p} \Psi({\bf r},t)\, d^3r,$$ where the $\Psi({\bf r},t)=e^{i({\bf k}\cdot{\bf r}-\omega t)}\,\,\,$ and $\hat {\bf p}=-i\hbar \nabla$.   2. Second one $$\int \Psi^*({\bf r},t)\hat {\bf p}^2 \Psi({\bf r},t)\, d^3r,$$ where the $\Psi({\bf r},t)=e^{i({\bf k}\cdot{\bf r}-\omega t)}\,\,\,$ and $\hat {\bf p}^2=-\hbar^2 \nabla^2$.


Is color confinement detected?

I'm a graduate student studying QFT. I'm quite interested that is color confinement detected or proved? (both directly and indirectly) Or it is just an assumption?


Sign convention for EMF

When we define the field generate by EMF, why there is not negative sign in $\mathcal{E} = \oint \vec{E} \cdot d\vec{l}$? Usually we talk about potential, there should be a negative sign, right?


Gravitational Anomalies and Topological Order

I wonder the relation of gravitational anomaly and topological order. Specifically: 1) What is the definition of gravitational anomaly here? 2) How are they related?


How do we measure the physical field of a particle?

In the renormalization procedure of quantum field theories, say $\lambda \phi^4$ theory for simplicity, we use the physical mass $m$, the physical coupling constant $\lambda$ and the physical field $\phi$ to fix the bare quantities $m_0$, $\lambda_0$ and $\phi_0$. It is clear that the physical mass $m$ and the physical coupling constant $\lambda$ can be found from experiments (and using the S-matrix). But how does one "measure" the physical field $\phi$? **Edit:** I would like to (hopefully) clarify my question. My question is related to field renormalization $\phi_0(x)=\sqrt{Z_\phi}\phi(x)$, and how we decide what value $\phi$ should be. For the physical mass and coupling constant, we can perform experimental measurements to determine their fixed values but we don't seem to do this for the physical field (as far as I understand; I still need to properly interpret JeffDror's answer). **Are we basically looking at the divergence of Feynman diagrams and choose $Z_\phi$ such that we can absorb this divergence?** If this is true, then this would mean that the "physical" field is a badly chosen word and it would make more sense to call it the "renormalized" field?


Why does the guy moving on spaceship look younger in twin paradox?

If there is no particular absolute choice of frame of reference, the guy who sits on Earth is also moving away from the guy on spaceship perspective and hence time on Earth should also dilate when viewed from the guy on spaceship perspective. But why does the guy moving on spaceship look younger in twin paradox? What am I misunderstanding terribly? ADDED:: Is time dilation symmetric? If one frame of reference are moving with constant velocity w.r.t other, we have to transformation relations $$\Delta t = \frac{\Delta t'}{\sqrt{1 - \frac{v^2}{c^2}}}$$ Does each one of them see clock on other tick slowly than their own?


Quantum mechanics, whats possible?

There is a thread in Physicsforums.com which states due to Quantum Mechanics, if you wait long enough diamonds will appear in your pocket, it also states its possible for all your atoms to spontaneously re-arrange themselves so you turn into a Boeing airplane. Surely this is fiction?


I'm confused by renormalization . If a lagrangian has a term with negative mass dimension , why can't the divergences be absorbed into lagrangian coefficients? What's the difference between divergences that can be corrected and those that can't


Why is a degree Celsius exactly the same as a Kelvin?

How on earth is it possible that the difference between two temperatures in Celsius and Kelvin is exactly the same. Given the historical definition of Celsius, I find it hard to believe that this is pure coincidence.


What is the importance of studying degeneration on $M_g$

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study the degeneration of certain functions on $\overline{M_g}$ as you approach its boundary. For which functions on $M_g$ is it interesting to physicists to study their degeneration as you approach the boundary? I know of theta functions, Green functions and delta invariants. Are there others? As in my other question, here is an article by Wentworth which I find interesting.


Where does a star's angular momentum go as its spin slows down?

