title: Can decoherence work when the environment itself is in a superposition without invoking collapse or splitting?

text: Decoherence is often presented as a program to solve the measurement problem using only the bare bones quantum mechanics framework of a Hilbert space and unitary Schrodinger time evolution. As often stated, there is no need to postulate a collapse or splittings into many worlds for decoherence to work. However, I have my misgivings. What do you guys think of the following example? The system is an electron and the environment contains a Stern- Gerlach apparatus. In configuration A, the apparatus measures the spin of the electron in the z-direction, and in configuration B, the x-direction. A standard decoherence analysis will pick out the z-basis as the pointer states for configuration A, and the x-basis for B. So far, so good. Now instead, set up a control qubit $\left( |0\rangle + |1\rangle \right)/\sqrt{2}$ using a Hadamard gate. A value of 0 sets up the apparatus in configuration A, while a value of 1 sets it up in configuration B. The crucial detail here is the environment is in a superposition. Now as far as I know, all the standard criteria for picking out system pointer states like for instance, diagonalizing the reduced density matrix, the predictability sieve, purification time, efficiency threshold, etc. , all give out crap for the pointer basis in this example. None of them can capture the fact that for a value of 0, the pointer states lie in the z-direction, but for a value of 1, they lie in the x-direction instead. Would it be fair to claim we need something extra like a collapse of the environment into configuration A or B, or a splitting into the worlds of configuration A and configuration B with each world treated in a different manner, to actually make decoherence work? Or are there loopholes allowing us to stick to a bare bones quantum mechanical framework? Most treatments of decoherence assume the system is in a superposition right after preparation but hardly any consider the scenario where the environment is also in a superposition during preparation. This, I think, is incomplete. And this is no idle matter either because the environment is always in a superposition. Structure formation leading to the condensation of matter into superclusters, galaxies, stars and planets have their origins in quantum fluctuations. Even major patterns here on earth like large scale weather events depend sensitively upon quantum fluctuations via the butterfly effect. The unpredictable, or at least not fully predictable, behavior of experimenters are also sensitive to quantum fluctuations in their brain. Without presupposing either collapse or a many worlds split, how can one get down to a proper decoherence analysis?


title: Reference-request: Computational science and physics

text: I'm interested in the following question: > _Is Nature computable at it's most fundamental level?_ Can anyone suggest any works (books, papers / articles, reviews) related to the above question? (I've also cross-posted on Computational Science SE.) * * * Some links (since this question is currently closed): (1) The Relation between Mathematics and Physics, P. A. M. Dirac


title: How do I find the initial velocity in this problem?

text: > An X-ray tube gives electrons constant acceleration over a distance of > $20\text{ cm}$. If their final speed is $2.0\times 10^7\text{ m/s}$, what > are the electrons' acceleration? I know this equation is relevant: $$v_\text{final}^2 = v_\text{initial}^2 + 2a(x-x_\text{initial})$$ Also I have that $x = .2\text{ m}$ (after conversion from cm), and $v_\text{final}$ is given. How do I find $v_\text{initial}$? I need that so I can find acceleration.


title: Heisenberg Uncertainty Principle

text: Question: The uncertainty in position is equal to the uncertainty in momentum. What is the uncertainty in velocity? What I did: I know that the uncertainty in position multiplied by uncertainty in momentum is equal to $\frac{h}{2\pi}$ As momentum is mass times velocity, and mass is constant and definite, the uncertainty in momentum is equal to the uncertainty in velocity. Thus the answer should be $$\sqrt{\frac{h}{m2\pi}}$$ However, this is incorrect. Where am I wrong?


title: Can closed loops evade the spin-statistic theorem in 3 dimensions?

text: The famous spin-statistics result asserts that there are only bosons and fermions, and that they have integer and integer-and-a-half spin respectively. In two-dimensional condensed matter systems, anyons are also possible. They avoid this result due to a topological obstruction. Suppose that "loops" were fundamental, rather than point-particles. In addition to normal relative motion, it would be possible for one loop to go through the hole of another, and a path of this sort would not be continuous with one that doesn't. Would this provide enough of a topological obstruction to be able to avoid the spin-statistics theorem?


title: How can two different 12V batteries have different amperage for same resistance?

