title: QCD phase diagram in the large N limit

text: I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying, so here goes: I am wondering what is known about QCD, or other field theories, in the regime of large density and low temperatures, specifically studied in the large N limit. I know of the qualitative picture at finite N, but lots of the instabilities (e.g. the superconducting ones) are suppressed at large N and replaced by other interesting phenomena. I am only aware at the moment of the DGR instability to form chiral density waves, and I am wondering what else exists in the vast and possibly quite old literature. Any pointers or entry points to that literature will be appreciated


title: How does a lower center of gravity affect fuel economy?

text: It's been a couple years since I've taken a physics class, and I have been wondering about this basically since I bought my car: I drive a 2010 Honda Fit, which has a street weight of approximately 2489 pounds (1129 kg). One feature of the Fit is that the back seats can sit upright like a normal car, or they can completely fold down (this image from this page shows a folded seat next to an upright seat). Since I rarely have more than one passenger in my car, I can leave the rear seats folded down almost all the time. Having the seats folded down will result in a lower center of gravity which should increase my fuel efficiency (I think!). I haven't been able to find an estimate on the weight of the rear seats, other than "pretty substantial" according to this forum post. But I'm not sure how much of a difference it will make overall, anyway. The first part of my question is, assuming that raising or lowering the center of gravity affects acceleration and deceleration, how does one quantify that? Then, given that I may be able to move, say, 100 pounds of seat from the "dead center" of my vehicle to, say, 1-2 feet closer to the wheels, how can I estimate how that will affect my fuel economy? Perhaps a more specific question would be, how can I estimate how that will affect my acceleration? (which is of course when I get the worst fuel economy, besides being stopped)


title: Matrix models and condensed matter physics

text: I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying. This one is also not terribly focused, but nonetheless here goes: I am wondering if there are some well-known and well-studied examples of large N matrix models (in which the fields are adjoint rather than vectors) which are of use in describing some condensed matter phenomena. There are lots of applications of matrix models in anything between nuclear physics to number theory, and there are well-known vector models which are useful in CM physics, but off the top of my head I cannot think about matrix models which are used to solve some condensed matter problems. Quite possibly I am missing something obvious...References or brief descriptions will be appreciated.


title: How does Cooper pairing work?

text: Cooper pairs are one of the models how superconductivity is explained. What still baffles me is how a vibration of the crystal lattice (the so-called phonon) can interact with the electron (an actual particle), in such a way that it then creates a coupled pair with an other electron... What is the explanation for this behaviour? What is the maths behind it?


title: Change of variable in harmonic oscillator time independent Schrodinger equation

text: I was revising the harmonic oscillator for my intro to quantum course and realised I'd sort of accepted a change of variable result without actually being able to get to it. It says: The stationary state Schrodinger equation of energy $E$ is $$-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\psi=E\psi\tag{5.2}$$ The first thing to do is to redefine variables so as to remove the various physical constants: $$\epsilon=\frac{2E}{\hbar\omega}, \xi=\sqrt{\frac{m\omega}{\hbar}}x$$ so that (5.2) becomes $$-\frac{d^{2}\chi}{d\xi^{2}}+\xi^{2}\chi=\epsilon\chi$$ where $$\psi(x)=\chi(\xi)=\chi(\sqrt{\frac{m\omega}{\hbar}}x).$$ So, I've tried working with the algebra but can't seem to get to this. I'm probably missing something really obvious, but it's getting quite frustrating! Can anyone help?


title: Making pink noise (1/f) using list of frequencies

text: I would like to see what type of noise I would get if I used just the frequencies in my voice. I created a matlab array using `fft` to get a `[frequency,amplitude,phase]` array of my voice. I then to reproduced my vocal signal using $A*\cos(ft+\phi)$ I would like to take this file/data and use it to create pink noise $1/f$. When I use $1/f$ for the frequency the numbers become very small. Does anyone have any ideas how to use my own vocal frequencies that I get from doing a fft in matlab to create $1/f$ pink noise?


title: If the axis of rotation is fixed, is it ok to say clockwise torque?

text: I know that the direction of torque is along the axis of rotation, but would it be acceptable to say, for example considering a vertical thin rod in the x-y plane with a force acting on the bottom end towards the left, 'it is a clockwise torque about the centre of mass' (even though strictly speaking the torque's direction is into the page?) It just seems to me easier to describe what's happening that way.


title: Wick's Theorem examples

text: Does anyone know of websites or texts that have an abundance of examples of computing time-ordered products of fields using Wick's Theorem for both bosons and fermions? I'm not just talking about the simple product of 4 fields, something such as $T(\phi^4(x) \phi^4(y))$ $T(:\phi^4(x): :\phi^4(y):)$ $T(\bar{\psi}(x)\psi(x)\bar{\psi}(y)\psi(y))$. These are actually the only examples I have and they're found in "Problem Book in Quantum Field Theory" by Voja Radovanovic.


title: Is it possible to create artificial gravity by magnetizing iron in the blood stream?

text: I have been thinking is it possible that using a rather strong magnet on the human body's blood stream we could enact a small case of artificial gravity on the human body via the iron in the hemoglobin in the blood stream. There are 4 Problems currently with this theory:   1. The could be residual effect on the human body    2. No one has ever tried to affect the blood stream directly using a powerful magnet   3. Can we create an electromagnet or a non normal magnet to cause slight magnetization but only of the feet and legs to ensure no abnormal strain is placed on the heart   4. If the iron in the blood is magnetized so will every other magnetic metal item on board such as tools or weapons


title: Lorentz Transformation

text: I have specific questions about Lorentz transformations specifically about length contraction.   1. Why does length contraction only occur in the direction of travel, (not in all directions) when approaching the speed of light?   2. From an outside observer's perspective watching an object traveling close to the speed of light, why does the obsever notice the object contract? These Ideas are accepted by physicists, but for 1 I don't understand how they know that length contraction is in only one direction without they themselves being in the reference frame of someone or something traveling close to the speed of light. For 2 Let's say that I'm driving in a car 5 meters long and I'm driving on a road that has white markers on it. Where the markers are in one line and each marker is 5 meters apart from the next one and the line of markers is parallel to my direction of travel. At normal speeds I would observe that at a given moment one marker would line up with back of the car and the next marker would line up with the front of the car (the car occupies 1 marker space). So now if I were to travel at speed such that my Lorentz factor is 2. I see one of two cases. In the first case I don't observe any change in myself or my car and I observe a length contraction of 2. Meaning that I would see a change in my environment not in myself or my car. So I would now observe that at a given moment the car would take up the length of 2 marker spaces. Therefore an outside observer would then see the car **expanded** by a factor of 2 not contracted. In the second case I do notice a change in myself. I contract with the space so things look very wierd in my perspective, Everything has contracted by a factor of 2 in the direction of travel. So my arms, my legs, the carseat, the hood of the car, etc. has all shrunk by a factor of 2. However because the road has shrunk by the same amount as the car, I take up only one marker space. Therefore an outside observer would not see any change in the length of me or the car. So in both cases the car doesn't contract from an outside observer's perspective.


title: Why does an ice cube not twirl along when twirling a drink glass?

text: When you have a drink with an ice cube and twirl the glass, the liquid itself seems to twirl but the ice cube stays roughly in the same place. Why is this?


title: Small object in air

text: What happens with very small spherical objects ($d=1\mu m$, e.g. a bacterium) in air? Do they fall? How quickly? Does it depend on their mass? We often see objects of little mass e.g. leaves falling from trees, but these object ususally have a very large surface, so their behaviour is very different to spherical objects. _Disclaimer: I'm a biologist._


title: Expressions of action and energy momentum tensor in bc conformal field with central charge equals one

text: I have a question with conformal field theory in Polchinski's string theory vol 1 p. 51. For $bc$ conformal field theory $$ S=\frac{1}{2\pi} \int d^2 z b \bar{\partial} c $$ $$ T(z)= :(\partial b) c: - \lambda \partial (: bc:) $$ with central charge $c=-3 (2 \lambda-1)^2 +1 =1$. Introducing $\psi$ and $\bar{\psi}$ to replace the anticommuting fields $b$ and $c$ as following $$b \rightarrow \psi =2^{-1/2} (\psi_1 + i \psi_2 )$$ and $$ c \rightarrow \bar{\psi} =2^{-1/2} (\psi_1 - i \psi_2)$$ It is claimed that $$S=\frac{1}{4\pi} \int d^2 z \psi_1 \bar{\partial} \psi_1 + \psi_2\bar{\partial} \psi_2 (2.5.18b)$$ $$ T=- \frac{1}{2} \psi_1 \partial \psi_1 -\frac{1}{2} \psi_2 \partial \psi_2 (2.5.18c)$$ I cannot obtain the above expressions of $S$ and $T$. Here is my derivations. First I try to recover the anti-commuting characters of fields $b$ and $c$ by $\psi_1$ and $\psi_2$. For $$bc+cb=0$$ I have $$\psi_1 \psi_1 + \psi_2 \psi_2 =0$$ Then for the action $$S=\frac{1}{4\pi} \int d^2 z \left( \psi_1 \bar{\partial} \psi_1 + i \psi_2 \bar{\partial} \psi_1 -i \psi_1 \bar{\partial} \psi_2 + \psi_2 \bar{\partial} \psi_2 \right)$$ (1) [Solved] Why the term $i \psi_2 \bar{\partial} \psi_1 -i \psi_1 \bar{\partial} \psi_2$ does not contribute to the action? (2) How to derive (2.5.18c)?


title: Why do these equations result an incorrect unit for acceleration?

text: ![](http://i.stack.imgur.com/BB7m9.jpg) Hello everyone. Imagine an object moving around a certain point on a circular orbit. Magnitude of the velocity is constant during the motion ($|v|$). The orbit radius is $r$. (I'd better notice that we're just talking about **kinematic** view of this motion.) According to the image I've uploaded, we'll have: $\large v_x(\theta)=|v|\cdot \cos\theta$ $\large v_y(\theta)=|v|\cdot \sin\theta$ Since perimeter of the circular path is $2\pi r$, and magnitude of the velocity is constant, we'll have: $\large\theta (t)=\frac{|v|\cdot t}{2\pi r} \times 2\pi =\frac{|v|\cdot t}{ r}$ Now we can combine these equations: $\large v_x(\theta)=|v|\cdot \cos(\frac{|v|\cdot t}{ r})$ $\large v_y(\theta)=|v|\cdot \sin(\frac{|v|\cdot t}{ r})$ By this point, everything is okay. But the problem occurs here, where we try to get derivative of $v_x(t)$ and $v_y(t)$ in order to find $a_x(t)$ and $a_y(t)$. As we know by differentiation we have: $\cos^{\prime}(x)=-\sin(x)$ $\sin^{\prime}(x)=\cos(x)$ And we know that acceleration(time) function is derivative of velocity(time). So: $\large a_x(t)=(v_x(\theta))'=|v|\cdot -\sin(\frac{|v|\cdot t}{ r})$ $\large a_y(t)=(v_y(\theta))'=|v|\cdot \cos(\frac{|v|\cdot t}{ r})$ Well, now something is wrong: These two equations result a $m/s$ unit (or something like that) for acceleration, but that's wrong. Acceleration unit must be $m/s^2$, (or something like that). The question is that: **Where does this problem come from?** I couldn't figure it out at all. I don't know, maybe some kind of misunderstanding about derivative concepts cause that. So please try to answer simple, clean as much as possible.