So we know that stars slow down as they age. But total angular momentum must be conserved. Where does that angular momentum go? The dissipation of Earth's tides _somehow_ transfers Earth's angular momentum to the moon (as shown in my answer at http://www.quora.com/Has-the-Earths- rotational-period-always-been-24-hours-If-not-what-was-it-before-and-what- caused-the-change). But where does a star's dissipation go?


Can you equate the diffusivity constant in random walks with the one in Brownian motion (Einstein relation)?

In an unbiased random walk in one dimension, the coefficient of diffusion is $D = l^2/2\tau$, where $l$ is the size of the jump and $\tau$ is time taken for that jump. In simple Brownian motion, Einstein's relation says that $D = \mu kT$ where $\mu$ is the mobility, $k$ is the Boltzmann constant and $T$ is temperature. In both cases, it turns out: $$ \langle x^2 \rangle = 2Dt$$ So my question is, can we equate the two constants? So, can we say that: $l^2/2\tau = \mu kT$?


Where does energy in a field come from?

Let us consider for example Earth's gravitational field. If we put a ball somewhere in this field, the ball starts to accelerate due to the gravitational force exerted on it. I understand the explanation for this : "Gravitational potential energy (GPE) is converted into kinetic energy". But why is it that this GPE seems to exist only when I introduce the ball into the gravitational field? Am I creating this GPE somewhere? Also, for my second question : If the gravitational field does work in accelerating the ball, then won't it have to lose energy in order to do all this work? Where does all this energy required come from? From all this thinking I come to the conclusion that a field is some sort of wizardy with an infinite amount of energy that can be transferred to objects placed in it. But I am perpetually confused because for all my life, I've been taught that energy cannot be created. So where do all these energies come about, really?


Fermi and Boltzmann distribution of carriers in semiconductor

What are the requirements for describing charge carriers (e.g. electrons) in a semiconducting material by \- Fermi distribution \- Boltzmann distribution When do we apply the one or the other? If the explanation to this question is in Ashcroft/Mermin, reference to the relevant chapter would be appreciated. I don't agree with both commentators so far. * * * Edit Consider Eq. 21 in Sze, PHysics of Semiconductor Devices: $$n = N_C exp\left(-\frac{E_C-E_F}{kT}\right)$$ or Eq. 2.16 in Pierret, Semiconductor device fundamentals.


Ideal gas law, pressure increase and temperature

If I had a container, full with air, and I suddenly decreased the volume of the container, forcing the air into a smaller volume, will it be considered as compression, will it result in an increase in temperature, and why? P.S: the container is strong enough to bear high pressure and not to explode.


How is a cathode ray tube different from beta minus radiation?

In beta minus the result is one neutron in the nucleus changing to a proton, plus an electron and an anti-neutrino being sent off. The antineutrino is indifferent to our health. So I guess what makes a beta source dangerous compared to a cathode ray tube must be a difference in the kinetic energy of the emitted electrons?


Qualitative discussion about entropy and disorder

Many discussions about entropy and disorder use examples of decks of cards, pages of books thrown in the air, two gases being mixed in a container, even the state of a nursery at the end of the day compared to the beginning of the day in order to explain the idea of order (and disorder). In all these examples it is pointed out that the disorder of the system increases, or that the system is in an ordered state and finishes in a disordered state after something has happened. Take the case of throwing the pages of the book in the air. You start with the pages numbered in sequence (I didn't want to use the for "order"), you throw them in the air, they land on the floor, you collect them up and notice the pages are not in sequence anymore. ANd the point is "They are not in the sequence _I_ call ordered. Nonetheless, they are in a new sequence." And, it appears to me that the probability to find the pages in this precise new sequence is equal to find them in the original sequence". In that sense, 'order' seems to be something that us humans define and it doesn't appear to be a property of the system. On the other hand, I can see that in the case of two gases mixing, empirically we find more states where the two types of molecules are occupying the entire volume of a container than one type of molecule in the left side and the other type in the right side of the container. Nonetheless, the precise state of each molecule, its position and therefore the entire state of the mixed up system is qualitative the same, isn't it? Isn't it equally difficult to make each molecule occupy that precise position in the mixed up state as in the unmixed state? Does this make sense? Thanks, Marcio


Action on Lard Oil

If water is mixed with lard oil and heated (creating some _super-critical_ liquid with water), how does this affect the volatility of the mixture in comparison with its _purity_..? So, My question is: What happens when a mixture of lard oil and water is compressed and then heated to over some $\approx800^o\text{C}$ almost instantaneously? **Note:** I'm no physicist or chemist, so simplicity would be appreciated :-)


Difference between spinor and vector field

How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument.


