title: Extra energy in quantum tunneling

text: In quantum tunneling, the probability of finding an electron inside the potential barrier is non zero (http://i.stack.imgur.com/v8Kr1.gif). So we can actually find an electron which had an energy E in a place where classically it should have an energy bigger than E. So if we find an electron in this potential barrier what will its energy be ?   * The energy of the electron is conserved so it's E and the fact that its kinetic energy is <0 is just a weird fact of quantum mechanics. (I don't like this answer)   * Since we know the electron is there we can treat it as a classical particle and because of that it's energy has rose to a value bigger than the potential of the potential barrier. The energy of the electron is not conserved unless the electron manages to pick up energy from nowhere. These are the answers I have in mind, I'm practically sure none of them is correct ... So how can we interpret this fact from an energy point of view ? The question may be silly but I'm just beginning quantum mechanics and I'm having a difficult time trying to understand it. I hope someone can help me. (Sorry for my bad english)


title: neutrinos by formation of "neutron pairs"

text: Here : http://www.newscientist.com/article/dn20084-neutron-star-seen-forming-exotic- new-state-of-matter.html are news on superfluidity in a neutron star. The necessary bosons they say are pairs of neutrons. So far, so good. But then they postulate the production of neutrinos in the formation of those neutron pairs. > When neutrons pair up to form a superfluid they release neutrinos which > should pass easily through the star,... What kind of reaction is that? n + n => neutrino + "n-n" + "XYZ" ?


title: Paradox of angular velocity

text: For a torque-free symmetric top, the Inertia tensor has an inverse $I^{-1}$, and $L=I\omega$. Which implies that $\omega=I^{-1}L$. But since $I, L$ are constants, $\vec\omega$ is a constant. However, $\vec\omega$ precesses. Why is there this paradox in argument?


title: Why don't we experience dilatations in Minkowski spacetime?

text: **The question** This question follows on from the use of projective coords for spacetime in Notation for Translation Group Generators . Under Felix Klein's Erlangen Program, Minkowski spacetime starts as a 4-d projective space. It is then made into an affine space (which has a notion of parallelism) by picking an invariant hyperplane at infinity; lines in the projective space which meet at infinity are parallel. An affine space admits dilatations which are expansions or contractions of space about the eigenpoint of the dilatation. When introduced projectively[1], dilatations are seen to be generalized translations because a translation is a dilatation in which the eigenpoint is on the hyperplane at infinity. Since one can smoothly change a dilatation into a translation by moving the eigenpoint onto the hyperplane at infinity, it is hard to avoid the conclusion that a dilatation is an active transformation on the same footing as translations, rotations and boosts. Yet we don't experience dilatations in everyday life. Why is this? **Some background observations** The general feeling in physics seems to be that dilatations are changes in the units of measurement, but this view is only possible because dilatations are introduced in an ad hoc way as scale transformations and the link with translations is missed. In order to see that dilations are active transformations like translations, the translation generators of Notation for Translation Group Generators become, $$ [P_{\Gamma}]^{\Phi}_{\ \Psi}=\delta^{\Phi}_{\Gamma}\delta^{4}_{\Psi} $$ where indices $\Phi,\Psi,\Gamma=0,1,2,3,4$. The translations are $P_{0},P_{1},P_{2},P_{3}$ and the dilatation generator is $P_{4}$. The attempt to construct classical relativistic Hamiltonian particle mechanics in Minkowski spacetime with dilatations fails unless the particles are massless. Massive particles only exist by removing the dilatations by hand in order to restrict the group to the ten parameter Poincare group. **Edit: An attempt to derive dilatations without having to read Coxeter [1].** Start with Minkowski spacetime as a 4-d projective space; points are rays in a 5-d vector space $x^{\Phi}\in V_{5}$ with upper case Greek indices like $\Phi=0,1,2,3,4$. The spacetime group is GL(5,R). Introduce parallelism by picking an invariant hyperplane $\delta^{4}_{\Phi}\in \tilde{V}_{5}$. Invariance means, $$ [D(g^{-T})]_{\Phi}^{\ \Psi}\delta^{4}_{\Psi}=\xi \delta^{4}_{\Phi} $$ where $\xi$ is a scale factor which is unimportant because the physical entities are rays. This implies that the group matrices are now have the block form, $$ \begin{equation} D(g)= \left[ \begin{array}{cc} \Lambda & a \\\ 0 & 1+a^{4} \end{array} \right] \end{equation} $$ where $\Lambda^{\mu}_{\ \lambda}$ is a 4x4 matrix and $a^{\mu}$ is a 4x1 vector and lower case Greek indices run $\mu=0,1,2,3$. These affine group matrices factor as, $$ \begin{equation} D(g)= \left[ \begin{array}{cc} 1 & a \\\ 0 & 1+a^{4} \end{array} \right] \left[ \begin{array}{cc} \Lambda & 0 \\\ 0 & 1 \end{array} \right] \ . \end{equation} $$ The first matrix is a dilation. In tensor notation, the pure dilatations are, $$ [D(g)]^{\Phi}_{\ \Psi}=\delta^{\Phi}_{\ \Psi}+a^{\Phi}\delta^{4}_{\Psi} $$ and a point transforms under a dilatation as, $$ x'^{\Phi}=[D(g)]^{\Phi}_{\ \Psi}x^{\Psi}=x^{\Phi}+a^{\Phi}x^{4} \ . $$ This shows that a dilatation acts like the identity on points in the hyperplane at infinity (the points which we can write as $x^{\mu}$ because $x^{4}=0$) and the eigenpoint of the dilatation is $a^{\Phi}$ with eigenvalue $1+a^{4}$. The last equation implies that a dilatation is an expansion or contraction of spacetime about the eigenpoint because a dilatation is a linear combination of the point and the eigenpoint. Consider two spacetime points $x^{\Phi}$ and $y^{\Phi}$. Transform them to $x'^{\Phi}$ and $y'^{\Phi}$ under a dilatation with eigenpoint $a^{\Phi}$. The dilatation moves the line joining $x$ and $y$ parallel to itself so that the line $xy$ (in an informal notation) is parallel to line $x'y'$. To see this, the point where the line $xy$ intersects the hyperplane at infinity is, $$ b^{\Phi}=\delta^{\Phi \Psi}_{[\Gamma\Delta]}x^{\Gamma}y^{\Delta}\delta^{4}_{\Psi} =x^{\Phi}y^{4}-x^{4}y^{\Phi} $$ and the dilatation acts like the identity on $b$ which is in the hyperplane at infinity, so line $x'y'$ also intersects the hyperplane at infinity at $b$ and so $xy$ and $x'y'$ are parallel lines. Now consider the quadrilateral $x'y'yx$. We have just seen that the sides $xy$ and $x'y'$ are parallel. The quadrilateral can be made into a parallelogram if the sides $xx'$ and $yy'$ are also parallel. These lines are parallel if they intersect at a point in the hyperplane at infinity. But, these lines both go through the eigenpoint $a$ of the dilatation. Therefore, $xx'$ and $yy'$ are parallel if the eigenpoint $a$ is moved onto the hyperplane at infinity. With the eigenpoint $a$ on the hyperplane at infinity, we've made a parallelogram by starting with a line $xy$ and moving it parallel to itself to $x'y'$ with $xx'$ and $yy'$ also parallel. This is how to construct a parallelogram with a translation. In other words, a translation is a dilatation with its eigenpoint on the hyperplane at infinity. This construction is why I cannot see a reason to exclude dilatations from the spacetime group, it seems dilatations must be just like rotations and boosts. [1] Coxeter, "Introduction to Geometry", chapter 5.


title: Index of refraction appearing in the radiative transfer equation

text: In this publication the Radiative Transfer Equation (RTE) (eq. (7)) contains the following emission term: $$n_i^2\kappa_{d,i}L_{b,i}(\mathbf{r})$$ where $n_i$ is the refractive index of component $i$, $\kappa_{d,i}$ is the discrete-scale absorption coefficient and $L_{b,i}(\mathbf{r})$ is the blackbody discrete-scale intensity emitted in direction $\hat{\mathbf{s}}$. The wavelength subscripts have been dropped for brevity. My question is, **how does the $n_i^2$ part appear in this term?** The referenced publications include three textbooks; I have checked the first two which include full derivations of the RTE and they both agree the emission term takes the form $\kappa L_{b}\left(\mathbf{r}\right)$; I do not have access to the other book. I'm guessing the answer has something to do with the fact that the radiative properties and intensity are labelled as 'discrete scale'. But google (scholar and web) has yielded no results for various searches including the terms `discrete`, `scale`, `refract`, and `radiation`. The objective of the paper is to spatially average the RTE, and for the mass, momentum, and energy transport equations, the starting point has always been the general (plain) form of the equation with all terms included. Here, the author is starting from a slightly different form and I would like to understand why. I've contacted the author about it and I am waiting on a response. I'm hoping someone here can riddle out how the emission term could contain the refractive index. Keep in mind assumption (iii): > the refractive index is constant in each phase


title: Applying a force on a rigid body which is used to create torque

text: ![Forces](http://i.stack.imgur.com/boy1P.png) In the above picture, I have a rigid body, in turquoise, which is connected to a point, the red circle. The dotted lines are used to divide objects or lengths, they are not part of the rigid body. The distance from the red circle to the application point of Force 1 is X+1. The length of the lower arm of the rigid body has a length of X + 1.5, and the distance of the red circle to the perpendicular line of the lower arm is X wide. The yellow arrow around the red circle is the direction of torque that we want to measure. If I produce a Force, Force 1 at the point shown, is has a certain magnitude and angle. Now the turquoise body is a rigid body that does not flex or bend. Am I right to believe that because this body is rigid and not rotating, that it is in Translation? If it is in Translation, should all particles of the upper arm be feeling the same force and angle of Force 1? And if so, am I safe to assume that the Force 2 traveling down the rigid body is the cosine of the angles between Force 1 and Force 2, thus decreasing the magnitude of Force 2 relative to Force 1 because of this force deflection? The final answer that I am grappling with is how do I correctly calculate torque at the red circle. I now that torque is Force x Distance. Which is the proper way to calculate torque? Is it: Force 1 x (X+1) - regardless of the deflections of Force 1 to Force 2? or Force 2 x (X+1.5) - which would be this decreased force multiplied by the lower arm length? I hope this wasn't too convoluted, but I was told that regardless of the elbow type fashion of this rigid body, the only force that matters when calculating the Torque is Force 1 multiplied by its distance to the red circle, X+1. If anybody can help answer this question and give me the physical proof as to why either method is correct, I would truly appreciative it. Thanks.


title: How can one calculate the central term of the conformal field theory algebra (and show it's really the virasoro algebra)?