title: How can dQ/T be interpreted as a system's level of disorder?

text: Long before statistical mechanics, entropy was introduced as: $dS = \frac{dQ}{T}$ At the time when entropy was introduced in this manner, was it known that entropy represents how "disordered" a system is? If so, how can one tell that entropy represents the disorder of a system from the above definition? If however it wasn't known that $S$ was related to the disorder of a system, what was the interpretation of $S$ at the time of formulation of the above formula?


title: Spin about an arbitrary axis

text: This is based off question 4.30 from Griffith's _Introduction to Quanum Mechanics_. It asks for the matrix $\textbf{S}_r$ representing the component of spin angular momentum about an axis defined by: $$r = \sin{\theta}\cos{\phi}\hat{i}+\sin\theta\sin\phi\hat{j}+\cos\theta\hat{k}$$ for a spin = $1/2$ particle.The problem is, I can't visualize how the spin vectors relate to spatial coordinates. $$\chi_+ = \begin{pmatrix} 1 \\\ 0\end{pmatrix}\ \text{ and } \chi_- = \begin{pmatrix} 0 \\\ 1 \end{pmatrix}. $$ So these form a basis of some kind of space, but this is a space I don't understand, for example, why does: $$\chi^{(x)}_+ = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\\1 \end{pmatrix} ?$$ I was going to try to use a rotation matrix with angles corresponding to my axis of rotation to try and "rotate" the spin vector onto it, but then I realized that not only do the dimensions not match up, but the angle is π/4, which doesn't make much sense to me. So I guess my question is, how does the 'geometry' of spin work and how can I transform the spins corresponding to a transform I do in space? Thanks.


title: Restriction on vector fields

text: The 2D vector field (x,-y) does not transform like a vector under rotation(Arfken Vol. 1)! Does this mean we cannot have such a vector field physically?


title: Does gravitational redshift imply gravitation time dilation?

text: The EEP is used to justify that if an observer on the ground shoots a beam of light towards a tower, then when the light reaches the tower, it will be red shifted. This is because of what happens in an accelerating spaceship. The books seem to say this implies time dilation, but I don't completely see why. Could it not just be like any other doppler effect? If I send a sound wave towards you and you are moving towards me, the frequency you observe will be greater than what I send out, but that doesn't mean your clock ticks slower (assume his speed is non-relativistic). Why does it necessarily imply time dilation? In the derivation used to derive this redshift, from the light beam on a spaceship, special relativity doesn't even seem to come into play, just newtonian physics is used. Anyway if the physics on the ground is indistinguishable from the physics in the spaceship, would that not mean that at the back of the spaceship a clock ticks slower, rather than just appearing to click slower, relative to someone at the front? Otherwise it would seem to me that the EEP would just say that we can't do any experiment to distinguish the physics between the rocket and the gravitational field, though the physics wouldn't really seem to be the same.


title: Wavefunction's inner product

text: When two wavefunctions are orthogonal we can write that $$\langle\Psi_n|\Psi_m\rangle=\delta_{mn}$$ This means that $$\langle\Psi_1|\Psi_2\rangle=\langle\Psi_2|\Psi_1\rangle=0$$ But if the two wavefunctions aren't orthogonal but $\langle\Psi_1|\Psi_2\rangle=d \in\mathbb{R}$ is real, can we then conclude that $\langle\Psi_2|\Psi_1\rangle=\langle\Psi_1|\Psi_2\rangle$?


title: Ohmic and Non-Ohmic devices

text: Why do some conductors follow Ohm's law and some do not? Isn't there any universal law that can explain the flow of current?


title: What is the Willmore energy of the Earth (or the geoid)?

text: Wikipedia defines the Willmore energy as: $$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$ where $H$ stands for the mean curvature of the manifold $\mathcal{M}$. What is the Willmore energy of the Earth, or the geoid?


title: Finding the coffecient of restitution

text: > A ball moving with velocity $1 \hat i \ ms^{-1}$ and collides with a > friction less wall, afetr collision the velocity of ball becomes $1/2 \hat j > \ ms^{-1}$. Find the coefficient of restitution between wall and ball. I approached it like: ![enter preformatted text here](http://i.stack.imgur.com/ANFIk.png) Now, $$e=cot^2\theta$$ but $\theta$ is not known. So,We equate velocity $$\sqrt{e^2 sin^2\theta+cos^2 \theta}=1/2$$ But this is hard to solve as $\cot^4 \theta$ get's involved. Is there any other method to do this or any easy method to solve these?


title: Gravity on flat object

text: I was wondering how gravity would behave on object of different shapes.   1. If the Earth was squeezed into a thin disk what would the gravitional acceleration be at the center of the flat surface? Would it be really low because the amount of matter beneath me would be small? If I stood on the edge would the gravitational acceleration be enormous because the amount of matter beneath me was huge?   2. If an object is lowered into the Mariana trench will the effect of gravity increase because it gets closer to the center of the Earth?


title: Debye Hückel Theory

text: In the derivation of the Poisson-Boltzmann equation, my textbook arrives at the expression $$ \rho_i = c_+ z_+ F + c_-z_-F = c_+^oz_+^oFe^{-z_+e\phi_i/kT} + c_-^oz_- Fe^{-z_-e\phi_i/kT} $$ where $c_\pm$ is the concentration (of positive or negative ions), $z_\pm$ is the valence of the ions, $F$ is the Faraday constant ($=eN_A$) and $c_\pm^o$ is the concentration in the bulk. The rest is standard notation. The book now writes that > Because the average electrostatic interaction energy is small compared with > $kT$, we may write $$ \rho_i = (c_+^o z_+ + c_-^oz_-)F - (c_+^oz_+^2 + c_-^oz_-^2)\frac{F^2\phi_i}{kN_AT} + \cdots $$ **How do they know that $ze\phi \ll kT$?** To make an estimate, I find the electric potential of a proton (or an $\text{Na}^+$ ion) in vacuum at 1 nm away. $$ V^{vac} = \frac{1}{4\pi\epsilon_0}\frac{+1e}{1nm} \approx 1.43 \text{V}. $$ In water ($\epsilon = \epsilon_0\epsilon_r = \epsilon_078$), $$ V^{sol} = \frac{1}{4\pi78}\frac{+1e}{1nm} \approx 0.02 \text{V}. $$ So in solution a $\text{Cl}^-$ ion would have the energy $$ w=(-1e)V^{sol} \approx -3.2 \cdot 10^{-21}J $$ while at room temperature $kT \approx 3.9 \cdot 10^{-21}J$. Is this roughly the idea behind the above assumption?


title: Schrödinger equation for a harmonic oscillator

text: I have came across this equation for quantum harmonic oscillator $$ W \psi = - \frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi $$ which is often remodelled by defining a new variable $\varepsilon = \sqrt{m\omega/\hbar}\,x$. If i plug this in the equation above I know how to derive this equation (it is easy all i needed was a definition of differential): $$ \frac{d^2 \psi}{d x^2} + \left(\frac{W}{\left( \tfrac{\hbar \omega}{2} \right)} - \varepsilon^2 \right)\, \psi = 0 $$ From this equation most of authors derive the energy equation: $$W = \left(\tfrac{\hbar \omega}{2}\right)\cdot (\underbrace{2n+1}_{odd???})$$ **QUESTION 1:** I don't quite understand how this is done. Could anyone please explain this? I dont know why energy has to be odd function nor how this comes into play. **QUESTION 2:** I am only guessing that if I would plug this in an equation above I would get I an equation below from which I could calculate possible $\psi$ functions for harmonic oscilator (please confirm).: $$ \frac{d^2 \psi}{d x^2} + \left( (2n +1)- \varepsilon^2 \right)\, \psi = 0 $$ **QUESTION 3:** Where do Hermitean polinoms (which are often mentioned) come into play?


title: Is there a center of the Universe if the Universe is finite?

text: Ok, let me make myself clear. I saw all the other questions related to the question, but none of them actually asks the question the way I would put it and therefore no one answers it the way I want it answered, so here it is, I'll try to formulate it. I perfectly understand that every single point in the Universe may be taken as the "location of the Big Bang", as it happened, because everything was in the very same place - the singularity (zero volume and infinite mass), in terms of 3-dimensional space. As far as I understand, once all this energy was released and the Big Bang occurred, it created and started expanding what we now perceive as the 3-dimensional space of our Universe. And as far as I understand it was expanding (and still is) equally in all directions. In **Wikipedia** the model given is that of a bread muffin, so that we see that all reference points expand equally, given a relative center, no matter where you are in the Universe. However, this is not true if you are close to the outer surface of the muffin, and the muffin has finite size and has an edge, where the Universe ends. Given the example with the muffin, with finite Universe, there is a single point that never formed a vector of movement, ever since the Big Bang took place. And this is the center of the muffin. **That would be the center of the Universe, given the possibility that it is finite.** However, if the Universe has no edge and no boundaries and is infinite in terms of 3-dimensional space, there would be no center in that space and I would be perfectly happy :) But if it is finite, as I described above, it would have a center. Or am I wrong? Please do not mark my question as duplicate, since no one puts it that way in the other questions, I've made my research :)


title: Black Hole Singularities

text: If two black holes collide and then evaporate, do they leave behind two naked sigularities ore? If there are two, can we know how they interact?


title: Virtual photon counting

text: How to calculate number of exchanged virtual pohotons per unit of time between two electromagnetically interacting objects?


title: How does this paper relate to standard QED?

text: This paper proposes a microscopic mechanism for generating the values of $c, \epsilon_0, \mu_0$. They state that their vacuum is assumed to contain ephemeral (meaning existing within the limits of the HUP) fermion/antifermion pairs. This affects the mechanism of photon propagation as follows: > When a real photon propagates in vacuum, it interacts with and is > temporarily captured by an ephemeral pair. As soon as the pair disappears, > it releases the photon to its initial energy and momentum state. The photon > continues to propagate with an infinite bare velocity. Then the photon > interacts again with another ephemeral pair and so on. The delay on the > photon propagation produced by these successive interactions implies a > renormalisation of this bare velocity to a finite value Now this description of what happens to the photon sounds awfully like a heuristic description you might give to the photon self energy contribution in standard QED perturbation theory, where, although it's used in the renormalization procedure, it most certainly doesn't make any changes to the velocity of light. Can someone explain how what's being proposed here relates to standard QED. I don't see how they fit together.