Sign convention in optics

Why is the sign convention used in the derivation of the lens formula and yet used again when it is applied in numerical problems? Won't the whole idea of sign convention be eliminated if it is used twice?


What if exactly half the Earth's population jumped at one instant? + Secondary Question

I read somewhere that when you jump, the sole effect caused by your jump on the earth moves it about $10^{-18}m$ (I don't remember the figure exactly, but I think it was that). However - obviously - with so many people running, jumping, etc. on the whole surface of the earth, the whole effect is canceled out, so it's safe to say there is no net displacement caused by us. This got me to wonder, what if the whole population of one longitudinal hemisphere of the earth jumped at the exact same time - while the other half remained stationary - **_would we be able to notice any discernable displacement in the short time before we all land back to the ground?_** Let's consider the earth to have a perfect distribution of people on both hemispheres, and let's consider the people of the Eastern Hemisphere to jump. Now the people jumping along the Prime Meridian and the International Date Line (these two lines together form the circle that divides the earth into the West and Eastern Hemispheres) would produce no effect, since they would simply cancel each other out. However, the closer you go to $0°N, 90°E$ (the centre of the East Hemisphere), the component of force which adds up will keep increasing. So there will be a net force in the direction of the Western Hemisphere. I was wondering if it would be noticeable. ![enter image description here](http://i.stack.imgur.com/6Pk9u.gif) * * * Another question which just struck me: Out of all the satellites/rockets/space-shuttles we've sent out, have **_they_** caused any displacement on the earth? They are quite massive in comparison to _us_ (not the earth, of course), and have large speeds. Many of them are continuously moving further and further away from the earth - so there's no question of them falling back to earth and thus shifting it to its original position again. Will the momentum be enough to displace the earth?


What bends fabric of space-time?

I know that mass can bend fabric of space-time, which causes gravity by making an object curve around a planet or star but is there anything else that can bend it? Other energy sources, forces anything?


Question about the Dirac notation for partial trace

I saw the following definition for the partial trace operator: > $\rho_A=\sum_k \langle e_k|\rho_{AB}|e_k\rangle$, where $e_k$ is basis for > the state space of system $B$. From what I know, in the Dirac notation, the meaning of $\langle v|A|u\rangle$ is the inner product of the vectors $|v\rangle$ and $A|u\rangle$, so I have two problems with this notation of the partial trace. First, how can an inner product be an operator? An inner product should be a complex number, so I guess that inner product represents an operator somhow. Second, what is the meaning of $\rho_{AB}|e_k\rangle$? $\rho_{AB}$ is a mapping on the space $A\otimes B$, but $e_k$ is a vector from the space $B$. So, I don't really know how to interpret the meaning of this notation.


how is the reference for the star maps in google determined?

If I go to google sky - http://www.google.com/sky/ Ihere is a longitude and latitude measure at the bottom of the page. Iow are these coordinates determined and who sets it? Do those coordinates move at all over time?