text: So I'm following Szabo's book "An Introduction to String Theory and D-brane Dynamics (2nd ed, 2011); still on the canonical treatment in chapter 3. After doing a mode expansion, we get (up to a constant) a set of nice ladder operators \begin{equation}[\alpha_m^\mu, \alpha_n^\nu] = m\delta_{m+n}\eta^{\mu\nu}\end{equation} where $\delta_{m+n}$ is unity if $m=-n$, zero otherwise; $\eta$ is the Minkowski metric in D dimensions, $m$ and $n$ are integer labels and the $\alpha$s are Fourier coefficients. I'm omitting the LM/RM sets and concentrating on the open string. We use these to define these composite operators: \begin{equation} L_n = \frac{1}{2}\sum\limits_{m=-\infty}^\infty{\alpha_{n-m}\cdot\alpha_m}\end{equation} which we want to use for generate conformal transformations. Cool. So we need some Witt-like algebra, and because we're quantising we're expecting some weirdness, and it turns out you get the Virasoro algebra. But I just can't get the central term. For a start, it doesn't look right: each pair of $\alpha$s can give me a factor of m, so having \begin{equation}[L_m,L_n]\end{equation}, we have four $\alpha$s and hence we'd expect to be able to extract some central term of order $n^2$. But the Virasoro central term is $n(n^2-1)$ - so where does the extra n come from? There are no conveniently-placed sums that might give rise to this. Every time I attempt to plug through the algebra I end up with something silly, for example $\sum\limits_{p=-\infty}^\infty p$. So no central term at all! So the root of my question is: where does the extra n come from? If I can understand that, hopefully I'll be able to do the rest of the derivation myself (don't take away my fun:)).


title: self-adjointness of an operator containing functional derivatives

text: I have given a Hamlitonian in curved spacetime containing a functional derivative and I wonder whether there exist techniques to find out if this operator is self-adjoint or not?! Any ideas? Thanks.


title: What is the relationship between kēV and kVp?

text: On X-ray topics we get used to talk about "energy", but what is kēV? and what is the relationship between kēV and kVp?


title: Does expanding space cost energy?

text: Does the cosmic inflation reduce the energy density (inversely) proportional to the volume, or does the inflation "cost" energy? Is space itself "something" created at the expense of energy?


title: $m_l=0$ in Hydrogen atom of Scrodinger equation

text: $m_l=0$ corresponds to s-orbital and the values associated are: $$\lambda=2\pi r/m_l\to\infty$$ It means a wave of of infinite wavelength has a constant height at all values of $\phi$(azimuthal angle). $$J_z=\pm hr/\lambda\to0\text{ also }J_z=\pm pr$$ which implies **zero linear and angular momentum** as $r\ne0$. Now what does **zero angular and linear momentum** signify/mean?(I think the electron is at rest but it cannot be)


title: Observers in Special Relativity

text: Suppose I am an observer in Minkowski space, observing a particle traveling with a constant velocity $v$. If I want to calculate the particle's Lorentz factor, given that the particle is traveling at a constant velocity, I can say $\gamma = \frac{\Delta (c\cdot t)}{\Delta \tau}$. My question is, to get a value for $\Delta t$, do I look at my clock, or do I look at the particle's clock?


title: Does this photograph portray double muon impact with nanogold atoms?

text: 1PHOTO 1: Macro-photograph of an NIH/FDA TEM of a nanogold dark stained biological sample projected onto Silver Halide (AgX) photographic gel paper. ![PHOTO 1](http://i.stack.imgur.com/1qSPj.jpg) On June 10 I questioned if PHOTO 1 depicted an electron-positron particle annihilation caused by TEM generated electron impacts. My engineering instincts led me to think that the simultaneous red and green on the left and right hemispheres might be related to a positron (which Dirac likened to an electron moving back in time) was associated with what I thought to be a collision of particles that generated the separation of the two green areas that appear to be undergoing some type of explosive event. I analogized the image in PHOTO 1 to the characteristics of the Roman era glass cup, the Lycurgus Cup which can be viewed in Wikipedia, reflects green and transmits red light depending on surface which the light photons initially strike. Anna.v (10-June-14) kindly explained that the energy required to generate an annihilation (511 Kev) would not be present in a 200 KeV TEM and suggested that such energies could be found during collision with a random muon but an accelerator would be needed for verification of nuclear emulsion type material which recorded the event. Unfortunately, that would be the film from which PHOTO 1 was generated which is not readily available, so I began a search for other explanations beginning with the definition of muon, a lepton, one of the four elementary particles - it contains no quarks or sub particles. The moons shadow from cosmic rays is depicted in Wikipedia under the definition of muon and the greens associated with the rays appear consistent with the greens in PHOTO 1. However, in the original image of PHOTO 1, the green areas appear to engulf dozens of smaller particles and in the thousands of photographs taken of these TEMS, some type of interference pattern projects from "particle emissions" generated by what I suspect to be electron impacts with nanogold atoms used to dark stain the targeted biological. No others throw off this distinctive green coloration. The imaging of PHOTO 1 probably inversely projects the green from the bottom to the top based on the angle of the light to the camera and the composition of the photographic gel paper (PHOTO 5). ![PHOTO 2](http://i.stack.imgur.com/iWtxx.jpg) Now I question if PHOTO 1 may be the image of a double muon impact, separated by the nanometer sized leg of the biological triangle. (PHOTO 2) If so, the deceleration of the muon would produce Bremsstrahlung radiation and decay into anti-particles. In this respect note the black streaks in PHOTO 1 which intersect at about 90 degrees above the two green areas which also align at approximately 90 degrees. ![enter image description here](http://i.stack.imgur.com/jsaMw.jpg) PHOTO 3: Screen photograph of enlarged areas of impact. Above the bottom vertices of the triangle and left of the right leg a circular particle emission spirals upwards possibly imaging expanding electron bands that have been pushed out from the impact area center. (Assuming the Rutherford orbital model is correct). Almost horizontally to the right of spiral and right leg another gold region appears to have been impacted. Could Bremsstrahlung radiation from the decelerations of the muons provided the shorter wavelengths needed to image an EM wave? In point is the picometer size waveform that transits from the lower vertices towards then away from two areas of impact on the right side of the right leg of the triangle in PHOTO 2. Might this be an errant displaced electron seeking a new home - it appears to traverse towards two spots of glowing gold that repel it - an indication of reversal of ionic attraction. ![PHOTO 4](http://i.stack.imgur.com/ShpXJ.jpg) PHOTO 4: Circular bands of color projected from suspected areas of particle emission/collision/expansion in PHOTOS 2 & 3. ![PHOTO 5](http://i.stack.imgur.com/C0qWA.jpg) PHOTO 5: PHOTO GEL COMPOSIITON: Photographic paper is comprised of four layer, a paper base coated with a layer of baryta/gel and photo gel, then about .5 MIL (12,500 nm) AgX (silver halide) and topped with a gloss coat which serves as somewhat of a projection screen with ellipsometric photography. This image portrays the reflection and "depth" and inverts the gold, possibly a reflection of gold dopant in the gel, from bottom to top. Some insight into this inquiry can be gleaned from the following project posted on the web. “Measurement of the Lifetime of Muons and Pions” “In competition to muon decay negative muons can be captured by nuclei in a similar fashions as electrons from the K-shell can be captured by a nucleus. The probability for a muon capture increases with the fourth power of the nuclear charge number. Muons can be captured into the K-shell of an atom forming a muonic atom. Since the rest mass of muons is larger by a factor of about 200 compared to electrons their orbit is approximately a factor of 200 closer to the nucleus. Consequently, there is a non-vanishing overlap of the wave function of the muon with the wave function of the nucleus and this can lead to a capture of the negative muon by the nucleus. For Z = 40 [Au=79] the Bohr orbit is already inside the nucleus. The capture probability for this condition is very close to 1.” (Advanced Laboratory Experiments, Universität Siegen, Dr. I. Fleck). Coherent suggestions, input and answers would certainly be appreciated. Thank you for considering these issues. Walter Kyle 29-Jul-2014


title: Time Varying Magnetic Field

text: What is said time varying magnetic field? I have heard a lot about it and the Internet is not willing to give me any answers. I assume time varying is a qualification, I was reading around here and heard about an induced magnetic field so I take it that is another qualification? What does that mean? Thank you.


title: Does one really need classical physics in order to understand quantum physics?

text: I want to start studying quantum mechanics, and then move to quantum field theory. I have a strong mathematical background, and I think this aspect of quantum physics won't be a problem to me. Though, Please note that I'm not a physicist and I have taken just one short course in university about classical physics, and I don't know anything about it. I ran over the Internet, and I saw that a lot of people debate over this. Some say that you don't need classical physics to understand quantum physics, others says you need to learn it in order to better understand quantum physics. Therefore, I thought that this is the perfect place to ask it. I would like to learn your opinion about this. And if you think that classical physics is a must or very useful for quantum physics, please specify me a single book, possible under 400 pages, which covers all the materials that I will need from classical physics.


title: What is the future of our universe?

text: Is the universe heading towards a Big Freeze, a Big Rip, a Big Crunch or a Big Bounce? Or is it part of an infinitely recurring cyclic model?


title: The Einstein-Hilbert Action On-Shell

text: If one consider the Maxwell action as $$S=-\int \mathrm{d^{4}}x\\! \ \frac{1}{4}F_{ab}F^{ab} \,$$ one find the usual Maxwell equation $$\partial_{a}F^{ab}=0$$ Then one can simply arrive the following the Maxwell **on-shell** action $$-\int \mathrm{d^{4}}x\\! \ \frac{1}{2}\partial_{a}(A_{b}F^{ab}) \,$$ Now my question is for Einstein Hilbert action. What is the expression of **on-shell** Einstein Hilbert action $$S=\int \mathrm{d^{4}}x\\! \ R \,$$ I know how to find Einstein equation from variational principle, which is given as $$R_{ab}-\frac{1}{2}g_{ab}R=0$$ How to write **on-shell** Einstein Hilbert action with above equation?


title: Is proper time an intrinsic value of Minkowski space?

text: What is proper time? Is it a part of Minkowski space (that is a mere spacetime interval)? Or is it an intrinsic characteristic of massive particles (a sort of "aging")? Example: In the following Minkowski diagram,   * A is the worldline of a particle moving with the observer's reference frame.    * B is the worldline of another particle with proper time between two events = 4 (space distance x= 3, time interval t = 5, proper time τ = 4, according to the equation $$ \tau^2 = \delta t^2 - \delta x^2 $$)   * C is just 2 events without any worldline between, and if a particle would travel through both events, its proper time would be 4 ![enter image description here](http://i.stack.imgur.com/3IKCG.jpg) Is there proper time in C even if no particle is travelling there? Or is in C only hypothetic proper time in case a particle is travelling this path?


title: Absolute value in the argument of a logarithm

text: I am wondering how the author rationalizes the removal of absolute value bars around the quotient argument of a natural logarithm. My take on this is that the potential at point $b$ MUST be greater than the potential at point $a$, or else the removal of the absolute value bars obtained through anti- differentiation would be unwarranted. This question has actually come up several times in problems relating to potential difference and any clarification would be much appreciated. Here are links to the example from my text. Figure: ![](http://i.imgur.com/IIIgEnF.jpg) Derivation: ![](http://i.imgur.com/pKB2WAn.jpg)


title: Binary stars' apparent magnitude

text: If you plot the apparent magnitude of a binary system (so you are unable to see 2 distinct stars) against time, it gives a repeating 'M' effect. ![Light curve for binary](http://i.stack.imgur.com/MneRp.gif) (from http://accessscience.com/) Assuming one star is brighter (and bigger in size) than the other. From the perspective of the observer will the apparent magnitude be higher when the bigger and brighter star is behind the smaller dimmer star, or in front? Alternatively worded, in the graph above, when the two stars are next to each other the magnitude will be at it's highest (the tops of the 'm'), but what will be the brighter star be in front or behind the dimer star in the center of the 'm', or at the ends on the 'm'?