text: My local car parts dealer presented me his inventory of car batteries. One 12V battery had a 'cold crank amperage' of 600amps. The other 12V battery had a 'cold crank amperage' of 585amps. **Questions:**   [1] If the resistance, my truck, is constant for both batteries, how can the amperage ratings differ?   [2] What is does a battery's 'cold crank amperage' really mean?


title: Do electromagnetic fields gravitate?

text: It's well known that electromagnetic fields contains energy but do they gravitate ? When we talk about the composition of the universe it's now accepted that the 74 % is dark energy , the 22 % is dark matter and then the 4 % is the rest of ordinary matter/energy that we can see or measure. Are the electromagnetic fields considered in this 4 % ?


title: What is a projection method?

text: Quoting from Solenthaler et. al. _Predictive-Corrective Incompressible SPH (ACM Transactions on Graphics, Vol. 28, No. 3, Article 40, Publication date: August 2009)_ (PDF link here) > These incompressible SPH (ISPH) methods first integrate the velocity field > in time without enforcing incompressibility. Then, either the intermediate > velocity field, the resulting variation in particle density, or both are > projected onto a divergence-free space to satisfy incompressibility through > a pressure Poisson equation. What does the author mean by "project?? Is there a simpler way of understanding this operation? I tried reading other articles, but they go even more deep saying solenoidal vectors etc. I am looking for a simple explanation at first to understand the concept.


title: Hammer vs large mass on nail

text: Why is a hammer more effective in driving a nail than a large mass resting over the nail ? I know this has to do with momentum, but cant figure it out.


title: Home experiments to measure the RPM of a pedestal fan without special equipment?

text: Is it possible to determine to an approximate degree, the revolutions per minute of a fan, for example a pedesal fan pictured below, without using some electronic/mechanical measuring device? ![enter image description here](http://i.stack.imgur.com/ekTtd.jpg) One thing that comes to mind is the markings on a old record player's turntable \- can that concept somehow be used to make marks on a fan blade to determine it's RPM? Or can something called a "strobe tuner" help: ![enter image description here](http://i.stack.imgur.com/2vZfO.jpg) Can I maybe make markings on a fan with a marker and figure out the RPM using nothing else but a stopwatch? Or maybe some other DIY technique that does not require the purchase of any measuring device? **Ps.** I'm not entirely sure if this is a physics or engineering question, so please feel free to move it to the appropriate site (I checked all 58 stackexchange sites and only the Physics site seemed to fit the question)


Does corona discharge charged insulator? I draw a diagram to make it simpler to understand. Electrons form negative side is accelerated due to electric field and accumulate on insulator surface. Gas on the positive side was ionize giving off electrons and flow to positive terminal. ![ ](http://i.stack.imgur.com/enf5G.png) Am I understanding this correctly, When I place plastic sheet between HV terminals and when I remove plastic sheet I got an electric shock, so I think corona discharge might charge plastic sheet.


title: Can we transfer burn to another planet at any time?

text: _Assume delta-v isn't a problem and circular orbits._ EDIT: Assume that you're already in orbit so you don't have to shift a massive load of fuel up, and the absolute ideal is something that has a calculable delta-V like a Hohmann transfer. If there was some kind of industrial accident on Mars and we needed to leave right away, is there always a burn we can perform to get to where we want to be? I can imagine that we could move into an orbit closer to the sun to 'catch up' Mars or further away than it to slow down, but would this be significantly quicker than just waiting for our ideal launch window? Is there a way to optimise the arrival, similar to a Porkchop Plot? **Background** \- I'm thinking about a management game with orbital mechanics where there's a clear 'optimal' time to travel - this is the equivalent of the launch window - but there's an ability to travel at any time if you're willing to pay the extra cost for the rocket fuel. So ideally the 'optimal' case is something like a Hohmann or Bi-elliptic transfer, and then the further you move away from your launch window, the more dV (and such the more cost) it will be. Clearly it's a game so I can throw realism out the window, but I was wondering if there's a specific realistic trajectory that a rocket would take if an emergency mission needed to be mounted - plus, waiting two yeard for the optimal trajectory to restock your Mars base would be a bit of a drag.