Power counting with a cutoff

In Effective Field Theory video lectures found here, the professor explained power counting in effective field theories and the difficulties of power counting associated with loop diagrams. He then mentions that introducing a cutoff ($\Lambda_{UV}$) to regulate our divergences does not preserve power counting due to the new scale that we are introducing. To see this he uses four-fermi theory with the diagram, $\hspace{6cm}$ ![enter image description here](http://i.stack.imgur.com/VwG04.png) We do our power counting (i.e., Taylor expansions) in powers of $m^2/M^2$ and then go on to consider the mass correction through, $\hspace{6cm}$ ![enter image description here](http://i.stack.imgur.com/xkHEw.png) Using a cutoff this gives a mass correction, \begin{align} a\frac{ m }{ M ^2 } \int _0^{\Lambda_{UV}}\frac{ \,d^4k _E }{ (2\pi)^4} \frac{1}{ k _E ^2 + m ^2 } & = a\frac{ m }{ ( 4 \pi ) ^2 } \left[ \frac{ \Lambda _{ UV } ^2 }{ M ^2 } + \frac{ m ^2 }{ M ^2 } \log \frac{ m ^2 }{ \Lambda _{ UV } ^2 } - \frac{ m ^4 }{ M ^2 \Lambda _{ UV } ^2 } + ... \right] \end{align} If I understand correctly this breaks the power counting because even if $\Lambda_{UV} \sim M$, the first term is an order 1 correction since its not proportional to $m^2/M^2$. So far so good. However, then the professor says that you can still use power counting with a cutoff if you fix the power counting order by order and that this can be done by introducing an intermediate scale, $\Lambda$. But I don't how this fixes anything... With an intermediate scale ($\Lambda$) we have, \begin{equation} a\frac{m}{M^2}\int _{ \Lambda } ^{ \Lambda _{ UV }} \frac{ \,d^4k _E }{ (2\pi)^4 } \frac{1}{ k _E ^2 + m ^2 } = \frac{a\,m}{ (4\pi)^2M^2 } \left\\{ \left(\Lambda ^2 + m ^2 \log \frac{ m ^2 }{ \Lambda ^2 + m ^2 } + ... \right) + \left( \Lambda _{ UV } ^2 - \Lambda + m ^2 \log \frac{ \Lambda ^2 + m ^2 }{ \Lambda ^2 _{ UV }} \right) \right\\} \end{equation} but how does this fix anything? For more context see my lecture notes here under Effective Field Theory (starts around equation 4.6)


A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?

I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector: $$\hat{A}=\frac{i}{\hbar}\left[\hat{p},\,\frac{1}{2}\hat{L}^{2}-k\left|\hat{r}\right|\right]=\frac{1}{2}\left(\hat{p}\times\hat{L}-\hat{L}\times\hat{p}\right)+k\frac{\hat{r}}{\left|\hat{r}\right|}.$$ But unlike the case of classical mechanics, even after a lot of tries I found no beautiful/short way to show it's commutation with Hamiltonian, the calculation are quit lengthy opposite to my first expectations, but something tells me that there is a shorter way, so dose anybody managed to find a simple way or trick to show that $$\left[\hat{H},\hat{A}\right]~=~0?$$


Why Counterclockwise starts from East?

In vector lessons, we have counterclockwise angle of rotation of the vector starts from East. "The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its "tail" from due East" http://www.physicsclassroom.com/class/vectors/u3l1a.cfm Why doesn't start from north or south or any other direction? Or is it just one of common rule between physicist.


Kubo Formula for Quantum Hall Effect

I'm trying to understand the Kubo Formula for the electrical conductivity in the context of the Quantum Hall Effect. My problem is that several papers, for instance the famous TKNN (1982) paper, or an elaboration by Kohmoto (1984), write the diagonal entries of the conductivity tensor in the form $$ \sigma_{xy}(\omega \to 0) = \frac{ie^2}{\hbar} \sum_{E^a < E_F < E^b} \frac{\langle a|v_x|b \rangle \langle b|v_y|a \rangle - \langle a|v_y|b \rangle \langle b|v_x|a \rangle}{(E^a - E^b)^2} .$$ This is the static limit $\omega\to 0$ and low temperature $T\to 0$. The sum goes over all eigenstates $|a\rangle$ and $|b\rangle$ of the single-particle Hamiltonian. $E_F$ is the Fermi energy. $v_x$ and $v_y$ are the single- particle velocity operators. However, these papers don't derive this equation, which is unfortunate because the Kubo formula is usually not presented in this form. I have found (and succeeded in rederiving) the following variation instead $$ \sigma_{xy}(\omega+i\eta) = \frac{-ie^2}{V(\omega + i\eta)} \sum_{a,b} f(E^a) \left( \frac{\langle a|v_x|b \rangle \langle b|v_y|a \rangle}{\hbar\omega + i\eta + E^a - E^b} + \frac{\langle a|v_y|b \rangle \langle b|v_x|a \rangle}{-\hbar\omega - i\eta + E^a - E^b} \right).$$ This is formula (13.37) from Ashcroft, Mermin, though they don't actually prove it. $f(E)$ is the Fermi distribution. A nice derivation is given in Czycholl (german). Now, my question is, obviously > How to derive the first formula from the second? I can see that the first equation arises as the linear term when writing the sum as a power series in $\omega$, but why doesn't the constant term diverge?