title: How does atmospheric seeing evolve over time?

text: I am writing an automated target selection application in which the seeing at the time of observation is an important factor: some fields are important to observe under good seeing, others, less so. Of course, it can only choose based on current conditions, not the conditions a few minutes in the future under which the observation will actually occur. In writing simulations to evaluate the effectiveness of different algorithms, I need to generate simulated seeing values. I have global statistics for the seeing at the site, but determining how the seeing changes over time is important.


title: How is wavelength actually related to space (/distance)?

text: Is a photon of 400nm in wavelength " **smaller** " than a photon of 1km in wavelegth?


title: Equation for the equipotential lines?

text: What is the equation for the equipotential lines in $x$-$y$ plane for a dipole oriented along the $x$ axis?


title: Rigid bar suspended by two ropes, tension of first rope after second rope is cut?

text: This is from a practice exam, I've been sitting here thinking about it for over an hour and can't convince myself of an answer, or write down any relevant exact equations. A bar of uniform density $\rho$, length $L$, and negligible width is suspended by two inextensible ropes of negligible mass. The second rope is cut, what is the tension of the other rope shortly after cutting? What is the tension's rate of change? I've been trying to work it out with numbers $g=10 m/s^2$, $\rho=30kg/m$ and $L=1m$ just to keep it simple. Obviously before the rope is cut the tension of both ropes is $T_1 = T_2 = 150N$ and after the bar falls down and is stationary, then $T_1 = 300N$ so there is some smooth function of the tension (I think). The angle between the first rope and the bar is $90\deg$ and increases to $180\deg$, and the force due to gravity can be divided into normal and tangential forces. The normal force increases the angular momentum, and the normal force is a maximum the moment the rope is cut. I'm having trouble putting this all together, but I think I am on the right track. Thanks in advance. edit: As stated below, this is the force of gravity components $F_N = \rho Lg Sin(\theta )$ and $F_T = -\rho Lg Cos(\theta )$ I still have trouble applying angular momenta to this problem.


title: Shear stress in directions other than the flow direction

text: Consider the flow of a newtonian fluid in a rectangular pipe. Consider a 3D coordinate system with the property that.that the flow of the fluid in the pipe is parallel to the x-axis. Let $V:\mathbb{R}^3\rightarrow \mathbb{R}^3$ return the velocity vector at each point of the 3D space. Assume that $V$ is differentiable with continuous partial derivatives. Consider a cuboid inside the fluid filling the pipe: $[x_0,x_0+\triangle x]\times [y_0,y_0+\triangle y]\times [z_0,z_0+\triangle z]$ . I am trying to compute the shear force on the face $\\{(x_0,y,z)|y_0\leq y\leq y_0+\triangle y,z_0\leq z\leq z_0+\triangle z\\}$ of the cuboid. My guess that it would be: $$\int_{y_0}^{y_0+\triangle y}\int_{z_0}^{z_0+\triangle z} \mu[v_x(x_0,b,c)+v_y(x_0,b,c)]dc \,db$$ (Note about the notation: $V(x,y,z)=u(x,y,z)\mathbf{i}+v(x,y,z)\mathbf{j}+w(x,y,z)\mathbf{k}$) Am I right? If I am not right could you please give the correct shear force on the face I am talking about. Sorry if the question is naive. If the question is naive it would probably be because I only had an engineering course in fluid mechanics and the course considered only shear stress parallel to the direction of flow. I don't know if this is because shear stress only acts in the direction of shear flow or because the book was assuming (without mentioning) that some components of $V$ are zero or have zero partial derivatives. Thus the only equation for shear stress I saw in the course is $\tau=\mu \frac{\partial v}{\partial y}$. I am looking for the most general case. Thank you **Edit:** After thinking again, I modified my guess to: $$\int_{y_0}^{y_0+\triangle y}\int_{z_0}^{z_0+\triangle z} \mu[v_x(x_0,b,c)+v_z(x_0,b,c)]dc \,db\,\mathbf{j}+\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_{y_0}^{y_0+\triangle y}\int_{z_0}^{z_0+\triangle z} \mu[w_x(x_0,b,c)+w_y(x_0,b,c)]dc \,db\,\mathbf{k}$$ However I am still unsure, and I would like someone to write the right answer for the net force acting on the face. **Reminder** : The main question of this post is the following: Is my second guess correct ? If not what is the right answer for the shear force ?


title: 2 entangled electrons in QFT

text: In field theory, by quantizing a dirac field, we can obtain a creation operator for a single electron of definite momentum, of definite spin up or down, these respectively are: $$a^\dagger_{+}(p)|0\rangle, {a^\dagger}_{-}(p)|0\rangle$$ Where we've defined the former to create a spin +1/2 electron, the latter to create a spin -1/2 electron. By addition and repeated-application of these creation operators we can write down a state of any number of particles, each having any superposition of spins. : $$ \int dp f(p)\prod_{i=0}^n(\alpha_ia^\dagger_{i+}(p)+\beta_i{a^\dagger}_{i-}(p))|0 \rangle $$ Where of course $i$ labels the particle, and $a,b,f$ are some distributions. **Question: For a given field theory, how does one write down a creation operator for a pair of entangled particles? (say electrons in a Dirac theory of spinors)** In quantum mechanics, an entangled state is one which lives in a tensor product Hilbert space, but does not have a tensor product decomposition. Since Fock space is essentially built up with a bunch of tensor products of Hilbert spaces, it doesn't seem unreasonable to demand that it contains such entangled states. But how does one explicitly write down such a state?


title: Massless fields in curved spacetimes

text: I read the following statement in one of Penrose's paper > zero rest-mass field equations can, with suitable interpretations, be > regarded as being conformally invariant. I take this to imply that if I would like to describe massless scalar fields (for example) in curved spacetimes I should couple them conformally. More precisely, the curved space generalization of the action is $$ \- \phi \partial^2 \phi \to - \phi \left( \nabla^2 + \frac{1}{6} R \right) \phi ~~~(d=4) $$ instead of the naive $\partial^2 \to \nabla^2$. Why is this the case? EDIT: I believe this holds for massive scalar fields as well, though we no longer have conformal invariance.


title: Non-locality and Bell's theory

text: Non-Locality – (just ) one more question? I have read comments that Bell’s theory proves quantum mechanics is non-local, and also comments that it does not. I have read a comment by a very eminent person who stated what it means is that a measurement of a particle at one location "influences the state of the other particle". So my question is this: Two particles in the singlet state (as always) go their separate ways. At time one each passes through a SG apparatus with orientation in direction $z$. At time two particle 1 passes through a second SG apparatus but with orientation direction $x$; nothing happens to particle two. At time three the spin of both particles is measured in direction $z$. What will be the outcome of the spin measurements?


title: Energy problem: What's wrong here?

text: > A car ($m=540\,\text{kg}$) engine, has a power of $60\,\text{kW}$. The > static friction coefficient between wheels and road is $k=0.6$. How long > does it take to reach the speed of $27.7\,\text{m/s}$, with constant > acceleration? I have tried the following: The energy to reach given speed is: $$\frac{mv^2}{2}=\frac{540\cdot 27.7^2}{2}=2.07\cdot 10^5\,\text{J}$$ In the meanwhile I dissipated (because of friction): $$-F_a\cdot s=-mgk\cdot\frac{1}{2}at^2=-mgk\cdot\frac{1}{2}\frac{\Delta v}{t}t^2=158,922\cdot t$$ The engine can do a work of $60,000\,\text{J}$ per second, so the work done is $60,000\cdot t$ So, I can do: $$2.07\cdot 10^5\,\text{J}+158,922\cdot t=60,000\cdot t$$ And I obtain a negative time. How is possible? What's wrong? Thanks a lot


title: Is it true that an isolated fundamental particle does not decay?

text: Is it true that an isolated fundamental/elementary particle does not decay? It seems logical to me.


1. How many photons can an electron absorb and why?   2. Can all fundamental particles that can absorb photons absorb the same amount of photons and why?   3. If we increase the velocity of a fundamental particle, do we then increase the amount of photons it can emit?   4. At constant temperature and mass of an isolated fundamental particle that does not and will not move (constant speed and 0 vibration ), is emitting a photon the only way to loose energy?


title: Displacement with zero velocity

text: I know that we can rotate a deformable object using internal forces only in space. Thus we can cause an angular displacement without the use of any external forces. The following youtube video shows are real life example to do so - http://www.youtube.com/watch?v=RtWbpyjJqrU **My question is** Can we do the same with linear motion also? By that I mean, can we cause a displacement in an object using only internal forces. I could not think of such a setup. And if we can what makes the angular displacement more special than linear displacement?


title: Is the step of analytic continuation unavoidable or can you model around it?

text: One sometimes considers the analytic continuation of certain quantities in physics and take them seriously. More so than the direct or actual values, actually. For example if you use the procedure for regularization, it sometimes seems like an ad hoc step. **Question:** In cases where analytic continuation is applicable, does this suggest there is another formulation of that theory, which leads to these result directly? That would be a theory where the "modified" interpretation of the mathematical quantities might be taken to be a starting point. For example if you define some fundamental quantity of your theory as an integral or sum, and it doesn't converge somewhere, and you make an analytic continuation to get some valuable results. Could this imply there is a formulation where that value comes naturally, i.e. a formulation where there never is this sum object which makes problems?


title: What is the pressure on an empty pneumatic tube?

text: At the residence I'm constructing a kitchen digester. The gas is to be collected into an old bicycle/scooter tube - initially as empty as may be after wringing it out.. Will the digester (total volume 50Litres, I plan to use no more than 2/3 of the total volume) i.e. about 7.5L gas at any time - be able to fill the tube against atmospheric pressure? Is it correct to assume the gas in the tube will be at 1 atm. pressure? EDIT: The photograph of the link shown is a very similar system; the digester I'm looking to construct is larger in volume (50L as against 1Gallon in the figure), and collects the gas over water. ![Sketch of a similar system](http://i.stack.imgur.com/O67dw.jpg)


title: Under what conditions, will a projectile motion follow a half circle?

text: What came to my mind is to throw an object far from the earth with a particular speed equal to $$V_{sat}=\sqrt{\frac{G\times M_{central}}{R}}$$like satellites. However, it would follow a full circle.


title: Can I express the heat flow of a fluid in terms of estabilshed characteristics of the velocity distribution?