title: Correct way to include constant external force in virial and pressure calculation

text: Halo, given a simulation cell with N particles where particles interact only with bond and pair potentials and periodic boundary conditions (minimum image convention) are used. On a subgroup of particles a constant (or non-constant) external force is applied. What is the correct way to include this force in calculation of virial terms and therefore pressure. Does the external force contribute to the virial, at all? Or do only interatomic interactions like bond and pair potentials contribute? Following standard literature one finds for the pressure $$ p = \frac{ 2\left\langle E_{kin} \right\rangle }{3V} + \frac{ 1 }{ 3V } \left\langle \sum_{i} {\mathbf f}_{i} {\mathbf r}_{i} \right\rangle $$ where $E_{kin}$ is the kinetic Energy, $V$ the Volume, ${\mathbf f}_{i}$ the toal force acting on particle $i$ and ${\mathbf r}_{i}$ the position of particle $i$. This reduces to $$ p = \frac{ 2\left\langle E_{kin} \right\rangle }{3V} + \frac{ 1 }{ 3V } \left\langle \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} \right\rangle $$ for pure pair potentials. Where ${\mathbf r}_{ij}={\mathbf r}_{j}-{\mathbf r}_{i}$ and ${\mathbf f}_{ij}$ is the force acting on particle $i$ due to particle $j$. Therefore I would assume that I simply have to add the external force to the virial, like this $$ p = \frac{ 2\left\langle E_{kin} \right\rangle }{3V} + \frac{ 1 }{ 3V } \left\langle \sum_{i<j} {\mathbf f}_{ij} {\mathbf r}_{ij} + \sum_{i} {\mathbf f}_{i}^{ext} {\mathbf r}_{i} \right\rangle $$ Where ${\mathbf f}_{i}^{ext}$ is the external force acting on particle $i$. But I've seen that ${\mathbf f}_{i}^{ext}$ is simply omitted in some popular ready to use md packages. Now I'm not sure what to do. Thank's in advance for help!


title: Electric Potential

text: A nonuniform linear charge distribution given by $\lambda = bx$, where $b$ is a constant, is located along an $x$ axis from $x = 0$ to $x = L$. What is the electric potential at a point on the $y$ axis? Set potential equal to zero at infinity. Express your answer in terms of the variables given and $\epsilon_0$ (epsilon not). I have calculated the answer to be: $$ b\frac{\sqrt{L^2 + y^2} - y}{4\pi\epsilon_0}. $$ The site I enter this answer onto tells me I am wrong. I have checked this with my dad, who is very good at physics, and he agrees with me on this answer. I would like to know if this answer is correct, and if not, why. Thanks.


title: Magnification in compound microscope

text: I was studying about compound microscope here I don't see why we multiply linear magnification of objective with angular magnification of eyepiece. Shouldn't it be both angular or both linear? Can we call this magnification angular magnification of the system? I tried to find about angular magnification of a optical ststem on the internet but it took me to the realms of some matrices which I do not understand as I have never read about such.


title: Change in Angular momentum of a spinning superball

text: I throw a superball towards the ground and when I do I intentional induce a spin on the ball so that the ball is spinning towards me on the descent. As soon as the ball hits the ground and comes back up, the direction of the spin has changed and its spinning away from me. I assume this tied directly to a conservation law but I am curious if anyone can give and explicit discussion as to the effect.


title: How can I calculate all of the tensions in the cords?

text: I have been stuck on this problem for the last half, and to be honest I don't even no where to begin. I'm being asked to calculate all of tensions for the cords. I know I'm supposed to use $\cos$ to calculate the $x$ tensions and $\sin$ to calculate the $y$ tensions, but that's about as far as I'm getting. Can someone please walk me through this and help me understand problem 51? http://postimage.org/image/nitqsyg6v/


title: where did I lose the negative sign when spring is in compression in a mass-spring problem?