Do low frequency sounds really carry longer distances?

It is a common belief that low frequencies travel longer distances. Indeed, the bass is really what you hear when the neighbor plays his HiFi loud (Woom Woom). Try asking people around, a lot of them believe that low sounds carry longer distances. But my experience isn't as straightforward. In particular:   * When I stand near someone who's listening loud music in headphones, it is the high pitched sounds that I hear (tchts tchts), not the bass.   * When I sit next to an _unamplified_ turntable (the disc is spinning but the volume is turned off), I hear high pitched sounds (tchts tchts), not the bass. So with very weak sounds, high frequencies seem to travel further? This makes me think that perhaps low frequencies do not carry longer distances, but the very high amplitude of the bass in my neighbor's speakers compensates for that. Perhaps also the low frequencies resonate with the walls of the building? Probably also the medium the sound travels through makes a difference? Or perhaps high frequencies are reflected more by walls than low frequencies? I found this rather cute high school experiment online, which seems to conclude that low and high frequencies travel as far, but aren't there laws that physicist wrote centuries ago about this?


DC current in ideal conductor and skin effect?

I know that the skin depth derived for AC current goes as $$\delta \propto \sqrt{\frac{1}{\omega \sigma}}$$ where $\omega$ is the angular frequency of the field and $\sigma$ is the conductivty. Now: for a conductor with **infinite conductivity** but with **DC current** , we find a $0\times\infty $ in the denominator and I do not know how to treat it. That said: 1) In a conductor with infinite conductivity and DC or AC current, does the current only flow on the surface? 2) What is keeping charge inside a real conductor? Shouldn't they always try to eliminate the electric field inside?


Find a central force given the orbit

I've been trying to solve the following problem for a long time. Let's consider a particle of mass $m$ in $\mathbb{R}^3$ with polar coordinates $(r,\theta,\phi)$. The particle moves on the orbit described by the equation $r=2a\sin\theta,\,\,a>0$, with velocity $|\mathbf{v}|=k/\sin^2\theta,\,\,k>0$. The task is to show that the force is central and to determine such a force. My idea has been to use the two equations given to prove the two components \begin{equation} F_\theta=m\left( r\,\ddot\theta + 2\dot{r}\,\dot\theta - r\,\dot\varphi^2\sin\theta\cos\theta \right)\\\ F_\phi=m\left( r\ddot\varphi\,\sin\theta + 2\dot{r}\,\dot\varphi\,\sin\theta + 2 r\,\dot\theta\,\dot\varphi\,\cos\theta \right) \end{equation} to vanish; unfortunately, I wasn't able to get to any interesting result. Can anybody give me some help on this?


The bar tender says we don't serve Tachyons around here?

![enter image description here](http://i.stack.imgur.com/iEOWe.jpg) What's the physics behind this joke?