title: Wilson Loops in Chern-Simons theory with non-compact gauge groups

text: VEVs of Wilson loops in Chern-Simons theory with compact gauge groups give us colored Jones, HOMFLY and Kauffman polynomials. I have not seen the computation for Wilson loops in Chern-Simons theory with non-compact gauge groups. I think that what keep us from computing them is due to infinite dimensional representations of non-compact gauge groups. Are there any references which describe this problem explicitly? What are the issues in computing Wilson loops in non-compact gauge groups? Is there any proposal to calculate them? Especially I am interested in the simplest cases of $SL(2,R)$ and $SL(2,C)$.


title: Help deriving the general linear wave equation $d^2y/dx^2=(1/v^2)d^2y/dt^2$

text: How do I derive the General Linear Wave Equation $$d^2y/dx^2=(1/v^2)d^2y/dt^2?$$ My teacher differentiated the general wave function $f(x + vt)+g(x - vt)$ twice with respect to both variables to get $d^2y/dx^2$ and $v^2d^2y/dt^2$, and then somehow combined them to get $$d^2y/dx^2=(1/v^2)d^2y/dt^2,$$ but I don't remember how he did it.


title: Which arguments for $m_u \approx 0$ are still in the market?

text: The RPP note on quarks masses has traditionally carried, and it is still there, the comment that > It is particularly important to determine the quark mass ratio mu/md, since > there is no strong CP problem if $m_u$ = 0. But in recent versions they have also added some remarks about how the calculations from the MILC and RBC collaborations show that this mass is no zero. So, does it still survive some argument for $m_u=0$? And, by the way, does the strong CP problem request exactly 0, or is it enough if it is approximately zero?


title: What is a simple intuitive way to see the relation between imaginary time (periodic) and temperature relation?

text: I guess I never had a proper physical intuition on, for example, the "KMS condition". I have an undergraduate student who studies calculation of Hawking temperature using the Euclidean path integral technique, and shamefully his teacher is not able to give him a simple, intuitive argument for it. What would it be? Added on October 21: First of all, thanks Moshe and S Huntman for answers. My question was, however, looking for more "intuitive" answer. As Moshe pointed out it may not be possible, since after all time is "imaginary" in this case. So, let me be more specific, risking my reputation. I should first say I understand there are formal relation between QFT and statistical mechanics as in well-known review like "Fulling & Ruijsenaars". But, when you try to explain this to students with less formal knowledge, it sometimes helps if we have an explicit examples. My motivation originally comes from "Srinivasan & Padmanabhan". In there, they says tunneling probability calculation using complex path (which is essentially a calculation of semi-classical kernel of propagator) can give a temperature interpretation because "In a system with a temperature $\beta^{-1}$ the absorption and emission probabilities are related by P[emission] = $\exp(-\beta E)$P[absorption]. (2.22) " So, I was wondering whether there is a nice simple example that shows semi- classical Kernel really represents temperature. I think I probably envisioned something like two state atom in photon field in thermal equilibrium, then somehow calculate in-in Kernel from |1> to |1> with real time and somehow tie this into distribution of photon that depends on temperature. I guess what I look for was a simple example to a question by Feynman $ Hibbs (p 296) pondering about the possibility of deriving partition function of statistical mechanics from real time path integral formalism.


title: Supersymmetry in Quantum Mechanics (Does it apply?)

text: Suppose we try to apply supersymmetry in quantum mechanics to a particular potential. If you come up with two partner potentials, and two partner Hamiltonians, and then look at the energy of the ground state, is it true that at most one of them can be zero? If they are both zero, does that mean that supersymmetry is just not preserved by the vacuum states of the partner Hamiltonians? Can we still apply supersymmetry to this system if the energy of both ground states are zero?


title: Can one define an acceleration operator in quantum mechanics?

text: It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator? I thought about doing it in the following way, starting from the definition of the momentum operator: $\hat{p} = -i\hbar \frac{\partial }{\partial x}$ Then we define a velocity operator in analogy to classical mechanics by dividing momentum by the mass $m$ $\hat{v} = \frac{-i\hbar}{m} \frac{\partial }{\partial x}$ In classical mechanics acceleration is defined as the time derivative of the velocity, so my guess for an acceleration operator in QM would be $\hat{a} = \frac{-i\hbar}{m} \frac{\partial }{\partial t} \frac{\partial }{\partial x}$ Is that the general correct definition of the acceleration operator in QM? How about relativistic quantum mechanics?


title: Vectors and motion in a plane

text: > A particle travels with speed $50 m/s$ from the point $(3,-7)$ in the > direction $7i-24j$ . Find its positional vector after 3 seconds. **My approach** : It has travelled a distance of 150m in the direction given by the unit vector $\frac{7\hat i - 24 \hat j}{25} $ . So, its position is now $210\hat i -720 \hat j$. But it started from $(3, - 7)$, so I have to subtract that to get $197\hat i - 713\hat j$. But I think this answer is awkward, and I may have gone wrong. So, please tell me if I am right, and if yes, can you suggest any shorter way of doing it?


title: Why do ice cubes make a cracking sound when placed in fizzy wine (Prosecco)?

text: When placing ice cubes in a fizzy drink such as Prosecco, ice makes a cracking sound, after which the fizzy bubbles more than usual. What is the physics of this phenomenon? ![enter image description here](http://i.stack.imgur.com/hKlxy.jpg)


title: Relationship of multiple particles under collision

text: Consider 3 particles. All 3 particles travel along the x-axis.   * The 1st particle possesses some mass, m, and its initial position is somewhere on the negative x-axis. It has some (positive) velocity v.   * The 2nd particle possesses some mass, A*m, and its initial position is at the origin of the x-axis (0). It has no velocity (at rest).   * The 3rd particle possesses some mass, B*m, and is initially situated somewhere on the positive x-axis. Like particle 2 it has no velocity (at rest). Find the relation between A and B, under which the 1st and 2nd particles will collide with each other more than once. note: it is assumed the collisions are elastic as well as there being no external forces considered. also the collisions are such that all particles remain on the x-axis


title: Having trouble weighing the sun

text: So I gather the way you (and Vera Rubin) calculate a galaxy's mass is by measuring a star's orbital velocity $v$ and its distance $R$ from the galactic center, and then plugging them into this equation derived from Newton's second law: $$M_{gal}=Rv^2/G$$ ($G=6.67\times10^{-11}$. Units of $v$ and $R$ are km/sec. and km., respectively) The value obtained for $M_{gal}$ this way famously disagrees with the value you would obtain by measuring the brightness of the galaxy, leading to the dark matter theory. I was playing around with some solar system data and found that if I calculate the mass of the sun by plugging this data into the equation above the result is too low by 9 orders of magnitude. $\sim 1.98 \times 10^{21}$ kgs. instead of the actual solar mass (according to wikipedia) of $1.98 \times 10^{30}$ kgs. Is there some "dark vacuum" in the solar system? Or where have I gone wrong in the calculation? Are there any datasets of $v$ and $R$ for stars in a particular galaxy out there that can be downloaded? I've heard that thousands of galaxies have now been observed to have stars "going too fast." Has any of that data been made available?


title: Noether's theorem and time-dependent Lagrangians

text: Noether's theorem says that if the following transformation is a symmetry of the Lagrangian $t \to t + \epsilon T$ $q \to q + \epsilon Q$ Then the following quantity is conserved $\left( \frac{\partial L}{\partial \dot{q}}\dot{q} - L \right) T - \frac{\partial L}{\partial \dot{q}} Q$. Suppose our Lagrangian is given by $ L = \frac{1}{2}m \dot{q}^2 - \ln t$ Then is not the Lagrangian invariant under the transformation given by $T = t$ $Q = 0$? Making this transformation contributes only an additive constant to the Lagrangian, which will not affect the dynamics, and so we should conclude that such a transformation is indeed a symmetry of the Lagrangian. However, the quantity $ \left( \frac{\partial L}{\partial \dot{q}}\dot{q} - L \right) t = \left(\frac{1}{2}m\dot{q}^2 + \ln t \right)t$ is clearly not conserved. The E-L equations imply that the kinetic energy is constant, and so this function is clearly an increasing function of time. Where is my error? Thank you.


title: Polchinski equation 4.3.16

text: I am trying to obtain the polchinski's equation 4.3.16 which is following $ Q_B^2 = \frac{1}{2}\\{ Q_B, Q_B \\} = -\frac{1}{2}g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M =0$ Where $ Q_B = C^I(G_I^m +\frac{1}{2}G_I^g)$ and $C^I$, $b^J$ are anticommuting(ghosts) and $[G_I, G_J]=ig^K_{IJ} G_K$,   $G_I^g = -ig^K_{IJ} C^Jb_K $ are ghost parts and $G_I^m$ are matter part and they satisfy above commutation relations What I have done are $\\{ Q_B, Q_B \\} = \\{ C^I(G_I^m +\frac{1}{2}G_I^g), C^J(G_J^m +\frac{1}{2}G_J^g)\\} =\\{C^IG_I^m, C^J G_J^m\\} +\frac{1}{2} \\{C^IG_I^m, C^JG_J^g\\} +\frac{1}{2} \\{C^I G_I^g, C^JG_J^m\\} +\frac{1}{4} \\{C^IG_I^g, C^JG_J^g \\} = C^IC^J [G_I^m, G_J^m ] +\frac{1}{2} C^IC^J [G_I^m, G_J^g] +\frac{1}{2}C^IC^J[G_I^g,G_J^m]+\frac{1}{4}C^IC^J[G_I^g, G_J^g] =C^IC^J [G_I^m, G_J^m ]+\frac{1}{4}C^IC^J[G_I^g, G_J^g] = C^IC^J ig^K_{IJ}G_K^m+\frac{1}{4}C^IC^J ig^{K}_{IJ}G_K^g =C^IC^J ig^K_{IJ}G_K^m+\frac{1}{4}C^IC^J g_{IJ}^K g^M_{KL}C^Lb_M $ compare with the textbook $\\{ Q_B, Q_B \\} = -g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M$ My calculation is something wrong. How can I fix it?


title: Number theoretical function applied in physics?