text: Most books just tells you that spring force is $-kx$ since it opposes motion and to just write $mx''=-kx$ regardless if the spring is in tension of compression. But when I try to derive this using $F=mx''$ and using free body diagram, and put the mass to the right of the equilibrium position, I get the correct equation of motion, but when the mass is to the left, I do not. Please tell me what I am doing wrong (and do not just say to use negative sign since it is a restoring force). I need to find the EQM myself from free body diagram. When the mass to the right we have                                    x        |             |<----->        |             |       k x (spring force since in tension        |-------------|------<----o mass        |             |        |            x=0      So applying $F=mx''$ on the above gives $mx''=-kx$ which is the correct EQM. I used minus sign here since the direction of the force vector $kx$ is negative and not because some book said to use negative sign all the time. Now lets look at it when the mass is to the left and the spring is in compression                             x        |      <------>        |             |              |---->o       |        |   kx       x=0        |      So applying $F=mx''$ gives $mx''=kx$ which is wrong. The force $kx$ is now in the positive direction. So how to make $kx$ become negative? $x$ is a extension amount. So it is always a positive number as far as finding the force in the spring is concerned. $x$ as a coordinate is negative, yes, but for Hooke's law, force in spring is proportional to extension. Extension is always positive regadless of which way it is. So when in compression, we have a force pointing in the positive direction and has an amount of $|kx|$. But I need to obtain $mx''=-kx$ even when the spring is in compression. I think my problem is in the $mx''$ term and not in the $kx$ term. What Am I missing? Where did I go wrong? How to get $mx''=-kx$ when mass to the left using just free body diagram? **adding** This is below is attempt to get the same EQM using D`Alembert's principle. I do not know it well yet, but actually now I get the correct EQM when the spring is in tension or compression using this. One is supposed to write $F-(mx'')=0$ where now $mx''$ is the so called fictitious inertial force that is always in the opposite direction to the resulting applied forces and acts on the same line. So using this: When the mass is to the right of the $x=0$ we get $F-(mx'')=0$ or $-kx-(mx'')=0$ or $kx+mx''=0$ so this works ok. Now lets try it when the mass to the left, we have $F-(-mx'')=0$ where I added a negative sign for $-mx''$ since now it is pointing to the left, i.e. negative. This is because now the force in spring is pointing to the right. So now we have $kx-(-mx'')=0$ or $kx+mx''=0$ which is the correct EQM ! Is the above correct way of using D`Alembert's principle on this problem?


title: Best way to move teaspoon to dissolve sugar in a cup of tea

text: I know that there is asked questions about something like cooling down cope of coffee..etc, but this one is little different. My friend was watching TV program of English eating etiquette, and they mentioned that it will be faster to move teaspoon back and forth in a line instead of in a circular way (what we doing usually) will force sugar to dissolve much faster! But I'm very suspicious with that, and my argument taken from my observation that all of the mixers I saw in industrial level was moving in a circular way, and I believe that they optimized their productivity as much as possible. So is there any physical reasons that explains which motion will produce the best vortexes for sugar dissolving?


title: Does increasing the density of a solution decrease the rate of temperature change?

text: I did an experiment to compare whether salt water (5% concentration of salt) or fresh water of the same volume took longer to heat up to a certain temperature. We found that salt water took longer to heat up than fresh water. Is this due to density? specific heat capacity? or should I have gotten different results.


title: Number of 2-electron integrals

text: Consider 2-electron integrals over real basis functions of the form $$ (μν|λσ)=∫dr⃗_1dr⃗_2ϕ_μ(r⃗_1)ϕ_ν(r⃗_1)r^{−1}_{12}ϕ_λ(r⃗_2)ϕ_σ(r⃗_2) $$ I am told that for a basis set of size $K=100$, there are $12,753,775$ unique 2-electron integrals of this form. Symmetry considerations mean that we have less than $K^4$ unique integrals, since we can exchange electrons and also exchange the basis functions for each electron without changing the value of the integral. How could one work out the number of unique integrals? I didn't understand very well this: Number of unique 2-electron integrals And I found that the number of 2-electron integrals is equal to $$ \frac{1}{8}n(n+1)(n^2+n+2),\hspace{3mm}\text{for}\hspace{4mm} n=100 \hspace{4mm}\Longrightarrow \hspace{4mm}12,753,775\hspace{4mm} \text{integrals} $$ but I don't know how to explain it.


title: Archimedes principle and specific gravity

text: A physical balance measures the gravitational mass of a body. I conducted an experiment to find out the specific gravity of a bob. I first measured the mass of the bob in air, and then in water. The mass of the bob in water was less than that in the air. Mass remains constant. How can it change in water? If it was not mass that decreased but it was the weight of the bob, then why did we use physical balance to find out the specific gravity of bob? Mass is measured in grams, or kilograms and weight in Newtons, slug or pound, etc. What I noticed was that the quantity (Mass or weight) decreased from 28.75 gm to 25 gm. Can kilogram or gram and Newton be used interchangeably?