Global Chern-Simons forms and topological gauge theories

I am reading the classic Dijkgraaf and Witten paper on topological gauge theories and something struck me that I didn't understand. For a trivial bundle $E$ on smooth 3-manifold $M$ with compact gauge group $G$ the Chern- Simons form is of course $$S(A)=\frac{k}{8\pi^2}\int Tr (A\wedge dA + \frac{2}{3}A\wedge A\wedge A).$$ The confusing sentence is the following: "If the bundle $E$ is not trivial, the formula for the action does not make sense, since a connection on a non- trivial bundle cannot be represented by a Lie algebra valued 1-form as in that formula." My concern is that this is wrong - of course we can represent a connection in a non-trivial bundle with a Lie-algebra valued 1-form! In fact, some texts define the connection in exactly that way (my trusty Choquet-Bruhat and DeWitt-Morette has that as one of their three equivalent definitions for a connection)- locally, a connection is represented as a linear map $\omega_p:T_p(E)\to \mathfrak{g}$. So are we supposed to read the "as in that formula" to mean "globally"? Since of course, if the bundle is not trivial there isn't a _global_ Lie-algebra valued 1-form that we can use for the Chern-Simons action. So that seems like a fine argument, but in QFT we are _always_ writing down actions with sections of bundles (fields $\phi(x)$, say) which are not trivial - we just make sure that the action is invariant under the gauge transformations. So, what is going on with that statement? Is it really saying we cannot represent a connection in such a way, or is it just stating the action is now local, and the rest of the paper is going to go on and tell me how to fix this with an element of $H^3(BG,\mathbb{R})$, etc.?


Path integral on matrix model

I was looking at a 0-dimensional matrix model, where the variables are $N\cdot N$ Hermitean matrices. It had a gauge symmetry, e.g. $U(N)$. And in the path integral, the Faddeev-Popov trick was used. So instead of integrating over the set of matrices, one was reduced to integrating over the $N$ (real) eigenvalues. Or, I tried counting the number of degrees of freedom and I as puzzled:   1. N by N Hermitean matrices have $N^2$ real independent variables ($N(N-1)$ values for the triangular part (non counting the diagonal ones), plus _$N$_ real variables for the diagonal ones, giving us a total of $N^2$   2. $U(N)$ has real dimension $N^2$, classic result from group theory   3. At the end, I arrive at $N^2 - N^2 = 0$ variables instead of $N$ real eigenvalues. I tried seeing it from another point but I can't derive what I'd like to. What I am thinking/doing wrong?


Simultaneous diagonalization of Hamiltonian and momentum operator

I'm looking at a translationally invariant problem with 3 atoms arranged in a circle each with one valence electron capable of tunelling to either of its two neighbors. With a tunelling rate of $-|A|/\hbar$, we have the Hamiltonian $$H = \begin{pmatrix} E_a & \- |A| & \- |A|\\\ \- |A| & E_a & \- |A|\\\ \- |A| & \- |A| & E_a \end{pmatrix}$$ which can be shifted by $+|A|$: $$\begin{pmatrix} E_a + |A| & 0 & 0\\\ 0 & E_a + |A| & 0\\\ 0 & 0 & E_a + |A| \end{pmatrix}.$$ Because of translational symmetry, $p$ is a good quantum number and $[H,\hat{p}]=0$. This means that we can diagonalize $H$ as well as $\hat{p}$ simultaneously and construct momentum eigenstates out of the Hamiltonian eigenstates. How do I do this? **EDIT:** Okay so I've shifted the matrix as follows (simply $-E_a$ on the diagonal): $$\begin{pmatrix} 0 & -|A| & -|A|\\\ -|A| & 0 & -|A|\\\ -|A| & -|A| & 0 \end{pmatrix}.$$ Eigenvalues are $-2|A|$ corresponding to the eigenstate (1,1,1), $|A|$ corresponding to (-1,0,1) and |A| corresponding to (-1,1,0). So $|A|$ is degenerate. Now I need to diagonalize $p$. I guess what my problem is, is that I'm not quite sure of the matrix representation of $p$ here. How do I then diagonalize it, if I can't find its matrix representation? I know each matrix element is $\langle \psi_i | p | \psi_j \rangle$, but I don't know what the wave functions look like. How do I proceed?


Precision of spectroscopy for astronomy

How precise can the measurements be when looking at spectral lines in astrophysics? For example, suppose I have a telescope in orbit, and I am looking at $H_\alpha$ lines coming from a star at 613 pc. Is it realistic to observer the lines move from 626.282 nm to 656.282002854 nm? this would correspond to a radial velocity of about 1.3 m/s (I am not an astronomer, and I just want to get a feel for whether this is realistic to measure, or out of the question)


In a non auto-ranging digital multimeter, what do the increasing numbers in the Ohm section represent?