text: I have a series of number theoretic phenomena (mathematics) that I can describe exactly by the superpositions or linear combination of the below function (I know it is an inverse Fourier type). Does anybody knows from parallels in physics (quantum, particle, waves) where exactly this function is applied? Are there perhaps other notations in physics of this function. The question is very important for me trying to investigate this equation. $$G(x)=\sum_{k=-j}^j \frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$$ while $i=\sqrt{-1}$ and angular frequency: $$\omega_j=\frac{2 \pi}{j}$$ PS: interestingly if $j\rightarrow \infty$ we get a ${\rm sinc}$ function that if squared then corresponding to pair-correlation function for the eigenvalues of a random Hermitian matrix a model of the energy levels in heavy nuclei. Note: of course taking the first and second derivative of the function $\frac{2 \pi}{j}\,e^{(i\,2\pi\,(k-1)\,x / j)}$ we can construct easily the differential equation of a harmonic oscillator.


title: Addes mass forces: can a force depend on acceleration?

text: My friend and I had a little discussion about added mass forces. I always interpreted $F=ma$ as a cause-effect relationship, so I find rather uneasy to accept that the cause can instantaneously depend on the effect. Is it fine to have a force which depends on an acceleration, in classical mechanics? I came up with some possible solutions to this:   1. It's perfectly fine for **F** =m **a** to be an implicit equation with respect to **a**.   2. The time derivative of the velocity appears as the result of an approximation of a time-delay.   3. It arises due to assumptions made on the nature of the fluid (i.e. incompressible).   4. None of them


title: Kronecker delta confusion

text: I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): $$g^{\mu\nu}g_{\mu\nu}=\delta_{\nu}^{\nu}=\delta_{0}^{0}+\delta_{1}^{1}+\delta_{2}^{2}+\delta_{3}^{3}=1+1+1+1=4.$$ But I've also seen (where the upstairs and downstairs indices are different): $$g^{\mu\nu}g_{\nu\lambda}=\delta_{\lambda}^{\mu}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\\ 0 & 1 & 0 & 0\\\ 0 & 0 & 1 & 0\\\ 0 & 0 & 0 & 1 \end{array}\right).$$ How can there be two different answers to (what appears to me to be) the same operation, ie multiplying the metric tensor by its inverse? Apologies if I've got this completely wrong.


title: 3D Vector Rotation of Point Mass

text: Triangle defined by points **OA** , **OB** and **OC** : (- **i** ,3 **j** ,-4 **k** ), ( **i** ,2 **j** ,2 **k** ) and (3 **i** ,7 **j** ,- **k** ) where **i** , **j** , **k** are unit vectors along x,y,z axis. Point mass is placed at **OA**. Triangle rotates with angular velocity **ω** about **BC** axis. Calculate linear velocity of mass as it passes **OA**. My Work: $$v=\omega \times r $$ Does **r** have to be the vector perpendicular to **BC** to the point **OA** or can it be any vector from the line **BC** to **OA**??? If so can you explain? I know the answer as my tutor told me I got it right but I didn't need to find the vector perpendicular to **BC** and I forgot to ask why...


title: Photons emitted at the event horizon?

text: While looking through the questions, a came across a section about black holes. I immediately though; what would happen if an atom is orbiting a black hole and emitted a photon perpendicular to the event horizon, going away from the black hole. How would light going away from a black hole react to the gravity? Photon "stuck" on the event horizon of a black hole actually talks about a photon stopping at the event horizon, but my question is about a photon just outside of the event horizon and if the photon is slowed down.


title: Why does centre of mass of ice-container system shift in absence of any net external force?

text: Consider a cube of ice in a flat based container(the base is very broad).The temperature of the system is at first fixed at a minus Celsius temperature, but then the system is left on a table with the top open to atmosphere.   The ice starts melting, and finally there is only water, spread over the container's base(water doesn't touch walls of the container,so it looks like water spilled on the floor)   Here, the centre of mass of the $H_2O$-container system moves downward because of the lowering of height of water molecules.   But,this system experiences no net external force on it,all the time.Hence, the centre of mass of this system shouldn't accelerate at all!   How is this contradiction sorted out?I feel that since melting is very slow, the centre of mass might move only very slowly, but still that doesn't explain things.For example if the room was very hot, melting wouldn't have been slow, right? Edit:Many people are probably getting confused regarding what perspective I'm taking.Let the container and water(solid/liquid) be a single system.We can think of them together as a point mass.This point mass is at equilibrium,and at rest,situated at the position of the centre of mass of ice+container system(let's say,at a height 'h' above the table.   There may be internal forces happening inside the point mass,but the net external force(resultant) is zero,all the time.Hence, according to Newton's first law,the point mass must remain at equilibrium and hence,at rest.But,when ice melts,the position of center of mass of ice+container system has moved down!   Hence the point mass has to move down, in the absence of any resultant external force.The gravitational force on ice+container is cancelled by normal reaction of table on container.


title: Supersymmetric generalisation of the bosonic $\sigma$ model in QM

text: I am reading some lecture notes which demonstrate how various models in **SUSY QM** can be used to obtain topological invariants such as the Euler characteristic from the Witten Index. The following lagrangian has been used directly, said to be the supersymmetric generalization of the bosonic $\sigma$ model. What is the motivation to consider this Lagrangian? How do I obtain the lagrangian? What is the sigma model being considered and how does it generalise? Please provide a direct answer or references as deemed necessary. (I googled to learn more, but most of the reviews start from lagrangians in TQFT which I have no knowledge about. I would like a more elementary explanation for the lagrangian. ) $\phi^i(t)$ are maps from $R$ or $S^1$ to a Riemannian manifold $M$ with metric $g_{ij}$. $$L = \frac{1}{2}g_{ij}(\phi)\dot{\phi}^i \dot{\phi}^j+\frac{i}{2}g_{ij}\bar{\Psi}^i \gamma^0 \frac{D\Psi^j}{dt} + \frac{1}{12}R_{ijkl}\bar{\Psi}^1 \Psi^j \bar{\Psi}^k\bar{\Psi^l} $$ $\frac{D}{dt}$ is the covariant derivative with $\dot{\phi}$ as a connection, and $\bar{\Psi}^i_{\alpha}=\bar{\Psi}^i_{\beta}\gamma^0_{\beta \alpha}$ How can I understand this from the physics of quantum mechanics? and what are the maps $\phi(t)$?


title: Shape Created by a Pile of Granular Objects Dropped Uniformly

text: I've observed many times that if you drop a lot of a 'granular' substance in one place and keep the nozzle out of which the substance flows, that the shape of the pile created very much resembles a bell curve. The situation is a bit hard to explain so for example image somebody holding a small pipe vertically above the ground and dropping a large amount of sand through that small pipe. The shape created by the sand on the ground largely resembles a bell curve wrapped around the vertical axis. I come from a math background and am really unsure of how to even start proving or disproving the proposition that the shape formed by the grains is a bell curve. Any help and input would be appreciated.


title: Calculating the Probability Current of a Travelling Wave

text: Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$ From my very poor and beginner's understanding of probability density current it is : $$\frac{d(\psi \psi^{*})}{dt}=\frac{i\hbar}{2m}[\frac{d\psi}{dx}\psi^{*}-\frac{d\psi^{*}}{dx}\psi]$$ By applying the RHS of the above equation : $$\frac{i\hbar}{2m}[-A^{2}ikxe^{-i(ωt-kx)}e^{i(ωt-kx)}-A^{2}ikxe^{i(ωt- kx)}e^{-i(ωt-kx)}]$$ This gives : $$\frac{-2iA^{2}ik\hbar}{2m}=\frac{k \hbar A^{2}}{m}$$ This is not the correct answer. :( What have I done wrong ? In the model workings instead of A in the complex conjugate of the wave function they have written $A^{*}$. Why is this necessary since $A$ is likely to be a real number anyways ?


title: Is it possible to extract energy from mass rotating in space?

text: Consider a single isolated rotating mass (for example planet), is it possible to extract energy out of its rotation? If yes, how could that theoretically be achieved?


title: What is the mean field value of a scalar field with spontaneously broken symmetry in a scattering event?

text: Consider you have a quantum field theory that undergoes spontaneous symmetry breaking at some critical temperature. It doesn't necessarily have to be a continuous symmetry that's broken, I don't think that matters for my question. For simplicity, think of the field being a scalar field. I (think I) do understand how symmetry restoration works in a background of some temperature. One looks at the mean field approximation, and the mean field then has statistical properties (temperature, pressure) and contributes to the Lagrangian. That adds to the symmetry breaking potential so that at the critical temperature the nature of the minima changes. That works fine eg in the early universe. But now consider you look at a highly energetic scattering experiment, high energy meaning a momentum transfer much much larger than the critical temperature (arbitrarily large if you want), with ingoing and outgoing asymptotic states and so on. The mean field approximation doesn't make much physical sense in this case because the scattering event itself is a large fluctuation. But the mean field value itself must still exist, or does it? I am thus wondering what's the mean field value of the field whose symmetry is broken in that scattering event. I do not mean the vacuum expectation value - if there is something that's scattering, it's not vacuum. And again, I know that if you have enough energy for a large perturbation, the mean field approximation isn't useful. I think I am missing something here. I am not even sure how the mean field would be defined for the scattering event.


title: Physical Explanation of Being Able to "Think"

text: This may look like a philosophical question, but I'm looking for physical explanations (if there's any), that's is why I'm asking it here. What is the ability of thinking? We are all creatures consist of flesh and bones. Our brains are also nothing but flesh and water? Why are we thinking? What happens to this thinking power when we die? How and why does it disappear? Does it really disappear, or go to somewhere else in a sense we don't know and understand yet? We observe that, different creatures have different capabilities of thinking. The creature which has the most advanced thinking ability is human. Besides human cats can think up to some degree; they know how to hunt, they decide where to hide and where to find food from. Ants think too, even if it is weaker then cats can do; they find food and quickly run away when you try to pick one of them from floor with your hand, because they sense danger and decide to escape from there. Cells can also think in some sense, but it is much less capable that we don't even call it "thinking". Most living creature sense themselves as separate being. What happens if we stop the time, we compile and exact verbatim copy of a normal human being, then start the flow of time once again? Would the second copy think too as the original one? Or would the second copy start his life in vegetative state? Or would it just be a stack of dead flesh? How does physics explain this? Is there any particle that causes us to think? One century ago, we didn't even dream that this many sub atomic particles existed. Can the ability of thinking be a cause of some unknown physical particle of a flow of vector field (like electromagnetic field) which hasn't been discovered yet? Do you expect that in the future some scientist(s) would discover this mystery? Is there any research going on this? What do we know about "thinking" today? (Note: There wasn't appropriate tags for my question. I would appreciate if another user with higher reputation could add some tags for this question and remove this note.)


title: Are there any known potentially useful nontrivial irreducible representations of the Lorentz Group $O(3,1)$ of dimension bigger than 4? Examples?

text: Are there any known potentially useful, nontrivial, irreducible representations of the Lorentz Group $O(3,1)$ of dimension more than $4$? Examples? A $5$-dimensional representation? EDIT: Is there some deep reason why higher-dimensional representations (other than infinite-dimensional representations) are less useful?