text: If $\rho$ is the mass density of a fluid and $A({\bf v})$ is an function of the velocity, which is distributed according to $f({\bf v})$, we have an averaging process $A\mapsto \langle A\rangle:=\int A({\bf v})\ f({\bf v})\ \mathrm d^3v.$ We now might define the pressure and energy in the frame of the flow as $P_{ij}:=\rho\ \langle (v_i-\langle v_i \rangle)(v_j-\langle v_j \rangle) \rangle,$ $E_\text{Kin}:=\rho\ \langle \tfrac{1}{2} ({\bf v}-\langle {\bf v} \rangle)^2 \rangle.$ I realize that $$P=\rho\ \text{Cov}(v_i,v_j),$$ $$E_\text{Kin}=\tfrac{1}{2}\sum_i \text{Var}(v_i).$$ So they are given by covariance (see also covariance matrix) and variance for $f$. Can I formulate the heat flow $Q_i:=\rho\cdot \langle \tfrac{1}{2} ({\bf v}-\langle {\bf v} \rangle)^2 (v_i-\langle v_i \rangle) \rangle$ also in terms of general statistical functionals of ${\bf v}$? E.g. some functional of skewness?


title: Experimental heat equation for a bottle of water

text: I'm not sure if this question belongs here, because I'm asking a question about experimental data rather than theoretic formula. Anyway, I'm trying to figure out the time-temperature relation of the (hot) water in an ordinary sports bottle. Assume the bottle is full with 500mL of water in it. Do you guys have any data for that? The equation doesn't have to be accurate, an error up to 20% can be accepted.


title: Is there still light in practical darkness? Do photons penetrate everywhere?

text: I always thought that, even in darkness, there would still be some light, making complete darkness, i.e. complete absence of photons as far as I know, just a theoretical thing. When I tried looking up whether or not this view was actually correct, I didn't find any clear information, however. Am I wrong?


title: Charge, velocity-dependent potentials and Lagrangian

text: Given an electric charge $q$ of mass $m$ moving at a velocity ${\bf v}$ in a region containing both electric field ${\bf E}(t,x,y,z)$ and magnetic field ${\bf B}(t,x,y,z)$ (${\bf B}$ and ${\bf E}$ are derivable from a scalar potential $\phi(t, x, y, z) $and a vector potential ${\bf A}(t,x,y,z)$), knowing that   1. ${\bf E}=- \nabla \phi - \frac{\partial {\bf A}} {\partial t}$   2. ${\bf B}= \nabla \times {\bf A} $   3. $U=q \phi - q {\bf A} \cdot{\bf v}$ ($U$ is the velocity-dependent potential) the Lagrangian is $$L=1/2 m v^2- q\phi + q{\bf A} \cdot{\bf v}.$$ Considering just the x-component of Lagrange's equation, how can I obtain $$m \ddot{x}=q(v_x \frac{\partial A_x}{\partial x} + v_y \frac {\partial A_y}{ \partial x} + v_z \frac{\partial A_z} {\partial x})-q(\frac{\partial \phi }{\partial x} + \frac{d A_x}{dt}) ?$$


title: Steepest descent for Mellin-type integration

text: Here I would like to see the behavior of a function as an integral when its argument (which is a parameter in the integral) goes to zero. If I try to evaluate an integral $$I(\lambda) = \int^{i\infty}_{-i\infty}\mathrm{d}z\frac{\mathcal{M}(z)}{z}\lambda^z$$ in which $\lambda$ is a number which approaches zero. Is the following way correct or not? First we write it as $$I(\lambda)=\int^{i\infty}_{-i\infty}\mathrm{d}z\frac{\mathcal{M}(z)}{z}e^{z\log{\lambda}}$$ where $\lambda$ is some meromorphic function, but on the exponential the first derivative of the exponent doesn't have any zero, therefore I pull the $1/z$ factor onto the exponent: $$I(\lambda)=\int^{i\infty}_{-i\infty}\mathrm{d}z\,\mathcal{M}(z)e^{z\log{\lambda}-\log{z}},$$ then the exponent $z\log{\lambda}-\log{z}$ is stationary at $z\sim0$ when $\lambda\to0$, then we just approximate the integral with the limit of the integrand when $z\to0$, which is $\mathcal{M}(0)\log{\lambda}$. Is this way of doing steepest descent reasonable?


title: Effect on length of day as the polar ice caps melt

text: If the polar ice caps of the earth melt, how will the length of the day be affected?


title: BRST transformation of adjoint spinor

text: in Yang-Mills-Theory with matter fields a dirac spinor $\psi$ transforms under BRST as $$\psi \to \delta_\Omega\psi=i\eta\psi $$ with $\eta$ being a ghost field. If I want to get the transformation of the adjoint spinor $\bar \psi$ I get by using the invariance of $\bar \psi \psi$ $$ 0=\delta_\Omega (\bar \psi\psi)=(\delta_\Omega\bar\psi)\psi - \bar\psi (\delta_\Omega\psi) \quad \Rightarrow \quad \delta_\Omega \bar\psi=i\bar\psi \eta$$ If I now want to get the transform directly, I get $$ \delta_\Omega \psi^\dagger \gamma_0=[\psi^\dagger,\Omega]_+\gamma_0=([\psi,\Omega]_+)^\dagger \gamma_0=(i\eta\psi)^\dagger\gamma_0=-i\bar\psi\eta $$ So I get different results. Where is my error? What I am not sure about, is whether if I have $(\eta\psi)^\dagger$ if this is equal to $\psi^\dagger\eta$ or $-\psi^\dagger \eta$ as the transpose part should be purely in the dirac space. Thanks in advance.


title: Phase transitions. Conceptual link of my intuitive notions and definition of Georgii's book in terms of probabilities

text: In his classic book O. H. Georgii (Gibbs Measures and Phase Transitions) in Chapter 2 p. 28 define the concept of phase transition follows. > **Definition** _A potencial $\Phi$ will be said exhibit a phase transitions > if $|\mathcal{G}(\Phi)| >1$._ In other words the set of Gibbs measures $\mathcal{G}(\Phi)$ for a potential $\Phi$ associated with a physical model has more than one Gibbs measure $\mu$. But for me this definition is'' hard to swallow "and does not seem intuitive.I think my difficulty is that being a mathematician I could not understand what exactly is a state of equilibrium in models of statistical mechanics. For example in the Ising model and Potts model.   * > The first intuitive notion that I have to equilibrium state (which may be a mistaken analogy) is the solid state, liquid and gaseous water for example. Also called phases solid, liquid and gaseous respectively. Therefore understand the term phase transition as the transition from one equilibrium state to another state of equilibrium. For example, the ice turning into liquid water. ![enter image description here](http://i.stack.imgur.com/SaYye.jpg)   * > The second intuitive notion that I have (which may be full of misunderstandings) about the phase transition is when the state of equilibrium of a model ferromagnetic spin moves to another state (equilibrium?) due to increasing temperature. My question assumes that the definition of Giorgii is just a technical way of describing my intuitive notions. _How do I make the conceptual link of my intuitive notions of phase trantions with the definition of Georgii's book in terms of probabilities?_ > **Question.** _More precisely wonder if there is a conceptual definition of > **equilibrium state** and **phase transition** that are acceptable in > statistical mechanics that can provide a link between my poor **intuitive > notions** and the concept of **phase transition** in terms of probabilities > in Georgii's book?_ Thanks in advance.


title: Scientist propose solution to quantum measurement problem, explain further

text: Link to Article The scientists > In their recent opus magnum, Theo Nieuwenhuizen (Institute of Physics, UvA) > and colleagues claim to have found a solution to the so-called quantum > measurement problem Their claim > After reviewing the literature on measurement models, they focus on the so- > called Curie-Weiss model for quantum measurements. The joint dynamics of the > tested system coupled to the measurement apparatus now produces results that > explain the measurement postulates. It shows that the off-diagonal terms > ("cat-terms") disappear in a physical process due to the coupling to the > apparatus, on a basis selected by the interaction Hamiltonian. The > registration of the measurement outcomes by the pointer of the apparatus > occurs because the measurement triggers the transition from the initial > metastable state to one of the stable final states; the apparatus being > large, this means an amplification of the signal. The transition of the > pointer variable from its initial metastable state to one of the final > stable states is a process with classical features, such as the dumping of > excess free energy in the bath. I am by no means a physicist, but I do enjoy trying to understand some of the issues they face. From what I've read this quantum measurement problem, well is a problem. Could someone explain why it is a problem, and how these scientist from the Institute of Physics propose to solve it? (as layman as possible)


title: Coulomb's Law- Why the Coulomb's law is valid only for point and static charges?

text: Why the coulomb's law is valid only for point and static charges? Is there is any definite reason?


title: Speed of light in general relativity

text: My question has a few parts concerning the speed of light in general relativity. Firstly, time changes in response to gravity and speed. Therefore, as gravity effects time in an area of space, should the speed of light "change?" When I say "change" I understand that light's speed is finite but wouldn't the length of second change and then change light's speed in meters per second? Secondly, does the speed of light actually change with gravity? Time stops at the speed of light but does that mean light cannot experience time? Can light experience slower time while travelling through warped space time?


title: N=4 SYM from Klebanov-Witten field theory

text: This is with reference to M. J. Strassler's lectures on "The Duality Cascade" pg. 46. I want to see how $\mathcal{N}=4$ SYM emerges when D3 branes, in the KW setup, are placed at smooth point of the conifold, purely from field theory point of view. That is: Given a $U(N) \times U(N)$ quiver with $A_i$ and $B_i$ in the bi- fundamental and the anti-bi-fundamental respectively and an $SU(2) \times SU(2)$ invariant quartic superpotential (which vanishes for $N=1$), in what manner do we higgs the theory so as to get $\mathcal{N}=4$ SYM? For the case of $N=1$ there is a comment in the lectures. It says, > ... suppose we just have one D3 brane and we allow $A_1B_1$ to have an > expectation value, so that the D-brane sits at some point away from the > singular point of the conifold. Then the gauge group is broken to $U (1)$, > and six scalars remain massless — the six possible translations of the > D3-brane away from its initial point — exactly the number needed to fill out > an $\mathcal{N}= 4$ $U(1)$ vector multiplet. I want to understand this statement. Please help me understand how to work out for case of $N=1$. I can try for the generic value of $N$.


title: Which point on a diffuse reflecting plane is brightest to the viewer?

text: Given is the plane y = 0. A point light source is positioned at (1,10,1). The      plane reflects only diffuse. The viewer is positioned at (4,6,4) and sees all      points on the plane with 0 <= x <= 8 and 0 <= z <= 8. At which point on the      plane does the viewer perceive the highest reflection?      I think the answer is (1, 0, 1), the projection of the point light source onto the reflecting plane. Because a diffuse reflection scatters light equally in all directions so it doesn't matter where the reflection is relative to the viewer. All that matters is where the reflection is relative to the point light source, because the point light source shines a varying amount of intensity per unit area on different parts of the plane. But the highest concentration of intensity will be shined along the plane's normal vector aka the nearest point on the plane. Is this correct and if not how do you do this? Thank you


title: Lie theory, Representations and particle physics

text: This is a question that has been posted at many different forums, I thought maybe someone here would have a better or more conceptual answer than I have seen before: Why do physicists care about representations of Lie groups? For myself, when I think about a representation that means there is some sort of group acting on a vector space, what is the vector space that this Lie group is acting on? Or is it that certain things have to be invariant under a group action? maybe this is a dumb question, but i thought it might be a good start... To clarify, I am specifically thinking of the symmetry groups that people think about in relation to the standard model. I do not care why it might be a certain group, but more how we see the group acting, what is it acting on? etc.