For my science fair project I am measuring the resistance of nichrome wires under different conditions. I am using an Innova 3300 digital multimeter to measure the resistance. Other websites have told me to move the pointer to the Ohms section. However, there are multiple numbers in that section. What number should I use and how does changing the number affect the results?


What does adding a scalar field component to the Einstein field equations mean for black holes and string theory?

If a scalar field component has to be added to the Einstein field equations (see below) to solve dark matter/energy, then how would string theory need to be modified and do black holes still exist? The proposed modified equations are (ignoring physical constants) $$ R_{ij} - \frac12 R g_{ij} = T_{ij} + \nabla_i\nabla_j \varphi $$ where $\varphi$ represents some kind of scalar potential. The conservation law for energy- momentum is proposed to be $$ \nabla^i(T_{ij} + \nabla_i\nabla_j\varphi) = 0 $$ Refs:   * Dark Matter, Dark Energy, and the Fate of Einstein’s Theory of Gravity http://mathgradblog.williams.edu/dark-matter-dark-energy-fate-einsteins-theory-gravity   * UNIFIED THEORY OF DARK ENERGY AND DARK MATTER, TIAN MA, SHOUHONG WANG http://www.indiana.edu/~fluid/paper/report.pdf


QM -group reps and transforming wavefunctions

QM texts seem to have many ways of motivating the angular momentum operators and deriving the l and m quantum numbers . But the connection between physical rotaions in 3 dim space and an operator in hilbert space that is called " angular momentum" is still somewhat unclear to me I found a set of notes on Lie Groups in Physics by Nobel prize winner Veltman - that seems to be particualry clear explanation of irreducible reps of SO(3) and angluar momentum operatiors see on page 10 of www.phys.uu.nl/~thooft/lectures/lieg07.pdf He begins with discussion of general linear diffrential equations that have a set of solutioins . He then says that when a 3 dim rotation takes place a solution of the DE is then transformed into a linear mixture of other solutions Is this true for all linear diffrential equations -?? or are these type of transformations amoung solutions in response to 3 dim rotations an assumption that is made about wavefunctions or about states in a Hilbert space as part of the assumptions of QM ? THanks Len


Question about streamline in boundary layers

I was wondering what the streamlines look like in a boundary layer. I found a picture that shows this: ![Streamline in boundary layer](http://i.stack.imgur.com/pgnq0.png) I would like to know if this presentation of the streamline is correct. I don't think that it is correct in the turbulent region, but I'm more interested in the laminar region. The streamline seems horizontal to me in that region, so it looks like there are no velocities in the y-direction, so $v_y=0$. I wonder if this is correct, because this seems to contradict the boundary-layer equations in which they use a $v_y$. I feel like that the picture isn't correct, but that the streamlines tend to follow the angle of the dotted line. So I wonder: is that feeling correct and do the streamlines have more or less the same slope as the dotted line? I also feel that the more we go in the y-direction the more the tendency to follow the dotted line becomes, so close to the plate they are horizontal, can anyone confirm or contradict this?


Is there a difference in handwritten nabla $\vec{\nabla}$ with an overset arrow and typeset nabla $\nabla$?

According to some physicist at KIT it is usual to write the following when using pen and paper: ![enter image description here](http://i.stack.imgur.com/cDBVO.png) whereas in typeset texts you write $\nabla$. Is that true? Are there sources for this convention?


What would happen to wet laundry in the vacuum of space?

Imagine for a second that I was in a college dorm room frustrated that all the dryers were broken or in use. I'm impatient and not a jerk so I don't wait or take out someone else's laundry, I shoot it into space with the ability to get it back and land it successfully on my campus. Ignoring practicality of course: How would my clothes react? Would the water in them freeze? Would the water just dissipate into the rest of space? Would it boil if it were in the light of the sun? I'm curious about three cases: One, where the clothes are in a general vacuum regardless of the effect of the sun. Two, where my clothes are specifically behind the Earth and so they are eclipsed by the Earth. Three, where my clothes are specifically between the Earth and the sun. For those willing to make the argument that this is a silly topic: would it or would it not have implications with space suits? Thanks!