title: What is the exact mechanism by which time dilates?

text: What is the exact mechanism by which time dilates for a fast moving object? Can the time dilation be explained by any theory other than relativity?


title: How do I publish a physical constant

text: I think what i've found is a physical constant that is a physical quantity, universal in nature and constant in time. **but It contrasted with a mathematical constant, which is a fixed mathematical numerical value and does not directly involve any physical measurement.**   1. Is it really necessary to consider fixed numerical value?   2. How do I publish a physical constant (that It contrasted with a mathematical constant)?


title: Is symplectic form in Hamiltonian mechanics a physical quantity?

text: Is symplectic form $dp_i \wedge dq_i$ in Hamiltonian mechanics a physical quantity? It feels to me to be something different than say energy, momentum or mass. Like just certain structure. The real reason why I'm asking lies not in mechanics, but in GENERIC. There apart of Poisson bracket the second bracket is added. This bracket is responsible for the irreversible evolution and depends highly on a physical system in hand. While I was musing about it I wondered whether I can attribute these two brackets as physical quantities and if no, how shall I at least call them as opposed to physical quantities?


title: Diving into a charged (Reissner-Nordstrom) Black hole

text: Apparently there are two event horizons in this type of black hole, where the second one is known as the Cauchy horizon. According to Carroll, if you go into the first one, you will fall until you reach the second one, at which point you are free to move around. From here you can choose to avoid the singularity and cross the Cauchy horizon again (I'm not sure I get how, isn't the horizon a null surface? doesn't this mean you have to travel at the speed of light to cross it?). From here you will necessarily move in the increasing radial direction until you are expelled from the black hole. Where do you end up in spacetime? How can an outside observer see you come out if he never saw you fall in? I've read about it being some kind of wormhole. Carroll says it's like exiting a white hole into another universe, but then he says you can choose to go back in, but I thought nothing can enter a white hole...


title: How does earth carry moon with it, if it can not force moon to touch it by gravitational force?

text: earth's gravitational force is acting on its moon in such a way that it forces moon to rotate round its orbit by centripetal force and carries it while rotating round the sun by gravitational force. I don't understand why in this condition moon doesn't come to the earth while earth is carrying it through gravitational force to rotate round the sun??


How to understand the work-energy theorem? I took a short lecture on physics for engineering last week. The lecturer emphasized that the work done on an object will cause the kinetic energy change as $$W = \Delta \text{KE}.$$ I know this concept might be so common to you but to me, as a beginner, it is pretty hard to understand the reason. My understanding is that 'work' is the energy an external object 'injects into' the object or is the energy an external object 'takes away' from the object. I think the work done by the object should equal to the total energy changed on that object, which could be in any form (heat, potential or kinetic energy.) Why does the theorem only explicitly refer to kinetic energy? Will this theorem work in some cases or in all cases?


title: Does the sun rotate?

text: As implied from the question, does the sun rotate? If so, do other stars not including the sun also rotate? Would there be any consequences if the sun and other stars didn't rotate? Me and my friends have differing views on this, and would like some clarification. Thanks!


title: Lenses and Magnification

text: A $2.05\text{ cm}$ tall object is placed $30.0 \text{ cm}$ to the left of a converging lens with a focal length $f_1 = 21.0\text{ cm}$ . A diverging lens, with a focal length $f_2 = -42.5\text{ cm}$ , is placed $30.0 \text{ cm}$ to the right of the first lens. How tall is the final image of the object? First I found the distance of the image created by the lens 1 which is $70\text{ cm}$. Then since distance of image is $70\text{ cm}$ the distance of the object to lens 2 is $10\text{ cm}$. So I used that to find the distance of image created by lens 2. Then I find the total magnification by multiplying both $M_1$ and $M_2$. Then use the $M_{total}$ to find the height of final object. But it's wrong.