title: Looking for the curve traced by a moving bicycle when its steering bar is fully rotated

text: I am looking for a curve traced by a moving bicycle when its steering bar is fully rotated either clockwise or anti-clockwise. How to model it mathematically? Is the curve a circle? My attempt is as follows: Let $\vec{r}_1(t)$ and $\vec{r}_2(t)$ be the position vectors for the tangent points (between road and tires) on the rear and front tires, respectively. I know that $|\vec{r}_2(t)-\vec{r}_1(t)|$, $|\dot{\vec{r}}_1(t)|=|\dot{\vec{r}}_2(t)|$ and $\dot{\vec{r}}_1(t)\cdot\dot{\vec{r}}_2(t)$ are constants. $\dot{\vec{r}}_1(t)$ is in the direction of $\vec{r}_2(t)-\vec{r}_1(t)$. Assuming the tires rolls without slipping then their linear velocity is the same.


title: Can gravity be absent?

text: Can gravity be absent? Not weightlessness as an astronaut experiences it because the astronaut's body still has gravity which will probably manifest in the presence of another smaller/larger body. For instance, given the mass of the moon it is subject to Earth's gravity holding it in orbit. I mean, total absence of gravity; can it happen? I think this question will need to be edited; reading it seems even more muddled than the thought ... I can't express it. Feel free to vote to close/delete


title: How much force needed to push a car on neutral?

text: The car is a 2002 Toyota Corolla SE I have in my garage on a stone surface, I.e., the floor itself. What I want to know is how to determine the force needed to get it in motion at all. The curb weight is 2,400 lbs. How could I determine the force needed to push it on neutral to get it in any kind of motion?


title: What does scalar phi represent in spacetime?

text: Trying to understand one-forms and vectors via Schutz's A First Course In General Relativity. ![contraction of one-form and vector\)](http://i.stack.imgur.com/3mXwV.gif) His example uses a spacetime diagram, a scalar field phi, a curve (worldline) parametrized using proper time and the tangent vector (4-velocity). I think I understand, but I can't see the physical significance of the scalar field phi. In plain English, in the spacetime diagram, what does or might phi represent?


title: Hollow Tube Stronger than Solid bar of same Outside Diameter (O.D.)?

text: I was listening to some co-workers talking about problems meeting stiffness requirements. Someone said that even with a solid metal rod (instead of the existing tube) we could not meet stiffness requirements. I started daydreaming... and went back in time over a quarter of a century to some class I was taking in college. Things are hazy when you go back that far; but I am sure that someone with a Ph.D., or some other letters after his name, said that you could actually _add_ stiffness to a solid rod by drilling out the center (and maybe by appropriately treating it, I forget). The reason, if I recall correctly, had to do with the added tensile strength of the inner surface. The reason I remember this from so long ago was that it was so counter-intuitive: I was stunned. I'm not a stress/structures guy; so I asked a co-worker about it, and he said that the solid rod would be stiffer, because it had the greater (bending) moment of inertia. I agree with the latter part of the statement, but my hazy daydream keeps me from agreeing with the preceding conclusion. My money is still on a series of concentric tubes, appropriately processed and internally supported, being stiffer than a solid rod of the same O.D. So, my question: Does anyone know of any references to this little structural trick (or engineering wives' tale). If so, can you quantify how much stiffness is gained?


title: how can you keep a gyroscope turning?

text: I'm trying to understand how do gyroscopes and cmgs used for navigating keep on spinning even though there's friction. Any engine/pushing mechanism I can think of will restrict the gyro's rotor to a fixed direction and defeat the purpose... how is it done without disturbing it?


title: Could heavy-lift ships be used as oil tankers to stop oil spills?

text: Years ago, I was watching a science programme about some law of physics that prevents liquid from spilling if it was a container underwater. I recall that it was proposed as a safe way to transfer liquids. What was the programme discussing? Could oil tankers use a similar idea to prevent oil spills? Additionally, could ships could be built with layers of a lightweight substance (like expanding foam) to ensure even if breached, the ship would never sink far enough for water pressure to be problematic?


title: Amplitude of power spectral density

text: Why is the amplitude of the Power spectral density higher for shorter period of time as compared to a longer period of time when calculated for any vibration data?


title: Potential energy of a rotating bar

text: I have a bar of length $L$, mass $m$, and negligible width. The bar can rotate along an axis through its center of mass (the rotation is in the $x$-$y$ plane), and also can move up and down (in the vertical axis). > My question is: What is the (gravitational) potential energy of this bar? I've found several exercises where $PE=mgy_{CM}$, where $y_{CM}$ is the position of the bar's center of mass, though I don't know why the assumption of just taking the CM. Since this is rigid body, I thought also in summing the potential energy of each particle (i.e, integrating the classical $mgy$ function with respect to the height $y$), but I don't know if this is correct, or necessary.


title: English translation of Heisenberg's paper ``Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik''

text: Is there any english translation of Heisenberg's paper _Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik_ online? I have found not. Thanks so much.


title: What is the power of the most powerful quasar found?

text: Trying to find an answer to this question, I came across many sources that are in complete in contradiction. For example Wikipedia states that a typical quasar has a power of $10^{40}$ watts while according to this, the most powerful quasar has a power of $10^{39}$ watts ($10^{46}$ ergs). So now which is correct, and what is the correct number for a _typical_ quasar ?


title: scaling of motor power

text: For car engines, the cylinder volume is often associated with the engine power, which suggests scaling of the power as $L^3$ where L is the linear size. Consider a system consisting of a motor and its energy supply, e.g., an internal combustion engine with a fuel tank, or an electric motor with a battery, or a steam turbine with a nuclear reactor etc. Now we scale the system geometrically, up or down by some factor, using same materials, and assume that in the scaled system nothing breaks or burns through, so it can function. What should we expect for the scaling of the output mechanical power? Would it be close to $L^3$? Can an example be constructed where the scaling is very different from $L^3$?


title: How do we calculate the Higgs potential?

text: What is the Higgs potential and how do we calculate it?


title: vacuum expectation value and Casimir effect

text: Is it correct to say that the VEV of the SM Higgs is 246 GeV? If so, is the VEV a reflection, or measure, of the Casimir Effect?


title: Relativity and Current in Wire

text: If an observer is stationary relative to a current-carrying wire in which electrons are moving, **why does the observer measure the density of moving electrons to be the same as the density of electrons if there were no current in the wire?** I read the explanation of magnetic force as a consequence of special relativity. That is, when the observer moves with respect to a current carrying wire in the same direction as the flowing electrons, then s/he observes the density to decrease due to Lorentz expansion and observes the density of positive ions to increase due to Lorentz contraction. The imbalance of charges results in a force that can be explained by Coulomb's law. That explanation mentions that when the observer is stationary, the density of positive ions and the density of electrons appears to be the same. That last point is what I am stuck on. Why does the density of _moving_ electrons appear to be the same as the density of _stationary_ positive ions? **Clarification:** I know that when the electrons are not moving, their charge density cancels out that of the stationary positive ions. When the electrons are moving, how could the charge density _still_ cancel out that of the stationary positive ions? **Important:** My assumption is that a current carrying wire will neither attract nor repel a charge that is stationary with respect to the wire. I am now not sure if that is right :-(


title: Energy used to stop / slow an object

text: I'm trying to workout how much energy (if any), I use (imagining me as an efficient machine rather than a complex bio-mechanical human) when I lower or catch a weight. I understand that when I push it up, I have used energy to do work as I have the force of the object (it's mass and gravity) and distance. So I've got two similar questions... If I am to stop a free-falling object, how much energy do I use? If I look at it from energy change I believe I need to use the same amount as the kinetic energy of the object at the moment before I catch? If I slowly lower an object how much energy do I use? In an ideal world would the total be the same as in the free falling situation, assuming the height at which the object fell was the same.


title: Hamiltonian is conserved, but is not the total mechanical energy

text: I wondering about the interpretation for the energy difference between the Hamiltonian and the total mechanical energy for systems where the Hamiltonian is conserved, but it is not equal to the total mechanical energy. For example, consider a bead (mass $m$) on a frictionless hoop (radius $R$) in the presence of gravity. The hoop is spun around an axis parallel to the gravitational acceleration at constant angular speed ($\omega$). This is the typical set up for this problem. The total energy for this system (using $\phi$ to denote the angle from the bottom of the hoop) is: $$ E = \frac{p_{\phi}^2}{2mR^2} + \frac{1}{2} mR^2 \omega^2 \sin^2 \phi + mgR (1- \cos \phi) $$ where $p_{\phi} = mR^2\dot{\phi}$. The Hamiltonian is: $$ H = \frac{p_{\phi}^2}{2mR^2} - \frac{1}{2} mR^2 \omega^2 \sin^2 \phi + mgR (1- \cos \phi) $$. So the difference between the total mechanical energy and the Hamiltonian is: $$ E-H = mR^2 \omega^2 \sin^2 \phi $$ which is twice the rotational kinetic energy, I think. I'm just trying to get a handle on what this difference means. Any help is appreciated.


title: Hard disk rotational frequency

text: I noticed the other day that the rotational frequency of typical consumer grade HDD are either 5400rpm or 7200rpm. These correspond to 90Hz and 120Hz respectively. Given that in parts of the world the grid frequency is 60Hz, and 120Hz is the first harmonic. What effects does this have on the platters. It would seem to me that this choice in rotational speed (7200 rpm) would be problematic in these certain countries.


title: Mathematical problems with impact on physics

text: Are there any purely mathematical, unsolved questions, whose resolution would have (great, or concrete) impact on physics? Eg. it could almost surely tell us whether particle x exist or not, assuming so and so well-accepted observational facts?


title: Find wavelength from relative permittivity and frequency

text: If i have a EM wave with frequency 1MHZ and εr=9 in a perfect dielectric ,is it possible to find the wavelength λ and wave propagation speed ? It seems impossible to me..


title: Causality in String Theory

text: For a point particle we have light cone: ![enter image description here](http://i.stack.imgur.com/Hnrw8.jpg) String Theory- with it's extended body concepts- however will not admit a light cone such as this. In particular the most problematic causal issue would be the following illustration:![enter image description here](http://i.stack.imgur.com/Dmt5v.jpg) From the illustration the past light cone from the endpoint of the string leading in it's direction of motion is seen as overlapping the future light cone from the opposite end. It is the rhomboid region that I wonder about- here it seems the past and future of the string would continually interfere with one and another.


title: What is the best material and design for building a parabolic dish?

text: I'm currently using a homemade metallic spaghetti strainer with a wifi adapter inside it to increase my wifi adapter's signal reception, but would think that a different material, like the aluminum mesh used for the Arecibo Radio Telescope could be better? Why is this radio telescope's reflector spherical and not parabolic?


title: Are there models/simulations of antigravitational antimatter-galaxies?