Resonance Raman spectroscopy vs fluorescence

In Resonance Raman Spectroscopy we often want to avoid the fluorescence. But what is the problem with fluorescence . What we want is a shift between exciting line and emitted radiation and both can show the vibration energy difference?? also from "What is the difference between Raman scattering and fluorescence?" I come to know the difference between Raman and fluorescence is that of lifetime of the molecule in the excited state, but how does a molecule come to know that it is studied for Raman spectroscopy so its lifetime in excited state is small or it is studied for fluorescence.


Is there a general law for physics?

Here is why I pose this question, as I advance in physics more and more this idea haunts me and in fact helped me figure out and memorize many physical laws. It seems to me that there is a relation between all the laws of physics, something that is a general law that determines how should all other laws be _generally_ stated. I can't express this law in a complete sentence and I'm not able to verify if it has an exception or not but I will try to explain this by examples to show what I really mean: ## Mechanics: Oscillations: It seems that the whole process happens because the object is trying to get back into its **initial state** i.e the origin. ![oscillator](http://i.stack.imgur.com/I7qc4.png) ## Electricity: Lenz's law: An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux. in other words: the current is trying to get its **initial state** by resisting the change of current. ## Atoms Radiation and Nuclear reactions happen because the atom is trying to be **stable.** Of course there are many other examples but I'm just stating few to make my point. The question is: is there something like that in physics? if not can you give me any counter example on this?


How to integrate over rapidity for Parton Luminosities in LHC?

I want to make a comparison of parton luminosities between Tevatron and LHC. According to Factorization theorem the cross section in hadron colliders, as long as the partonic cross section has $\hat \sigma \sim \frac{1}{\hat s}$ dependence, could be written like that : $$\sigma =\sum_{12}\int dx_1 dx_2 f_1 \left(x_1,\mu_F^2\right) f_2\left(x_2,\mu_F^2 \right) \hat{\sigma}_{12} \left(x_1 x_2 s,...\right)$$ here $f_{i}(x_{i},\mu_F^2)$ represents the parton distribution function for $x_i$ fraction of the incoming hadron, $i$ represents the parton flavor and $\mu_F$ is the factorization scale. I wanted to reproduce the Figure 4.4 (page 82) in book Physics at The Terascale or the Figures in LHC Physics Potential vs. Energy. **My Questions?** I appreciate for any reply.   1. When I am scanning over the $\sqrt{\hat s }$ (partonic-center-of-mass energy) what value should I set the factorization scale? $\mu_F = \sqrt{\hat s } $ or $\mu_F = \sqrt{s}$ (center-of-mass energy)?   2. How could I integrate over the rapidity? What I mean is on page 82 it says that "The Luminosity $dL_{ij}/d\hat s$ integrated over y is shown in Figure 4.4a" and I don't understand what changes after integration over y. In Equation 4.11 the parton distribution functions are not function of rapidity thus I could simply move $dy$ to the right side and carry out the integration and it will contribute as a factor (though I don't know the bounds $-\infty<y<\infty$ or something specific for the collider $-5<y<5$ ) however there is relation between the rapidity($y$) and $x$ like this $y=\ln(x1/x2)/2$. Thus I don't understand that line?   3. If I just ignore the rapidity and use the Equation (1) in "LHC Physics Potential vs. Energy" I got some results but I am still off compared to the Figure 1 in the same paper, so I concluded that I don't know how the integration over rapidity affects the over all Parton Luminosity. Equation 4.11 $$\frac{dL_{ij}}{d \hat{s} dy}=\frac{1}{s}\left(x_1,\mu\right) f_2\left(x_2,\mu \right)$$


Harmonic oscillator with light damping

My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$ x = Ae^{- \gamma t} \cos (\omega \, t)$$ for $\gamma = \dfrac b {2m}$. It says this implies the amplitude follows the relationship: $$x_{max} = Ae^{-\gamma t}$$ I am not understanding where this second equation comes from. Why does the cosine term disappear if $\omega$ is a constant?