title: Diffraction grating zeroth order intensity

text: I don't understand how the zeroth order intensity maximum works. The intensity of a transmission diffraction grating is: $$I=I_0\sin^2(β/2)/(β/2)^2\sin^2(Nγ/2)/\sin^2(γ/2)$$ Where $γ =kd\sin(θ)$, $β=kb\sin(θ)$, $d$ is the grating constant and $b$ is slit width. For the zeroth order intensity maximum, $θ =0$, so $γ =β=0$. I understand how $\sin^2(β/2)/(β/2)^2$ goes to $1$ as $θ$ tends to zero. But what happens with $\sin^2(Nγ/2)/\sin^2(γ/2)$? Answer: Taking L'Hôpital's rule twice ($lim γ→0$) I get: $$ [0.5N^2(dγ/dt)^2(cos^2(Nγ/2)-sin^2(Nγ/2))]/[0.5(dγ/dt)^2(cos^2(γ/2)-sin^2(γ/2))] = [N^2(1-2sin^2(γ/2))]/(1-2sin^2(γ/2)$$ Which approaches $N^2$ as $γ$ goes to zero.


title: A brief explanation needed regarding newton's law in particle

text: So based on the topic, this is the question. "A particle with mass $m$ is moving along x-axis with $v_0$ at $t=0$ and $x=0$. The particle is acted by an opposing force with magnitude proportional to the square of velocity. Find out the a) velocity b) position and c) acceleration of particle at any time ($t>0$)" I have the answers and solution for this but hardly understand them (I dont understand at all actually. Plus where does β come from?) Basically I just need a clarification on question a) as b) and c) are related to the first question. This is a self study for final exam. Answer provided by my friend but she also copied it from a source. So without understand it we do it correctly. But it seems to be useless. So I hope you guys can help me. This is the answer. a) velocity ![Part a](http://i.stack.imgur.com/nU9kp.jpg) b) position ![Part b](http://i.stack.imgur.com/PknJE.jpg) c) acceleration (t>0) ![Part c](http://i.stack.imgur.com/cY0hx.jpg)


What kinds of inconsistencies would one get if one starts with Lorentz noninvariant Lagrangian of QFT? The question is motivated by this preprint arXiv:1203.0609 by Murayama and Watanabe. Also, what is the basic difference between   * a theory that is non-relativistic approximation to relativistic theory, and    * a fundamentally Lorentz non-invariant theory?


title: Why does magnet attract iron but not other metals?

text: While searching, i found this page: http://timesofindia.indiatimes.com/home/stoi/Why-does-a-magnet-attract- iron/articleshow/4298171.cms but it does not have full explanation. So Please tell full explanation of why this happens?


title: M-theory no lagrangian?

text: Is there any formulated lagrangian (density) for M-theory? If not, why is there no lagrangian? If not, is this related to many vacua existing? Thnx.


Why are ceramic containers used for hot beverages? For example, is it dangerous to use plastic? Are there any other reasons other than to protect hands from heat?


title: Electromagnetism and the principle of relativity

text: I'm reading the book "Fundamental Physics 2: Electromagnetism" by Alonso and Finn. I understand everything up to the point where everything is "unified". The following example is given in the book: > _Take 2 observers $O$ and $O'$ that are moving with a constant velocity $v$ > relative to each other and 2 charges $q$ and $Q$ that are at rest relative > to $O'$. For observer $O'$ there's only an electrical interaction between > $Q$ and $q$; he measures the force $\vec{F'}=q\vec{E'}$ where $\vec{E'}$ is > the electrical field intensity caused by $Q$ in $q$, as measured by $O'$. > Because $O$ sees the charge $Q$ moving, he observes an electrical field > $\vec{E}$ and a magnetic field $\vec{B}$ caused by $Q$ and because he sees > $q$ move with a velocity $\vec{v}$, the force exerted on $q$ by $Q$ > according to $O$ equal to $\vec{F} = q(\vec{E}+\vec{v}\times\vec{B})$._ Now a bit further it says that we choose the x and x' axis so that $\vec{v}=\vec{e_x}v$.   Because of this: $\vec{v}\times\vec{B} = -\vec{e_y}v\vec{B_z}+\vec{e_z}v\vec{B_y}$. I don't understand how this last conclusion is made.


title: Not so simple problem using momentum, energy and angular velocity...?

text: I have an object in free space (no gravity) with angular momentum $ = \omega_i $, and some velocity vector $=\vec{V_i}$. To simplify we will say it has a mass-less rigid rod length $ = \ell $, connecting two small masses both of mass $ = M $. The masses are small in the sense of a radius equal to the rod radius both much smaller than $ \ell $. Of course for simplicity keep this in the 2D plane. I know precession can play a role, but accounting for it can make the problem easier, I've already done that. Now, we want to change the velocity of the object by $ \Delta V $. We must do this by taking a bit of the matter of mass $ = m $ off the object and get it moving away at a velocity $ = \vec{V_m} $. While keeping $M>>m$ What will be the most energy efficient mechanism to do this? What will be the most momentum efficient mechanism to do this? The final answer, like with the precession related solution, gives some $ \vec{V_m} $ value parallel or perpendicular to $\vec{V_i}$, and some mass "$ m $" proportional to "$ M $". The goal will be to obtain maximum $ \Delta V $ with minimum energy or momentum. While keeping $M>2m$ Right now I am getting different solutions based on minimizing energy vs. momentum, is this logical? Why? This has helped me get to my solution thus far. Thanks JCooper and Maksim Zholudev. **EDIT:** The answer should be at least partially derived using formulas. Some relation between $ \Delta V $ and the input energy or momentum must be shown.


title: Confusion regarding a method of writing constraint equations

text: I came across a method for writing the constraint equations known as "The Virtual Work Method".I am quoting the exact language of the text(well,not exactly the exact)- > Consider the atwood machine shown.When one holds both the masses such that > the string is slack,there is no tension.Once released,the string will become > taut and it will oppose the motion of one block where as it will support the > motion of other one. > > ![fig](http://i.stack.imgur.com/UT1Oh.png) > > Thus it is transferring the effort of one block to the other one.so it will > not do work on its own.Since sum of the work done by string on each block is > 0 > > $$\vec T\cdot\vec x+\vec T\cdot\vec y=0 $$ $$\Rightarrow \vec x+\vec y=0$$ > _Differentiating twice_ > > $$\vec a_x+\vec a_y=0$$ Well here i do not not understand why sum of the work will be 0 and another thing is why is the work called virtual?


title: Explanation of Lorentz-Force

text: In high-school level books (for example the german standard text: "Dorn- Bader") I have often seen an explanation of the Lorentz force as on the following picture: ![enter image description here](http://i.stack.imgur.com/eHiQk.gif) The textbooks consider the superposition of the circular field of the wire and the homogenous field of the magnet (sure the homogenity doesn't matter here). Then the net field as you can see on the picture above on the right is larger on one side of the wire (here on the right) and smalle on the other one. So far so good. However why does this **explain** the occurence and direction of the Lorentz force. Do do so, one would need another principle for example that the wire always wants to go to the weaker field regions or something like this. And this principle should be somehow more evident than the Lorentz-force itself (which you can "see" experimentally). But how is this needed principle exacly forumlated? Why it is correct? Is there any good reason that it is more evident than just taking the Lorentz- force as an experimental fact? Would be great if someone could clarify the logic of this, evaluate the soundness of the argument and embedd it conceptually and mathematically in the big picture of electromagnetic theory. Additionally I want to know if the above cited "explanation" has any common name and if there are university level textbooks which proceed in a similar way. I feel that this argument goes back to Michael Faraday (just by the style of reasoning) - so if someone has a reference to the orgin of the argument I would be interested in it, too. By the way: The magnetic field in the above cited book ("Dorn Bader") is introduced by the interaction of permament magnets...


title: Fluidic gravity, TOE, what's your oppinion on this research?

text: It's interesting because we don't normally consider the "vaccuum of space" as a fluid, but it's becoming more apparent that it's an ocean of subatomic stuff. Here's a link to a book: Unified Fluid Dynamic Theory Of Physics Do you think this is crack-pot science or a worth-while book to study? Thanks.


1. Can anyone outline the theory of plane wave Born approximation for direct nuclear reactions in detail?   2. Also What are the modification introduced in the distorted wave Born approximation? I was unable to find the right material/books/web resources on this.


title: Can a scientific theory ever be absolutely proven?

text: I personally cringe when people talk about scientific theories in the same way we talk about everyday theories. I was under the impression a scientific theory is similar to a mathematical proof; however a friend of mine disagreed. He said that you can never be absolutely certain and a scientific theory is still a theory. Just a very well substantiated one. After disagreeing and then looking into it, I think he's right. Even the Wikipedia definition says it's just very accurate but that there is no certainty. Just a closeness to potential certainty. I then got thinking. Does this mean no matter how advanced we become, we will never become certain of the natural universe and the physics that drives it? Because there will always be something we don't know for certain?


Why are all observable gauge theories not vector-like? Will this imply that the electron and/or fermions do not have mass? How is this issue resolved? Background: The Standard Model is a non-abelian gauge theory with the symmetry group U(1)×SU(2)×SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons. Massless fermions can either have their spin pointing along their direction of travel or opposed to it, these two types of massless fermions are called right-handed and left-handed. A fermion mass can turn a left-handed fermion into a right-handed fermion (this is technically called a Dirac fermion mass). The ordinary electron for example has both left-handed and right-handed pieces. For this to be allowed in a gauge theory the left-handed and right- handed fermions must have the same charge. If all the fermions in the gauge theory can be paired up in this fashion so that they are all allowed to have masses, then the gauge theory is called vector-like.


title: Why are the solution coefficients for a harmonic oscillator proportional to minors of the determinant?