text: In the comments to another question's answer, I started wondering: Assuming antimatter possessed _negative gravitational_ mass§ (which is not _proven_ impossible to date, though deemed unlikely), basically entire galaxies consisting of antimatter could form. Due to the repulsion of ordinary matter, they would rarely collide with usual ("pro-matter-")galaxies% and actually drift apart, maybe even partly contributing to inflation. But I don't want to speculate too much, so my questions is > Have entire galaxies consisting of negative-gravitational-mass antimatter > been considered in any sound theory, model or simulation? * * * § I.e. while such antimatter gravitationally repulsed conventional matter, it would still _attract_ other antimatter. % Unless there exists e.g. a (currently unknown) force that attracts antiparticles to their exact counterparts


title: EMF in a half-ring shaped conductor around a solenoid

text: A half-ring shaped conductor is being placed around a solenoid. This solenoid has a changing magnetic field. a) There is a current and EMF (Electromagnetic force) in the half-ring shaped conductor b) There is a current but no EMF in the half-ring shaped conductor c) There is no current but there is an EMF in the half-ring shaped conductor d) There is no current but there is no EMF in the half-ring shaped conductor My thoughts about this: First of all the field outside a solenoid isn't very big in contrast to the field inside the solenoid. But I assume that it can't be neglected. I think there won't be a current as we would need a full ring- shaped conductor for this. I do think that there will be an EMF in the conductor, and this will generate some eddy currents in the conductor. But there will be no "full-on current" so to speak. So I would say the answer is C. Is this correct? A Little side-question; Is it at all possible for a non-conducting material to have an induced EMF. For example when we move a non-conducting material (ring- shaped) into a magnetic field. The magnetic flux in the ring changes, but will there be an induced EMF? I know that there won't be a current in the loop, but I'm not certain about the EMF. Thank you very much!


title: Deviation from Earth's orbit

text: How much orbital deviation is required for the Earth to get knocked out from current orbit so it either moves away from Sun or towards the Sun?


title: Addition of spin?

text: There is something I don't fully understand. When I have a problem like this: Let's look at a system with $j_1=1$ and $j_2=2$ spins. Let our system be in the state in which the total spin is $2$ and the projection onto the $z$ axis is 0 ($S_z$). What values can the $S_z$ of the $1st$ particle take? When I have a problem like this, I always start by creating this fig:![enter image description here](http://i.stack.imgur.com/JfWRc.jpg) It represents the eigenstates. It's easy, I just have to find the simpliest way to the $(J=2,M=0)$. But when the problem is that I just have to add, for example, 1/2 and 1/2 spins, then how can I find the eigenstates? There must be a basic formula for it, I've been looking for it but I can't find it.


title: Change in intensity of electric field with constant velocity

text: Consider a +Q charged particle is travelling towards another test charge +Q. Now what would be the difference in electric field experienced by the test charge(avoid the gradual decrease in distance between them)? Would the field lines look compressed and effective field strength increased for the test charge?


title: Relation between intensity of light and refractive index

text: The intensity of light (as calculated from time average of the poynting vector) is given by $I = (1/2) \epsilon v E_0^2$. Here the intensity is dependent on the velocity of light in the medium. The refractive index also depends on the velocity of light. So is it safe to say that the intensity of light depends on the refractive index of the medium?


title: Does the mass of a star change as it collapses into a black hole?

text: I know (I think!) that when a really big star collapses on itself it creates a black hole. My question: When a star collapses, is the mass equal to the mass of the star when it's not a black hole? Or does it change while collapsing? This question came to me and my friend while studying Newton's law: $$F=G \frac{m_1 \cdot m_2}{r^2}$$ If the mass of the star doesn't change, then it can't have enough force to "eat" light (unless it has that force in the first place). Does the force change because of the density?


title: What is the shape of the Sun's orbit around the Milky Way?

text: Here is a question for you really bright folks. How does the motion of the sun along the midpoint of the plane of the galaxy look like? I suspect it to travel above and then below the plane but I would think its path would be in the shape of a screw.


title: The Spin Connection

text: 1. Why do we need to introduce the spin connection coefficients $\omega_{\mu \space \space b}^{\space \space a} $ in General Relativity? To me, they just look (mathematically) like the Christoffel symbols and so I'm assuming they aren't tensors? If I'm right in saying that, then do these $\omega_{\mu \space \space b}^{\space \space a} $ transform the same way under general coordinate transformations as the Christoffel symbols, and if so, what's the need in them then?    2. Finally, what's the relation between the Vierbein $e_{\mu}^a$ and the spin connection (non-mathematically)?


title: What are the eigenvalues of the Lorentz matrix?

text: The Lorentz matrix defines the transformation of a four-vector between different frames of reference, such that $$ p^{'\mu} = \Lambda^{\mu}_{\ \ \nu}p^{\nu} $$ where in this example $p^{\mu}$ is the four-momentum. 1) Are Lorentz transformations of this form only valid for **constant** (not changing in magnitude) velocities? I guess so, since $\gamma$ is a function of $v^2$. How can we transform between accelerating frames? 2) Is **Lorentz invariance** a law of nature? Which physical quantities should we expect to be invariant (forces? charge?)? 3) What are the **eigenvectors** and the **eigenvalues** of the general Lorentz matrix? I mean what is their physical significance? They do not change under Lorentz transformations? (I know the ones for the boost in the z direction are something like the Doppler shifted frequencies, but what does this mean? They are the same in all frames? What about the eigenvalues for the boost in a random directiom matrix?)


title: Polar Decomposition of a Complex Scalar Field

text: People often write a complex scalar field via polar decomposition. What does this parametrization precisely mean? To be more explicit consider the following Lagrangian of a complex scalar field with a $ U(1) $ symmetry, \begin{equation} {\cal L} = - m ^2 \left| \phi \right| ^2 - \frac{ \lambda }{ 4} \left| \phi \right| ^4 + \left| \partial _\mu \phi \right| ^2 \end{equation} We can then make a field transformation, \begin{equation} \phi (x) = \rho (x) e ^{ i \theta (x) } \end{equation} We typically refer to $ \rho (x) $ and $ \theta (x) $ as real scalar fields, but this is a strange for a couple reasons. Firstly, the $ \rho (x) $ field is positive definite. This is a boundary condition that we don't normally see in QFT. Its especially weird if $ \rho $ is quantized around $0$ since it can't ``go in the negative direction''. The second reason I think calling $ \rho (x) $ and $ \theta (x) $ quantum fields is strange is because after we write out the Lagrangian we get, \begin{equation} {\cal L} = - m ^2 \rho ^2 - \frac{ \lambda }{ 4} \rho ^4 + \partial _\mu \rho \partial ^\mu \rho + \rho ^2 \partial _\mu \theta \partial ^\mu \theta \end{equation} The $ \theta (x) $ field doesn't have a proper kinetic term and so can't be a propagating field! If these objects aren't quantum fields then how should I think about them?


title: Particle antiparticle annihilation-do they have to be of the same type?

text: I read that a particle will meet its antiparticle and annihilate to generate a photon. Is it important for the pairs to be of the same type? What will happen when for example a neutron meets an antiproton or a proton meets a positron? Are there any rules to determine what happens when such particles meet?


title: How to calculate calorie expidenture from lifting a weight?

text: Am I correct to say that when a human lifts a dumbbell from, say, 2 feet off the ground to 6 feet off the ground, he will have increased the potential energy of the weight and thus will have burned calories to supply that energy increase? Will two different people, Arnold Schwarzenegger in his prime and a scrawny kid for example, expend the same amount of calories moving the dumbbell from 2 feet straight up to 6 feet? Even though it's much easier for Arnold to lift it up, is he still burning the same number of calories as the scrawny kid? If one lifts the weight up very slowly, isn't that going to expend more calories? When you prolong your exercise, but still do the same number of repetitions, you sweat a lot more and feel more tired, which seems to say that more calories are burned. But there's no time variable in the formula for potential energy (U = mgh). What other energy equation must we use then?


title: Singularities and quantized space time

text: Discrete space time quanta would solve the problems of infinite densities for singularities in General Relativity and Quantum Gravity by imposing a non zero limit on the minimum radius of black holes. This would also prevent the "Violet Catastrophe" where the size of "Quantum Foam" energy fluctuations approach infinity as the scale approaches zero. Despite this, there appears to be a great deal of reluctance to even mention this as a possible solution to these problems in popular science magazines and documentaries, despite the idea having been around for half a century or more. Is this because it is inconsistent with some popular theories, or are there other implications which make it unappealing?


title: Why does gravity forbid local observables?

text: I heard in a conference that gravity forbids to construct local gauge invariants like $\mathrm{Tr}\left\\{−\frac{1}{4} F_{μν}^{a}F_{a}^{μν}\right\\}$ and only allows non-local gauge invariant quantities like Wilson Loops: $\mathrm{Tr}\mathcal{P}\exp\left[\oint_{\gamma} A^{a}dx_{a}\right]$. Could someone explain me where does it come from?


title: How is the atmosphere composition of astronomical objects in solar system determined?

text: How is the atmosphere composition of astronomical objects, for example triton, the moon of neptune, determined?


title: What shape of track minimizes the time a ball takes between start and stop points of equal height?

text: I was at my son's high school "open house" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end points of the second track were the same but it was bent into a dip with a decending slope and an ascending slope. The game was to guess which ball bearing wins the race. At first blush ("the shortest distance between two points is a straight line") one might think the straight wins, but in fact the bent track wins easily (because the ball is going very fast for most of the time). His point was that nature doesn't always work that way our antediluvian brains think it should. So my question (and I do have one) is what shape of track give the absolute shortest time? I think it's a calculus of variation problem or something. For simplicity, let's assume a) no friction b) a point mass and c) that the end point is a small delta below the start point. I played around with some MATLAB code and it seemed to give a parabola of depth about 0.4 times the distance between the start and stop point. The value of g and the mass didn't seem to matter. But I don't know if my code was right or wrong and I couldn't figure out the significance of 0.4. I Googled around for an answer but couldn't find one. Thanks for any pointers!


title: Higgs vs phonons

text: Jim Baggott's "Higgs" quotes David Millers' prize-winning one-page explanation of the Higgs mechanism (the one that evokes Margaret Thatcher crossing a room). I've heard that part many times, but not what follows: an analogy between the Higgs mechanism and lattice distortions in solid state physics:   * Lattice distortions increase the effective mass of moving electrons, like the Higgs mechanism.   * Waves of clustering (phonons) can occur even in the absence of moving electrons and behave like bosons, like the Higgs particles. As I understand it, this analogy is **not** to superconductivity, just garden- variety solid state physics. My question: how good is this analogy and where does it break down? * * * Edit: Courtesy of Daniel's answer to the related question linked by Qmechanic, Miller's entire explanation can be read here.


title: Harold White's work on the Alcubierre warp drive

text: I've read a bit on Harold White's recent work. (A paper on Nasa's site) I haven't been able to find any comments by people claiming to know anything about the physics involved. Is this really serious? From a blog post (first link in this question): "But interesting things have developed since the original Alcubierre paper. Running quickly through what White told the Houston audience, Chris van Den Broeck was able to reduce the energy costs of a warp drive significantly and other theorists have continued to drop the numbers. White’s team has been examining ways to continue that progression, but what is eye-catching is that he is working on a laboratory experiment to “perturb spacetime by one part in ten million” using an instrument called the White-Juday Warp Field Interferometer to create the minute spacetime disruption. "


title: What do you feel when crossing the event horizon?