title: Dipping a Dyson Ring below the event horizon

text: The basic assumption about inescapability beyond the Event Horizon is that the necessary escape speed (orbital speed) would exceed speed of light, therefore no object can achieve it. Now, would it be possible to escape it by other means, say, by mechanical push against the gravity, through application of force without increase of speed? * * * First, short introduction: What is a Dyson Ring? (feel free to skip or skim this section if you know it): Dyson Ring is a hypothetical, artificial, ring-shaped structure surrounding a massive body (e.g. a star), rotating - essentially orbiting it - rotating at speed slightly higher or lower than orbital speed necessary to maintain orbital equilibrium - state of freefall near its surface. This way, its rotation creates centripetal force simulating gravity for inhabitants. It would create a habitat much larger than any planet, given some "rim" it could hold atmosphere, and essentially is a neat sci-fi alternative for a planet - similar conditions, vastly more surface. The one property of it is that tensile strength of its construction allows it to exist at distance from the central body (star) that is off from normal orbit matter moving at this speed would take. Most theoretized Dyson Rings move slightly faster than needed, stretched out, with the inner side inhabited, and the Sun in zenith for all of their surface. It is possible though (if not recommended due to buckling risks) to make one that moves slower, the star's gravity not fully overcome by rotation, compressive force applied to the construction. * * * Now - technological problems aside - of building a Dyson Ring (not habitable and much smaller; a scientific device of ~30km radius.) and bringing it close enough to a Black Hole. Let's imagine a Dyson Ring that can change its circumference (say, built of segments connected with expanding/contracting actuators) - and, as effect, radius within certain range - specifically between outside and inside of Schwarzschild radius of a selected Black Hole. It is also supplied with power sources that allow to vastly increase its rotary speed. Now, with the actuators expanded the ring is placed around the black hole, outside the even horizon, rotating so fast that its segment orbit the black hole. Its angular velocity so high the linear velocity of its surface is near to speed of light. Normally the centrifugal force would tear it apart, but it remains in equilibrium with centripetal force of gravity of the black hole. Now, actuators contract. The angular velocity rises a little, with accordance of law of conservation of momentum, but since we're at relativistic speeds, the increase of angular speed is minimal; it's mostly the ring gaining some mass. Now, the ring is below the Event Horizon. Since at this altitude orbital speed would exceed the speed of light, any object there would quickly spiral into the black hole. Still, the ring does not, due to its own tensile strength preventing further decrease of its own radius. It moves slower than the orbital speed, but the mechanical strain overcomes the gravity. (note, while below the Event Horizon, we're still pretty far from the singularity, so the forces exerted are not yet extreme enough to cause collapse of matter, and the rapid rotation overcomes great most of the compressing gravitational force) Expand the actuators, increase the radius of the ring - and it is forced back outside the Event Horizon, able to broadcast its findings. Possible physically, or am I missing something?


title: Intuitive understanding of the entropy equation

text: In thermodynamics, entropy is defined as $ d S = \dfrac{\delta q_{\rm }}{T}$. This definition guarantees that heat will transfer from hot to cold, which is the second law of thermodynamics. But, why do we denote entropy as$\dfrac{\delta q_{\rm }}{T}$ other than $\dfrac{\delta q_{\rm }}{T^2}$,$\dfrac{\delta q_{\rm }}{e^T}$,or something else? Is there an intuitive explanation for this $\dfrac{\delta q_{\rm }}{T}$?


title: Could dark energy make a large black hole less black?

text: Theoretically, if a black hole grew to a huge scale such that the effect of dark energy was large, could the black hole become 'normal' space again (i.e. no horizon or singularity)? What I'm trying to understand is, does dark energy 'uncurve' space-time?


title: quantum optics of a polarizing beam splitter

text: I would like to use the Heisenberg picture in quantum field theory to model a polarizing beam splitter. Is there an easy way for someone to show me how the field operators ($a^\dagger_{input1}$, $a^\dagger_{input2}$etc) transform across a polarizing beam-splitter?


title: What's the relationship between glow and arc discharge?