text: I'm studying the oscillations of systems with more than one degree of freedom from Landau & Lifshitz's _Mechanics_ Third Edition (for those who have the book, my question corresponds roughly to explaining the last paragraph before equation 23.9). First I give a brief background to my problem to show what I understand. So far I understand that given the Lagrangian, the equation of motion is: $${m_{ik}\ddot{x}_k + K_{ik}x_k = 0}\tag{23.5}$$ Using the usual _ansatz_ $x_k = A_k e^{i \omega t}$ then, we get the set of equations: $$(-\omega^2m_{ij} + k_{ik})A_k = 0\tag{23.7}$$ where it can be easily seen that the section in brackets must have determinant 0, leading us to the _characteristic equation_ : $$|k_{ik} - \omega^2m_{ik}| = 0\tag{23.8}$$ where furthermore it can be easily shown that for each solution to the characteristic equation $\omega^2_\alpha$, $\alpha = 1, \ldots, S$ where $S$ is the degrees of freedom of the system, $\omega_\alpha$ must be positive and real. I now quote in full the paragraph I'm having difficulty with: > The frequencies $\omega_\alpha$ having been found, we substitute each of > them in equations (23.7) and find the corresponding coefficients $A_k$. If > all the roots $w_\alpha$ of the characteristic equation are different, the > coefficients $A_k$ are proportional to the minors of the determinant (23.8) > with $\omega = \omega_\alpha$. Let these minors be $\Delta_{k\alpha}$. A > particular solution of the differential equations (23.5) is therefore $x_k = > \Delta_{k\alpha}C_{\alpha}\exp{(i\omega_\alpha t)}$ where $C_\alpha$ is an > arbitrary complex constant. I think my main problem here is my poor knowledge of linear algebra. I don't understand why the roots all being different implies that the coefficients are proportional to the minors of the determinant. Furthermore I'm slightly confused by the wording "minors of the determinant" since to me, a determinant is a scalar value, it seems like they should be saying "minors of the matrix $(k_{ik} - \omega^2m_{ik})$" Can someone explain the linear algebra behind all this to me? I've taken a Linear Algebra course but I was quite poor at it.


title: Superposition of Negative and Positive Energy States

text: This is a question about the negative energy solutions to the free particle Dirac Equation in the first quantized picture. We need both the positive and negative energy solutions to have a complete set of states. For example, the bound states of the hydrogen atom consists of superpositions of positive and negative energy free particle solutions. So then, what is the physical interpretation of a superposition of positive and negative energy states? Suppose we had a hydrogen atom, and we immediately removed the nucleus. If we measured the energy of the resulting free electron, what would we see?


title: What are the optimal shapes, if any, for fractal antennae?

text: A DIY fractal antenna project is described at http://www.htpc- diy.com/2012/04/diy-flexible-fractal-window-hdtv.html Mainly idle curiosity, but I was wondering what the principles behind these might be, and if there was an optimal shape for them. I'm guessing the idea is that self-similarity, up to a point, optimizes sensitivity over a wider frequency range.


title: Continuity equation for compressible fluid

text: A question is given as > Consider a fluid of density $ \rho(x, y, z, t) $ which moves with velocity > $v(x, y, z, t) $ without sources or sink. Show that $ \nabla \cdot \vec J + > \frac{\partial \rho }{\partial t} = 0 ;$ where $ \vec J = \rho \vec v > \hspace{0.5 cm}$ ( $\vec v$ being velocity of fluid and $ \rho $ density). In the solution it assumes, $ - \nabla \cdot \vec J $ is the change in $ \vec J$ within the volume element which should be equal to $ \frac{\partial \rho}{\partial t}$ _(why equal?)_. I don't understand this part and I doubt if this question is correct. I think the question should be $$ \rho (\nabla \cdot \vec v) + \frac{\partial \rho}{\partial t} = 0$$ Is the question correct? If it's correct help me to understand the solution (if it's right) or please provide me correct answer. Thank you!!


title: Friction on an object moving with momentum over a surface

text: I'm familiar with the equations for friction for a static object and an object moving at steady speed over a surface from high school physics. But we never learned how an object moving only due to momentum experiences friction. This is something I've modeled several times while building simple 2D games, but I have no idea if what I have made matches reality in any way. Is the force of friction dependent on the speed of the object across the surface? Or is the force constant as the object decelerates? Is there a simple equation giving the acceleration (deceleration) of the object depending on it's mass and velocity?


Does the Casimir Effect violate the Quantum Inequalities? From what I understand, the Casimir Effect is able to produce negative energy densities for an indefinite amount of time, or is that wrong?


title: Energy provided to piston in compression and exhaust stroke?

text: In the compression stroke of a petrol engine, the mixture is compressed by the upward movement of the piston. From where does the piston get energy to compress the mixture? Similarly, in the exhaust stroke, the piston again moves upward to expel the gases. Where does the piston get energy from?


title: $\langle B|A \rangle$ expressed in terms of the Partition Function

text: Say you have an electron departing from point A and reaching poing B after a time t. According to some helping friend, the _Partition Function_ for that electron going from point A to B can be written as $$Z = \int_{A \to B} [\mathcal{D}x]~ e^{iS[x]}$$ where $\mathcal{D}x$ is the measure that sums up over all paths going from $A$ to $B$, and $e^{iS[x]}$ is the weight of each path, $S[x]$ is the action. That friend states then " _From this partition function all desirable quantities can be obtained_." Not having much idea about Feynman's Path Integral Formulation of Quantum Mechanics, I have looked around a bit, and I would like someone to confirm the following statement I make: The amplitude for the electron to go from A to B in the time t can be found in terms of that Partition Function $Z$ given above, as $$\langle B|e^{-iHt}|A\rangle = Z$$ Did I catch it right?


title: Photon number conservation during scattering

text: I was reading this writeup on the Kompaneets equation and the Sunyaev- Zel'dovich effect. On page 3, section 2 the author states > There is no way to increase the mean energy of a planckian distribution > without changing the particle number. But as far as I understand, photon number is conserved during a scattering process, and therefore isn't it possible that for a given Planckian distribution all the photons gain the same energy through scattering and thus increase the mean energy of the Planckian distribution? Or am I missing something here?


title: A question about polarization in quantum mechanics

text: We start our question we a definition A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if \   1. $P$ is Lagrangian   2. P involutive   3. dim$P\cap\bar P \cap TM$ is constant Now, Introduce an hermitian form on $P$ defined by  $$b(X,Y)=i\omega(X,\bar Y)$$ . Note that when $P$ is real then $b$ is vanishes identically on $P$. Cinsequently, $b$ projects onto a non-degenerate form on the quotient $P/{(P\cap \bar P)}$ and we denote it by $\bar b$. $P$ is said to be of type $(r,s)$ if and only if $\bar b$ has signature $(r,s)$ i.e. its matrix is $$diag(\underbrace{1,1...,1}_{r},\underbrace{-1,-1...,-1}_{s} )$$ for $0\le r+s=n-dim_{\mathbf C }P\cap \bar P$ . Then, $P$ is said to be positive if $s=0$. In the case if $r=s=0$ then $P$ is real Philosophically the main goal in defining polarizations in quantum mechanics is to find the wave functions that are covariantly constant along its directions. When the form $b$ is not positive show that there are no globally defined wave functions


title: Planck's Law-Where does it come from?

text: It seems that Planck's constant was made from fitting a curve for blackbody radiation, is it just experimental-further more his assumption that energy comes in quanta seems to have been a guess. Why would energy come in quanta? Wikipedia says that he didn't think about it much, but I don't know why harmonic oscillators would even be suggested to only have a quantized number of energy modes, moreover what an energy mode would be. With a set frequency, how does one vary the energy? _The main question I have, is what the reasoning behind the quantization of oscillators?_


title: In an ideal gas, if temperature increases, will pressure or volume increase?

text: This question came up when I was talking about the atmosphere. Someone had mistakenly claimed that as temperature increases, the density of the atmosphere should increase as well. I reasoned from the ideal gas law that, as temperature ($T$) increases, then so should $V$, leading to lower density. But I realized that pressure could increase instead; the atmosphere is not in a sturdy laboratory container where volume is a constant. Under these circumstances, how can one predict whether pressure or volume (or both) will increase?


title: Volume of Matter in the Universe

text: What would the size of the universe be if it were physically possible to remove all of the empty space, leaving only matter?


title: Why were space physicists wrong about the location of the heliopause?

text: The heliopause is now estimated to be something around 100 AU (1 AU = Astronomical unit = about the earth sun distance). See the wikipedia article:   http://en.wikipedia.org/wiki/Heliosphere From reading a book on NASA's Voyager mission, I learned that before the launch of these spacecraft, the expectation was that the heliopause was at around Jupiter or Saturn's orbit which is about 5 to 10 AU. The book says that as the spacecraft continued to move away from the sun, the space physicists kept increasing their estimate of the position of the heliopause. If it's about an energy or flux balance, then the estimate would be defined by a certain area of the sphere where the heliopause occurs, and since areas are proportional to the square of the radius they got the number wrong by as much as $(100/5)^2 = 400$. That's a huge underestimate of the sun's output or a huge overestimate of what goes on in interstellar space. The book doesn't explain why it is that early estimates were wrong and I didn't see an explanation. Perhaps someone knows and will give a nice intuitive explanation for the estimates.


title: Can we create a small black hole?

text: Suppose we create very large spherical body by using gamma rays generator and they will concentrate on a single point at the centre of sphere.We will place this spherical body thousands of kms above the sun so that it can draw energy from sun.Can the centre of the sphere create a Mini Black hole by the energy from the concentration of gamma rays? [Just applying my 14 yrs old brain..if im wrong then please correct me...]


title: Can black holes be created on a miniature scale?

text: A black hole is so powerful to suck everything into itself. So is it possible that mini black holes can be created? If not then we could have actively disproved the rumors spread during LHC experiment.


title: What is the net charge of the Earth?

text: This question arose in a seminar today about the solar wind... This is my vagueish understanding of the problem - please correct if you see errors! The 'classical' picture of atmospheric electricity is that the Earth as a whole is neutral, but that thunderstorms maintain a voltage of around +300kV at the electrosphere with respect to the Earth's surface, with a current of around 1 kA slowly discharging around 500 kC of total charge separation. The solar wind is supposed to neutralise any net charge that might be there between the Earth as a whole and the solar wind. However, positive and negative charges in the solar wind are differently trapped in the van Allen belts, from which they can then descend to the Earth's atmosphere, which implies that a net charge can be developed due to this differential leakage. This begs the question of whether there are any estimates of the total net charge. I've hunted in the literature but have found little useful material other than Dolezalek's 1988 paper: http://www.springerlink.com/content/u057683112l148x5/ Can anyone offer an explanation, or point me to some more relevant papers?


title: Is "Causality" the equivalent of a claim that the future is predictable based on the present and the past?

text: In classical (Newtonian) mechanics, every observer had the same past and the same future and if you had perfect knowledge about the current state of all particles in the universe, you could (theoretically) compute the future state of all particles in the universe. With special (and general) relativity, we have the relativity of simultaneity. Therefore the best we can do is to say that for an event happening right now for any particular observer, we can theoretically predict the event if we know everything about the past light cone of the observer. However, it tachyons (that always travel faster than the speed of light) are allowed, then we cannot predict the future since a tachyon can come in from the space-like region for the observer and can cause an event that cannot be predicted by the past light cone. That is, I believe, why tachyons are incompatible with causality in relativity. Basically, the future cannot be predicted for any given observer so the universe is in general unpredictable - i.e. physics is impossible. Now in quantum mechanics, perfect predictability is impossible in principle. Instead all we can predict is the probability of events happening. However, Schrodinger's equation allows the future wavefunction to be calculated given the current wavefunction. However, the wavefunction only allows for the predictions of probabilities of events happening. Quantum mechanics claims that this is the calculations of probabilities is the best that can be done by any physical theory. So the question is: "Is the predictability of the future to whatever extent is possible (based on the present and the past) equivalent to the principle of causality?" Since prediction is the goal of physics and science in general, causality is necessary for physics and science to be possible. I am really not asking for a philosophical discussion, I want to know if there are any practical results of the principle of causality other than this predictability of the future of the universe. Please don't immediately close this as being a subjective question, let's see if anyone can come up with additional implications for causality besides future predictability.