text: I have heard the claim over and over that you won't feel anything when crossing the event horizon as the curvature is not very large. But the fundamental fact remains that information cannot pass through the event horizon so you cannot feel your feet when they have passed it. Is there a way to cross the even horizon at reasonable speed in radial direction (below say 0.01*c )? So what would it really be like?


title: What path would advanced spaceships take to move between planets?

text: Right now, space travel is all about carefully moving between orbits. If you want to go from Earth to Mars, you wait until the two planets are correctly aligned, and then place yourself into an elliptical orbit around the sun so that the apoasis of the orbit hits Mars. You only have to make a single, long burn at the start of the orbital change, and if you do it right you'll fall into a neat Martian orbit. Using this method, it takes quite a long time to get to Mars! It seems that most of the work getting there is done by gravity - the craft's engines only do a little. It's as if the craft is a transistor - its engines provide a little seed current so that gravity can do the rest. I suppose we do it like this because fuel is hard to get into space, and our engines are not very good. 100 years from now, when these are no longer issues, how will spacecraft move from one planet to another? Will they still think about orbital changes, or will they just point towards the target and hit the accelerator?


title: Why water is not superfluid?

text: My question is in the title. I do not really understand why water is not a superfluid. Maybe I make a mistake but the fact that water is not suprfluid comes from the fact that the elementary excitations have a parabolic dispersion curve but for me the question remain. An equivalent way to ask it is: why superfluid helium is described by Gross-Pitaevsky equation and it is not the case for water?


title: Applying Lagrange's equations ignoring normal reaction

text: A small bead is sliding on a smooth vertical circular hoop of radius $a$, which is constrained to rotate with constant angular velocity $\omega$ about its vertical diameter.   $\theta$ is an angle between the downward vertical and the radius to the bead. I've calculated that: Lagrangian for this motion is $L=\frac{1}{2}m(a^2\dot\theta^2+a^2\omega^2\sin^2\theta)+mga\cos\theta$   There are 4 positions of equilibrium when $g<a\omega^2$: $\theta_1=0$, $\theta_2=\pi$, $\theta_{3,4}=\ arc\cos(\frac{g}{a\omega^2})$ and 2 positions of equilibrium ($\theta_1=0$,$\theta_2=\pi$) when $g>a\omega^2$. For what values of $\omega$ is the equilibrium position at the lowest point of the hoop stable? Why can we apply Lagrange's equations to this problem , thus ignoring the normal reaction between the hoop and the bead?


title: Why don't cancelling wavefunctions for two different particles give zero total wavefunction?

text: Let $\left|a\right>=e^{i(kx-\omega t)}$， $\left|b\right>=-e^{i(kx-\omega t)}$ be two neutral particles in the 1D free space without any interaction. Then $E_a=\left<a\left|\hat{H}\right|a\right>=\hbar\omega$, $E_b=\left<b\left|\hat{H}\right|b\right>=\hbar\omega$. Hence $E_a+E_b=2\hbar \omega$. That means the energy of the system $\\{a,b\\}$ is $2\hbar \omega$. However, in another way, $\left|a\right>+\left|b\right>=0$, hence $E_{a+b}=(\left<a\right|+\left<b\right|)\hat H(\left|a\right>+\left|b\right>)=0$. That means the total energy is $0$. These two method seems both make some sense, but they are contradict to each other. What's wrong with them?


title: "Classical" limit of Quantum Hall Effect

text: Imagine a partially filled $\nu=1$ state of the integer quantum Hall effect (IQHE). One way to think about it is to imagine a gas of electrons where each particle is locked to the lowest quantum state of their circular motion (a Gaussian wave-packet in both $x$ and $y$ with the characteristic length being the magnetic length $l_0$). When this gas becomes degenerate, we get a fully filled $v=1$ Landau level and the bulk conductivity vanishes. I wonder, how does a single-particle theory of a partially filled$\nu=1$ IQH state state look like in terms of such particles? What is the kinetic energy term? I seek a kind of field theory for this, valid on the scales of $k \ll l_0^{-1}$ and regularized in a consistent way on the scale of $l_0$. Perhaps a tight-binding-like lattice model? I would bet it has been done (if possible at all) by some "high-flying" quantum field theorists out there , but I have not seen such concept being used in "down-to-Earth" experiment-oriented calculations of IQHE.


title: From where this number of 14 Tev has arised for proton-proton head on collision in LHC?

text: I am interested in finding out, why are collisions at 14TeV done in the LHC, instead of some other energy?


title: If I put 3 bottles of water next to each other in the fridge, which one is cold first?

text: I was wondering: If I put three bottles of water next to each other in the fridge, which one is cold first? Does it matter? Is it the one in the middle because it gets refrigerated by the other two? Thx! :)


title: Lee-Yang circle theorem

text: what is Lee-Yang circle theorem and what is it used for ?? , i mean given a measure how can you know that is Ferromagnetic and hence all its zeros lie on a Circle ?? the Lee-Yang circle theorem proof is it only valid for the measure $ \delta (x-1) + \delta (x+1) $ how can you proof the Circle theorem for any measure ?? if any measure is positive and EVEN does Lee-Yang circle theorem hold ??


title: Can the short-time dynamics of an open quantum system be approximately unitary?

text: Considering the physics of an open quantum system described by a Hamiltonian $H=H_S+H_E+H_{SE}$, where the subscript $S$ refers to the system of interest, $E$ to the environment and $SE$ to the interaction between them, It is possible that at short-times and for sufficiently weak system- environment coupling, the evolution of the system of interest is approximately unitary? If yes, what defines the time-scale of unitarity? Does the Hamiltonian that describes the short time evolution is $H_S$ or some renormalized Hamiltonian? Finally, is there any relation between my question and the phenomenom of decoherence?


title: How do photons travel at a speed that should be impossible to attain?

text: If it requires infinite amount of energy to travel at the speed of light then how photon attains this speed? Its source is never infinitely sourced.


title: what does it mean for a particle with no size to have angular momenta?

text: I recently was reading about higgs boson and particle spin recently and I stubble upon an question that contains an answer to what a spin is. It explains that electrons etc. have no size yet they have angular momenta. I dont understand what exactly is meant by that. Does it refer to the angular momenta of the magnetic field? I just dont imagine something with no size having any sort of angular momentum.


title: Calculating the dispersion relation of dirac lagrangian in curved spacetime

text: I am trying to calculate the dispersion relation for a fermion in a gravitational field. So far, I have computed the equation of motion, but I am stuck trying to figure out a determinant I just can't get.... I am having trouble calculating the final determinant. My derivation is below: Equation of motion: \begin{equation} \left(i\gamma^a\partial_a - m -\frac{1}{2} \gamma^a\gamma^5B_a\right)\Psi = 0 \end{equation} If we take $\Psi = u(\vec{p_a})e^{-ip_ax^a}$ as our ansatz we get the result \begin{align} \left(i\gamma^a(-ip_a) -m -\frac 12\gamma^a\gamma^5B_a \right)u(\vec{p_a}) &= 0 \\\ \left(\gamma^ap_a -m -\frac 12\gamma^a\gamma^5B_a \right)u(\vec{p_a}) &= 0 \end{align} We multiply this by expression by $\left(\gamma^ap_a +m -\frac 12\gamma^a\gamma^5B_a \right)$ and expand to get \begin{equation} \bigg[\gamma^b\gamma^ap_ap_b -m^2 + m(\gamma^ap_a - \gamma^bp_b) - \frac{m}{2}(\gamma^b\gamma^5B_b-\gamma^a\gamma^5B_a) - \gamma^b\gamma^a\gamma^5B_aP_b ~~ - \notag \end{equation} \begin{equation} -~~ \gamma^a\gamma^5\gamma^bB_bP_a + \frac 14 \gamma^b\gamma^5\gamma^a\gamma^5B_aB_b\bigg]u(\vec{p}_a) = 0 \end{equation} \begin{equation} \bigg[p^ap_a -m^2 + \frac 12[\gamma^a,\gamma^b]\gamma^5B_aP_b - \frac 14B^aB_a\bigg]u(\vec{p}_a) = 0 \end{equation} This is a matrix times spinor equal to zero, meaning the determinant of the matrix must be zero (since the spinor being zero is of no interest to us). Therefore: $$ \text{det}\bigg[(p^ap_a -m^2 - \frac 14B^aB_a){\mathcal{\large\mathbb{1}}}+ \frac 12[\gamma^a,\gamma^b]\gamma^5B_aP_b\bigg] = 0 $$ Anyone have any tips on calculating such a monster? Or even a different way of finding the relation?


title: Dirac Lagrangian density in curved spacetime

text: I'm trying to derive this form of the Dirac Lagrangian density in curved space-time: $$ \mathcal{L}~=~\det\left(e\right)\bar{\Psi}\Bigg (\frac{i}{2}\gamma^{a}\partial_{a}-m+\gamma^{a}\gamma^{5}B_{a}\Bigg)\Psi $$ starting from this form: $$ \mathcal{L}~=~\sqrt{-g}\Bigg (\frac{i}{2} \bar{\Psi} \gamma^{a} \stackrel{\leftrightarrow}{D}_{a} \Psi-\bar{\Psi}m\Psi \Bigg) $$ I can get the first term, however, in the last term I get a factor of $ \quad - \frac{1}{2} \quad$ out the front, i.e. $$ \mathcal{L}~=~\det(e)\bar{\Psi}\Bigg (\frac{i}{2}\gamma^{a}\stackrel{\leftrightarrow}{\partial_{a}}-m-\frac{1}{2}B_{a}\gamma^{a}\gamma^{5}\Bigg)\Psi $$ I've tried deriving it in several different ways and always end up with the above expression. The problem is, in all of the papers that I've read on the subject, it is given in the form as presented at the top of this page. If it helps, here is a link to my derivation in full: http://we.tl/mUgiw0BkMh Would really appreciate any help in solving this problem.


title: Tachyon and Photons

text: Is there a particle called "tachyons" that can travel faster than light? If so, would Einstein's relativity be wrong? According to Einstein no particle can travel faster than light.


title: What constitutes work?

text: I'm asked: "An object is moving along a floor at constant velocity. Thinking about this situation as realistically as possible, is work being done on the object?" And the solution is: "Work is being done on the object. To move at a constant velocity, a force must be pushing the car so as to overcome friction, air resistance, etc." However, since work is defined as $W = F · x$, and $F = ma$, wouldn't work be zero? Because acceleration is zero, and hence the sum of the forces acting on the object is zero, and hence the dot product of **F** and **x** is zero? Or do we count work even when it is cancelled out y equal and opposite work? Thanks! EDIT: I can see that the frictional force is not doing any work (since the object is not moving in the direction of the friction) but that the push-force IS doing work, since the object is saying in the same direction. However, the _net force_ on the object is still $0$. So I understand that we can attribute work to the push-force -- but is the object experiencing any OVERALL work? How is this possible if acceleration is $0$?