title: Currents and magnets

text: I've watched this video on YouTube by Sixty Symbols entitled "Currents and Magnets". In the video, the professor demonstrates the expansion of a wire due to current heating it up and he also demonstrates how the current interacts with a magnet. I want to ask about the magnetic interaction. At about 2:28 he shows how the orientation of the magnet relative to the wire/current determines whether the wire is repelled or attracted. It appears that, mathematically, whether the wire is pulled or pushed is determined by the cross product between the vector of the flow of the current, $\vec{c}$, and the vector of the magnetic field, $\vec{m}$. However, the cross product is not commutative (order of the operands is important). What determines whether I should cross $\vec{c}$ with $\vec{m}$ or whether I cross them the other way around?


title: Equivalent definitions of primary fields in CFT

text: I have come across two similar definitions of primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible but I can't seem to be able to show it. By compatible I mean definition 1 $\iff$ definition 2. I will write both definitions in the two-dimensional case and restricting to holomorphic transformations. Def #1 from Francesco CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(\omega)=\left( \frac{d\omega}{dz}\right)^{-h}f(z)$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$. Def #2 from Blumenhagen Intro to CFT: A field $f(z)$ is primary if it transforms as $f(z) \rightarrow g(z)=\left( \frac{d\omega}{dz}\right)^{h}f(\omega)$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$. Can someone show me how they are indeed the same?


title: Infinite Energy from Bobbing

text: Could this device theoretically continue in motion forever? If not, why not? (click below for images): Device description. Device process.   1. The device is less dense than air, so it rises. The propeller spins slightly (maybe) charging the device battery.   2. After rising some distance X, the device compressor turns on to deflate the device.   3. The device becomes more dense than air and falls quickly spinning the propeller, charging the battery.   4. After falling X, the compressor releases and the device becomes less dense than air, going back to step 1. The logic here is that there must be some distance X that the device can rise than will generate more energy than what is needed by the compressor. Here is the underlying math to help: $$PE = mgh$$   * $m$ = mass of the device   * $g$ = coefficient of gravity   * $h$ = height device has traveled up    * $PE$ = potential energy of the device as it travels up $$CW = nRT(\ln(V_b) - \ln(V_a))$$   * $n$ = number of moles of gas in balloon of device   * $R$ = ideal gas constant   * $T$ = temperature of the gas   * $V_b$ = volume of the balloon after compression   * $V_a$ = volume of the balloon before compression   * $CW$ = work to compress the balloon As $h$ increases $PE$ increases but $CW$ stays the same resulting in energy gain of the system.


title: Why do rainbows have distinct colors?

text: When I searched on the Internet for the reason of formation of rainbows, I got many explanations like this one & this. All the explanations consider only one spherical water droplet (like this one). Why don't the refracted waves coming from the adjacent droplets interfere with each other? For example, why doesn't a red ray coincide with a violet ray coming from another droplet at a suitable position? How are we able to see rainbows with distinct colors? Since there are thousands of raindrops in the sky when it is raining, shouldn't it be a mess due to overlapping of different wavelengths? Please explain.


title: Are there cosmic rays in a nuclear reaction?

text: If the range of energies of cosmic rays is not so far away respect to gamma's, why those are not found commonly in a nuclear reaction?


title: Does the inability to postselect to a given branch argue against MWI?

text: It has been argued the exponential size of the wavefunction can be interpreted as many parallel worlds, and this explains how quantum computers can factor large integers and compute discrete logarithms (although on closer examination, Shor's algorithm doesn't work at all by trying out a superposition of all candidate solutions. It works by using number theory to find a _different_ function with periodicity, and period finding.). However, unless we can examine specific individual branches of our choice, why should we argue for the ontological reality of the branches? Measuring which branch we are in "collapses", i.e. picks out a branch sampled at random, and not a branch of our choice. To be more specific, let's say we wish to invert a one-way function f with no periodicity properties, i.e. solve $f(x)=y$ for a specific y. We can easily prepare the state $C\sum_x |x\rangle\otimes |f(x)\rangle$ where C is an overall normalization factor. However, unless we can also postselect for the second register being y, how can we interpret this as many parallel worlds? Doesn't this inability argue for a more "collectivist" interpretation where the individual branches don't have "individual" existence, but only the collective relative phases of all the branches taken together have any real existence? To take a less quantum computing example, consider Schroedinger's cat. Suppose we have N cats in N boxes, and we perform independent cat experiments on each of them. Let's suppose I pass you a prespecified N bit string in advance, and I ask you to "subjectively" take me to the branch where the life/death status of the cat in the ith box matches the value of the ith bit. That, you can't do unless you perform this experiment over and over again for an order of $2^N$ times. **PS** Actually, there might be a way using quantum suicide. Unless the life/death status of each cat matches that of the string, kill me. However, this rests on the dubious assumption that I will still find myself alive after this experiment, which rests upon the dubious assumption of the continuity of consciousness over time, and that it can't ever end subjectively.


title: When can a classical field theory be quantized?

text: Given a classical field theory can it be always quantized? Put in another way, Does there necessarily need to exist a particle excitation given a generic classical field theory? By generic I mean all the field theory variants, specially Higher Derivative QFT(particularly Lee-Wick field theories).   I ask this question because, several times in QFT we come across non physical particles when we try to quantize some field theory for e.g. ghost fields and ghost particles. These fields have opposite sign in front of the kinetic energy term. Such terms are common in higher derivative field theories. Hence we have to ask should we talk about particles in such situations.   Now if the answer is No! **Then we have to ask what is more fundamental in nature, particles or fields?**   _Schwinger disliked Feynman diagrams because he felt that they made the student focus on the particles and forget about local fields, which in his view inhibited understanding._ -Source Wikipedia


Does quantum fingerprinting really argue for the exponential size of wavefunctions? Quantum fingerprinting is the idea that an exponentially long classical string can be encoded in a linear number of entangled qubits using quantum fingerprints. To an exponential degree of accuracy, but not exactly, the fingerprints of two different exponentially long classical strings will be nearly orthogonal. However, the whole idea of quantum fingerprinting rests upon the ability to tell if two different pure quantum states are identical or orthogonal. Can such a comparator exist? Let's work with comparing qubits first. Our hypothetical comparator has the property that if two pure qubits are identical, it always outputs YES. If they are orthogonal, it always outputs NO. For other cases, it may output either. Then, it would definitely output NO for $|0\rangle|1\rangle$ and $| 1\rangle|0\rangle$. By the superposition principle, it would also have to definitely output NO for the linear superposition ${1\over\sqrt{2}} \left[ |0\rangle|1\rangle +|1\rangle|0\rangle\right]$. It would also have to output YES for both ${1\over 2}\left(|0\rangle + |1\rangle\right)\left(|0\rangle + |1\rangle\right)$ and ${1\over 2}\left(|0\rangle - |1\rangle\right)\left(|0\rangle - |1\rangle\right)$, and hence, also YES for their linear combination ${1\over\sqrt{2}} \left[ |0\rangle|1\rangle +|1\rangle|0\rangle\right]$. Contradiction. If we can't compare whether or not two quantum fingerprints are identical or nearly orthogonal, how then are they supposed to work? Is their apparent exponentiality "fake"?


title: Gravitational compression / compression in general

text: How exactly does gravitational compression, or compression in general, increase temperature? It seems counter-intuitive seen as temperature usually increases from the solid to the gas phase.


title: Photons and Black holes

text: How many photons in one Planck volume would it take to form a tiny black hole? A photon doesn't have mass but it does have energy, $1.0101 \times 10^{-37}$ Joule for red $650$ nm wavelength light if I'm correct. A photon is a point- like boson so an infinite number of photons can fit into any given area. So more photons is equal to more energy which would bend space-time even more till they themselves wouldn't be able to escape from their own bend in space- time.


title: Hidden particles in higher derivative field theories

text: Given a higher derivative classical/quantum field theory with say one scalar field, particularly the Lee-Wick standard model. It has been shown that such a field theory encompasses two kinds of fields, one normal and other a ghost field, in a particular limit, generally the high mass limit of the ghost field. We started with a single field but in ended up with a two particle field theory. This makes me ask:   **Is there any general mathematical theorem by which we can show that a $n$th order derivative theory(scalar, spinor or vector) can be quantized into $n$ different kind of particles?** Has it got to do something with the symmtries involved in the theory?   And what is the physical importance of the constraint? And how exactly is the system behaving when that constraint is not satisfied?(It seems to be in some sort of a enmeshed state of the two particles.) My question mainly relies on the paper in the following link [The Lee-Wick standard model]: http://arxiv.org/abs/0704.1845 . In section 2 (A Toy Model) he considers a higher derivative Lagrangian eq.(1) for a scalar field $\phi$. And then introduces an auxiliary field $\tilde{\phi}$. And then it is shown that given the condition $M > 2m$, the Lagrangian sort of gets decoupled into two different fields. One normal field and other a ghost field, both interacting with each other.


title: Linear Algebra for Quantum Physics

text: A week ago I asked people on this forum what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I have some questions.   1. How is this going to be applicable to quantum physics? I have learned about matrices (addition, subtraction, multiplication and inversion) and about how to solve multiple equations with 3 unknowns using matrices, and now I am starting to learn about vectors. I am just 1 week in, so this is probably not even the tip of the iceberg, but I want to know how this is going to help me.    2. Also, say I master Linear Algebra in general in half a year (I'm in high school but I'm extremely fast with maths), what other 'types' of math would I need to self-study before being able to understand rudimentary quantum physics mathematically?


title: Will a one year undergraduate course of Linear Algebra be enough for QM?

text: > **Possible Duplicate:**   >  Linear Algebra for Quantum Physics Can you get all/most of the knowledge you need of Linear Algebra for QM in a one year course? I know for certain my course also includes eigenvectors and eigenvalues towards the end of the year..


title: Journal for high school physics papers

text: I am currently a senior in high school. I have spent the past four years participating in physics research at a local lab. Are there any journals in which I as a high schooler can publish my research?


title: Calculating temperature of water in the freezer

text: Assuming water volume ($V$), initial water temperature ($T_0$) and environment temperature ($T_e$) are known, what is the easiest way to calculate temperature of water in given time ($T$)? For the sake of question, let's assume water container is so thin it doesn't matter. Also, water container is airtight (for example, soda can or sparkling wine bottle). Environment temperature is constant and air is not moving remarkably. I already experimented with sparkling wine bottle (filled with water and temperature logger). Blue line is for fridge, red for freezer and greenish for ice water (without salt). But it's not very practical to repeat same experiments with different and different containers. ![Temperature of sparkling wine bottle](http://i.stack.imgur.com/iJ2H6.png) Why? I thought it would be nice to make a simple mobile application for giving alarms when your drinks are chilled to whatever temperature you want.


title: Integral form of Gauss's law for magnetism from Stokes' theorem?

text: How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?


title: Stellar winds from neutron stars

text: It seems that this question has not really been explored in the literature. Do isolated neutron stars (which do not accrete material) emit stellar wind? If yes, what composition would it have? If yes, what will be the rate of mass loss for the star? One might also think up that the process of Hawking radiation might possibly be applicable to neutron stars, where negative energy particles get trapped in the nuclear star atmosphere (instead of crossing the horizon in case of black holes), which would also lead to some sort of evaporation and corresponding wind. However, in this question I am more interested in 'classical' winds.


title: Why can a qbit be used as a classical bit if information about the measurement axis is needed?

text: If Alice wants to send one **bit** of classical information she can use a **qbit**. Then Bob needs to know which axis to measure to get the information. This needs an extra agreement between Alice and Bob before starting communication. I can figure out why up or down agreement has no importance here since 0 and 1 are just different and need the same level of agreement (two choices available, one for 0 another for 1). Using **qbits** , an extra agreement is needed before sending information. A lamp could be on or off without any need for more agreement (which axis in `qbit` is an extra agreement). Has this extra contract any importance?


title: Magnetic field associated with edge currents in IQHE?

text: According to some theory, currents in IQHE are due to the edge states. The edge current on the upper edge flows from left to right, that on the lower edge flows from right to left. Did anyone detect or try to detect the magnetic field associated with the edge currents? Is this a way to test the theory?


title: If angular momentum corresponds to linear momentum, what corresponds to energy?

text: Angular momentum is defined from linear momentum via $\vec L = \vec r\times\vec p$, and is conserved in a closed system. Since energy is the time part of the linear four-momentum, is there a quantity defined from energy that's also conserved?


title: How can the torque a bicycle experiences be calculated based on the center of gravity, weight and a force?

text: The position center of gravity of a bicycle and its rider is known, and the distance from it to the point of contact of the front wheel with the ground, in terms of horizontal and vertical distance (x cm and y cm), are both known, as well as the total weight of the bike and the rider. The brakes are then applied, making the bike turn around the point of contact with the ground of the front wheel, and this force is also known. Is there a way to calculate the amount of torque generated from this information?


title: How can you have a negative voltage?

text: How can you have a negative voltage? I don't really understand the concept of negative voltage, how can it exist?


title: Hydrogen radial wave function infinity at r=0

text: When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $\psi(r,\phi,\theta)= R(r)Y_{l,m}(\phi,\theta)$ I understand that the radial part usually has a singularity for the 1s state at $r=0$ and this is why you remove it by writing: $R(r) = \frac{U(r)}{r}$ But what is the physical meaning of $R(r=0) = \infty$. Wouldn't this mean that the electron cloud is only at the centre of the atomic nucleolus? Thanks in advance!


1. What exactly are crystal planes and how do they reflect x-rays?   2. Are crystal planes real physical planes or just an abstract concept?    3. What are these planes made of?    4. If they are an abstraction, what do the x-rays hit and get reflected by? Individual atoms?    5. Then where is the concept of a plane coming in?    6. And why do different crystal structures have different 'active' reflecting planes?    7. Could someone please clear up the concept of these planes and Bragg's Law for me?


title: Question regarding the direction of electric field

text: Suppose that there are negative charges (e.g. electrons) only. There are more negative charges on left than on right. How would electric field be constructed? (So, What would be the direction?) And how would electrons flow (because of electric field)? The text says that it flows from left to right, but I am not getting this.


title: Deriving Feynman Rules (with the presence of a gluon field strength tensor)

text: If I have a Lagrangian of the form: $$ \mathcal{L} = k \bar{\psi} \varepsilon^{\mu \nu} \lambda^a \phi G^a_{\mu \nu} + h.c. $$ [where $\phi, \psi$ are fermions, $\lambda^a$ are Gellmann matrices, $\varepsilon^{\mu \nu} $ is some antisymmetric tensor and $G^a_{\mu \nu}$ is the gluon field stength tensor.] And I want to have the Feynman rule for the vertex and it's conjugate I would usually compute the following: $$ \frac{\delta S}{\delta \psi \delta \phi \delta A_\mu^a} $$ [where $A_\mu$ is a gluon field]. Following that prescription, I have $$ S = k \int d^4 x \bar{\psi} \varepsilon^{\mu \nu} \lambda^a \phi G^a_{\mu \nu} $$ ignoring color for a moment, $$ \implies \frac{\delta S}{\delta \psi \delta \phi } = 2k \int d^4 x \gamma^0 \varepsilon^{\mu \nu} \left( \partial_\mu A_\nu + i g_s A_\mu A_\nu \right) $$ using the convention that $G_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + i g_s \left[ A_\mu, A_\nu \right] $ Taking the Fourier transform to do the last variation I get a leftover $A_\mu$ in the expression though, don't I? Something seems off here. With a nicer field strength tensor, like $F^{\mu \nu}$ I get a nice expression at the end only in terms of the momentum of the photon field...here I have a leftover term involving an $A_\mu$. The initial variation shouldn't be with respect to the field strength tensor, should it? I'm messing up something trivial, could someone point me in the right direction?


title: Potential of a Body

text: I have a doubt about the electric potential of a body. Well, I know that given a continuous distribution of charge we can find the potential at a point $a$ using the following relation: $$V(x,y,z)=\frac{1}{4\pi\epsilon_0}\iiint_{S} \rho(a') \frac{1}{|a-a'|}dV'$$ However, this gives a function that at each point tells me the electrostatic potential energy divided by charge. The difference of potential between two points can be found then calculating this function at those points and subtracting. My problem is that I've found exercises that asked something like: "find the difference of potential between two concentric spheres with uniform charge density and radii $a$ and $b$". However, what does it means calculate the difference of potential between two spheres? How do we deal with problems like this? And last but not least, what it means the potential of a sphere? The potential of a charge as the difference of potential between infinity and the place of the charge I understand, but what should be the potential of a whole sphere? Thanks in advance for the help.


title: Mean-field theory and spatial correlations in statistical physics

text: In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by requiring that : Eq.(1)$\hspace{75pt}$$\langle S_i S_j \rangle = \langle S_i \rangle \langle S_j \rangle $ for $i\neq j$ Where $S_i$ is the local spin observable supported at site $i$ of a given lattice (in the classical Ising case, it is just $\pm1$). I divide my questions/comments into two parts : Part 1 (a): I know that there are more sophisticated ways of formulating much more rigorously mean-field theory in statistical physics, but is the above relation an equivalent definition for the particular case of a spin model ? Part 1 (b) : Given that the above relation is an equivalent definition of MFT for a spin model, is it true to say that : "Mean-field theory is equivalent to taking out any spatial spin correlations of our system." ? I think this follows from the Eq.(1). Part 2 : However, and here is what is confusing me : Why can we define a correlation length $\xi$ and a corresponding critical exponent $\nu$ (c.g. $\nu=1/2$ for MFT applied to the Ising model) from the connected two-point correlation function ? Eq.(2) $\hspace{75pt}\langle S_iS_j \rangle - \langle S_i \rangle \langle S_j\rangle\sim e^{-|i-j|/\xi}$ To me, Eq. (1) and Eq. (2) look contradictory for distances $|i-j|$ smaller than the correlation length, yet there are both derived from MFT...


title: Relationship between resistance and voltage drop

text: In a series or parallel circuit, if two bulbs have the same resistance, do they have the same voltage drops? The problem I am asking about is below. Do A, B, and C have the same voltage drops since they have the same resistance? If so, how do I determine the current? ![Problem](http://i.stack.imgur.com/2wyv1.png)


title: Can modern twistor methods to calculate scattering amplitudes be applied to renormalization group calculations?

text: As explained for example in this article by Prof. Strassler, modern twistor methods to calculate scattering amplitudes have already been proven immensely helpful to calculate the standard model background in searches for "new physics". If I understand this correct, the "practical" power of these methods lies in their ability to greatly simplify the calculation of scattering processes, which are due to limited computer power for example, not feasable applying conventional Feynman diagrams. Depending on the system considered, a renormalization group transformation involves the calculation or summation of complicated Feynman diagrams too, which usually has to be simplified to obtain renormalization group equations which are numerically solvable in a finite amount of time. So my question is: Could the new twistor methods to calculate scattering amplitudes be applied to simplify investigations of the renormalization group flow, in particular investigations of the whole renormalization group flow field beyond a single fixed point, too? Are such things already going on at present?


title: Simple model of the solar system. Parameters? Accuracy?

text: I was thinking of making a simple 2D model of the solar system, with planets moving along ellipses like $$x(t) = k_x \sin(t + k_t) (\sin(k_\phi) + \cos(k_\phi))$$ $$y(t) = k_y \cos(t + k_t) (cos(k_\phi) - \sin(k_\phi))$$ and, for earth at least, a angle that some longitude (say the Greenwich Meridian) is facing in the $xy$ plane: $$d(t) = k_dt+k_e$$ or something equally minimal. Two questions:   * Where can I find the appropriate constants in the most cut-and-paste-able form?   * How long will this kind of model be accurate for? (If I want to use it to vaguely look in the right part of the sky for a particular planet)


title: Determing aperture size for space-based-solar-power laser transmitter

text: I have hit a bit of a roadblock in my simulation. In the equation I require the aperture size for the laser source. $$I_a ≈ \frac{\lambda d}{A}$$   * $I_a$ = aperture atop the atmosphere ($m$)   * $\lambda$ = wavelength of laser (in this case 1023nm)   * $A$ = aperture of the laser transmitter (...? $m$)   * $d$ = distance from SPS to atmosphere (39,000km) But all the articles I have read from NASA, JAXA, ESA, sciencedirect concepts, no one specifically talks about the details of laser transmission if used for SBSP. I just need some reasonable value for the aperture size. So we are talking about GW scale laser transmission from a single main source, if anyone has knowledge of optics/photonics/lasers, could you recommend me a sensible size along with a citation. Thanks


title: Heating coffee by yelling?

text: Is it a myth that yelling to a coffee mug will heat it? I have been hearing my friend saying that screaming will heat coffee or water.