title: How does a train, airplane measure its speed?

text: I always get the doubt about this. I know that a bike measures its speed based on the motion of its front wheel. So what is the case with train? Is it same principle? Then what about an airplane? Is it by radars?


title: How to distinguish female and male voices via Fourier analysis?

text: What makes one, without looking, be able to identify the gender of the talker as male or female? I mean if we Fourier analysed the voice of males and females, how the 2 spectrums are different which account for that distinction in sounds?


title: Conservation of electric charge in Feynman diagram

text: Here is a Feynman diagram showing the mutual annihilation of a bound state electron positron pair into two photons: ![A Feynman diagram showing the mutual annihilation of a bound state electron positron pair into two photons. ](http://i.stack.imgur.com/cJIsN.jpg) Is the electric charge conserved at the point A (or B)? What is the "charge" of the virtual fermion "moving" between A and B?


title: A die versus a quantum experiment

text: Let suppose you roll a die, and it falls into a hidden place, for example under furniture.   Then although the experiment has already been made (the die already has a number to show), that value can not be known, so the experiment was not fully realized.   Then till you see the die's top side, the probability remain p = 1/6.   I see no difference between this and the wave function collapse, at least as an analogy.   Can someone explain a deeper difference? Thanks


title: Renyi entropy in physical systems

text: We know that the Shannon entropy $H(P)=- k_{\mathrm{B}}\sum_i p_i \ln p_i$ is mostly the entropy of the thermodynamic systems. Does the Renyi measure $H_{\alpha}(P)=\frac{1}{1-\alpha}\log \sum p_i^{\alpha}$, $\alpha\neq 1$ also actually measure the entropy of some physical system?


title: What is the information geometry of 1D Ising model for a complex magnetic field?

text: Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by $$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = 1}^N\sigma_i$$ where $\sigma = \\{\sigma_i\\}_{i = 1,\dots, N}\in\Omega := \\{\pm 1\\}^N$, $\\{J_i\\}_{i = 1,\dots, N}$ are nearest neighbor interaction strength couplings, and $h \in \mathbb{R}$ is the magnetic field. Let's consider the ferromagnetic case, that is, $J_i \geq 0$ for $i = 1, \dots, N$, and for the sake of simplicity (though this doesn't matter in the thermodynamic limit), take periodic boundary conditions. Neither in the finite volume, nor in the thermodynamic limit does this model exhibit critical behavior for finite temperatures. On the other hand, as soon as we allow $h$ to be complex (and fix the temperature), even in the finite volume $N$, the partition function has zeros as a function of $h$. In the thermodynamic limit these zeros accumulate on some set on the unit circle in the complex plane (Lee-Yang circle theorem). Now the question: let's consider information geometry of the Ising model, as described above, when $h$ is real. In this case the induced metric is defined and the curvature does not develop singularities (obviously, since there are no phase transitions). Now, what about information geometry of the Ising model when $h$ is complex? This is a bit puzzling to me, since then the partition function attains zeros in the complex plane, so that the logarithm of the partition function is not defined everywhere on the complex plane, and the definition of metric doesn't extend directly to this case (the metric involves the log of the partition function), let alone curvature. Is anyone aware of any literature in this direction? I thought it would be a good idea to ask before I try to develop suitable methods from scratch.


title: Quantum Wave Mechanics

text: I am studying QM-I these days. Now, I just think of the wave function as just a mathematical function that defines the state of the particle at an instant and from it you can extract various observables. I have been told that it is just a misnomer calling it an actual wave, and there is no wave there, its just that the wave function satisfies the general wave differential equation in 3d space. Now is this true? Is there really no wave? What about QFT, when we say electron is an excitation in a Quantum Field, do we mean to say, it is sort of a pulse in that field medium? Like a photon is a pulse in an EM wave and essentially electron is just a pulse, which we see as a particle? Or is there no wave there ? Is this related to $E=\gamma mc^2$ by relating energy in a wave to a particle.


title: Effect of linear terms on a QFT

text: I was always told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \phi _i \phi ^\mu \phi _i - \frac{m ^2 }{2} \phi _i \phi _i - \frac{ \lambda }{ 4} ( \phi _i \phi _i ) ^2 \end{equation} with $m ^2 =-\mu^2 > 0$, adding a linear term in one of the fields $\phi_N$, does change the final results as you no longer have Goldstone bosons (since the $O(N)$ symmetry is broken to begin with). Are there any other effects of linear terms that we should keep in mind or is this the only exception of the "forget about linear terms" rule?


title: How does it seem to be that space/time always equals c?

text: Given the way objects move, they seem to be going all the same "velocity" so to speak, that velocity being the speed of light. Except, velocity is displacement/time, so if something goes faster, the displacement is equally balanced by the reciprocally changing time factor so that it always balances out to 299,792,458 m/s. Well, that's the way it appears to be. The faster something displaces, the slower time moves for it; the slower something displaces, the faster time moves for it. These two factors always balance out to equal _c_. Why is this? Is there something obvious and mathematical I'm missing about it, or is this a little deeper? Explanations perhaps? Deeper insight on this factor? It seems very interesting. I could be completely wrong though; please clarify if so, but honestly, $E = \frac{mc^2} {\sqrt{1-\frac{v^2}{c^2}}}$ – it's freakin' weird.


title: Bulk modulus of Liquid helium and first sound

text: Does anyone know where to find the bulk modulus of liquid helium ? I've been looking all over the internet but everywhere I get N/A. Any tips ? I'd need it to estimate the speed of first sound in liquid helium which is given by: $c_1=\sqrt{\left.\frac{\partial p}{\partial\rho}\right|_\sigma}$


title: First integral of relativistic Euler-Lagrange equations

text: Connsider a pseudo-Riemannian ($4$-dimensional) manifold $M$ with a pseudometric $g_{ab}$. The Lagrangian of a free particle in $M$ (in analogy to the flat case) is $$\mathcal L=\frac{1}{2}g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}$$ where $\tau$ is the proper time and $x^a$ are the coordinates respect the charts. Now on my relativity book I read: > The four-velocity identity $g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}=-c^2$ > is a first integral of the relativistic Euler-Lagrange equations for free > material particles. Now what does this phrase mean? What is a first integral for the Euler- Lagrange equations? Moreover why $g_{ab}\frac{dx^a}{d\tau}\frac{dx^b}{d\tau}=-c^2$? Thanks in advance


title: Counting Problems in Physics

text: What are some _classic_ counting problems in physics? I'm trying to think of interesting examples to give in a math class on the matter, and I feel as if physics should have some ones to offer.


title: Does gravity stack?

text: If I am standing on a planet on the opposite side from its sun will I feel the downward pull of both the planet's gravity and the star's gravity? In other words, does gravity "stack" (Any D&D players out there?)


title: Show that getting parallel transported does not change angle between them- Tensors

text: I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to tensors or not. The question goes like thiS: > _1\. If a vector $u^{i}$ gets parallel transported along a curve S, then > $$u^{i}{}_{; j}~ \frac{dX^{j}}{dS}~=~0.$$ 2\. If the angle between $u^{i}$ > and $v^{j}$ is $\theta$, show that getting parallel transported does not > change angle between them._


title: What is relation between electrons and photon?

text: What is the relation between electrons and photons? Why do atoms get excited when their electrons come in contact with photons? Why do electrons go from a higher to lower energy level when emitting a photon?


title: P-N junction voltage under polarization

text: All the books that deal with the p-n junction under applied bias asume that the same equations used for the equilibrium case (no bias) can be used for the biased case provided that the juntcion voltage $V_{ic}$ is substituted bt $V_{ic}-V$, where $V$ is the applied bias. Although this can seem plausible I don't see any reason for why the junction voltage drop should be the $V_{ic}-V$, i.e., why the voltajes should obey a sort of superposition principle. Is there any reason on the ground of fundamental electrostatic / thermodynamics? Thanks.


title: Ampere's law and Biot-Savart law gives different terms for magnetic field in middle of a current running in a loop

text: ![enter image description here](http://i.stack.imgur.com/ofJjE.jpg) I would like if someone could clarify this issue for me: When dealing with a current $I$ running in a loop with radius $R$ and looking for the magnetic field in the middle of the loop. By using Ampere's law, i know that the current $I$ runs through a loop with the same radius $R$ ,we get that: $$\int(B*dl) = \mu I $$ $$B= \mu I/2\pi R $$ and when using BIO-SAVART we get that $$dl*r = |dl||r|sin90$$ and therefore i get that: $$B = \mu I/2R$$ Which is not the same result as with ampere's law. I obviously miss something, maybe i can't use Ampere's law? Anyway, if someone could help me out here I would really appreciate it. Thanks.


title: What are all the approaches that have been tried for a theory of quantum gravity?

text: I am aware that that the most researched approach is that of string theory. I have also heard about quantum loop gravity. What other approaches are there to unify gravity and QFT? Also, please include a brief description of that approach, and how it is different from the others.


title: General Physics Short question

text: Does any one know where this formula come from ? $\alpha (t) = -\omega^2\ \Theta(t)$ Thanks


title: Asymptotic limit of the two kink solution of the sine-gordon equation

text: I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: $$\phi=4\arctan\left(\frac{\sinh\frac{1}{2}(\theta_1-\theta_2)}{(a_{12})^\frac{1}{2}\cosh\frac{1}{2}(\theta_1+\theta_2)}\right)$$ where $\theta_1=m\gamma_1(x-v_1t)+\delta_1$, and similarly for $\theta_2$. Here $a_{12}, m, \gamma, \delta$ are all constants. The author says that the limit as $t \to \pm \infty$, is $$\lim \phi(x,t) \to 4\arctan(\exp [\theta_1+\eta_1^{\pm}])+4\arctan(\exp[\theta_2+\eta_2^{\pm}]) $$ where $\eta_1^{\pm}=-\eta_2^{\pm}=\pm\frac{1}{2}\ln(a_{12})$ I am having a hard time proving this. I did the expansion of the hyperbolic functions in terms of the exponents, and tried to limit the appropriate terms to 1, but I am not getting an answer.


title: Can you tell if a particle is in superposition?

text: This may be an easy answer for anybody. Is it possible to detect if a particle A is still in a superposition via the sending a group of particles B through a box containing particle A?


title: Why are generalized positions and generalized velocities considered as independent of each other?

text: I'm confused how $$\dot{\mathbf{r}}_{j}=\sum_{k}\frac{\partial\mathbf{r}_{j}}{\partial q_{k}}\dot{q}_k+\frac{\partial\mathbf{r}_{j}}{\partial t}$$ leads to the relation, $$\frac{\partial\dot{\mathbf{r}}_{i}}{\partial\dot{q}_{j}}=\frac{\partial\mathbf{r}_{i}}{\partial q_{j}}$$ Sources suggest that while differentiating the first equation the generalized velocity and generalized positions are considered as independent of each other, i.e., $(\dot{q_1},\dot{q_2},...,\dot{q_n})$ is independent of $(q_1,q_2,...,q_n)$. I don't understand how they are independent.