text: I'm trying to understand the differences between an arc and glow discharge. Most texts I have found on the Internet only describes the observed differences but doesn't really explain what happens on the atomic level in the two types of the discharge. So what I know:   * Glow discharge happens on higher voltage in the mA current range. It has recognizable structure (eg. the Faraday dark space and the Crookes dark space). It's faint and cool.   * Arc discharge happens on lower voltage, and it's in the Amp current range and higher. It runs from electrode to electrode without dark spaces. It's often blindingly bright and hot.  A characteristic of the normal glow discharge is that the voltage is stable, only the current changes, hence it's used as a voltage stabilizer. As you increase the current through the glow, it tends to cover more area of the cathode. When it covered all of it, the voltage starts to increase to supply more and more energetic electrons, this is the abnormal glow. **What happens at point`I` in the graph below that causes the voltage drop again?** ![enter image description here](http://i.stack.imgur.com/6CYxA.png) Glow discharge is often demonstrated with discharge tubes. Countless of youtube videos show amateur experimenters pumping down their tubes, causing initially an arc discharge which turns into a glow discharge as the tube pumped down, **what causes this mode transition?** Whenever you turn on a compact fluorescent bulb it glows for a second then the discharge turns into an arc as the cathodes heat up and the bulb brightens up. I know hot electrodes emit more electrons, but **why more electrons cause transition into the arc mode?** I suppose it's due to the same reason for all cases. Can anybody help me put the things together, so I can get the point?


title: Leaving a stellar black hole's orbit

text: Say Earth is orbiting a black hole 1900 km in diameter, 1.5E33 kg in mass, and has a gravitational pull of 110,858,725,761.77 m/s^2. It's around 40 AU away Earth's orbital speed is 129,305.15664 m/s. (I did the calculations, it makes sense) I'm writing an extra credit assignment for my 10th grade physics class. I need the earth to leave its current orbit and fly outward at a speed around its current orbital velocity (but not caused by a major collision because I want the Earth to stay intact). I'm too far invested in my assignment and I don't want to change anything too major, we haven't learned much about black holes and special relativity so keep that in mind in any answers. Thanks, please correct me if I made a foolish mistake.


I'm new to this field hence this weird question. Why would we need to ground an AC source? Why wouldn't it be enough to have just one pole to get an AC current going? I understand why it wouldn't work in DC case where current is flowing in one direction. However, in case of AC source where the current is not flowing anywhere but rather just oscillating back and forth it's not that clear to me why connecting load to only one pole wouldn't work? Thanks.


title: Quantum mechanical proof of Conway's SPIN and TWIN axioms?

text: I think I understand correlations of EPR-state and GHZ-state which deal with spin-1/2 particles and (-1, 1) measured values. Conway's state is spin-1 particle state with (-1, 0, 1) measured values. Which quantum mechanical state vector correspond to Conway's state? And how to prove SPIN and TWIN axioms of Conway's free will theorem from quantum mechanics?


title: Why doesn't current flow when the wire is open?

text: I realize this may be a hard to answer question but we are currently studying current in our school. One thing that struck me was why the heck doesn't it flow when the wire's closed? When you connect the positive terminal of the battery (But do not connect the negative terminal) to a conductor, the loosely bound / free electrons in the conductor should be attracted to the battery's +ve terminal. This should create a chain reaction till every atom in the conductor has donated the maximum it can and no further flow is possible due to the atoms now strongly attracting their remaining electrons. Why doesn't this happen? There should be momentary current and the wire should become positively charged. Why not? Why does it only flow when the circuit is closed?


title: Charge Distribution in Reissner-Nordström Black Holes

text: In an electrically charged black hole, such as the one described by the Reissner-Nordström metric (i.e. with no angular momentum), where would the electric charge be situated (neglecting any charged particles falling into the black hole)?


title: Can the correlation for the Potts model be bounded?

text: I am studying a $d$-state Potts model. A configuration $\sigma$, which assigns for each $x\in \mathbb{Z}^2$ a value $\sigma(x)\in [1,2,\ldots,d]$, with the probability on a finite lattice defined as $$ P_{\Lambda}(\sigma) = \frac{1}{\bf Z_{\Lambda}} e^{-H_{\Lambda}(\sigma)}, $$ where $H_{\Lambda}(\sigma)$ is a Hamiltonian and $\bf Z$ is a normalising constant. The form of the Hamiltonian is $$ H_{\Lambda}(\sigma) = -\beta\sum_{\\{d_{\infty}(x,y)=1\\} \cap\\{ y\in \Lambda\\}\cap\\{ x\in \Lambda\\}} \delta\bigl(\sigma(x),\sigma(y)\bigr) + \sum_{x\in \Lambda} \sum_{i=1}^d h_{x,i} \delta\bigl(i,\sigma(x)\bigr), $$ where $x \in \Lambda$ where $\Lambda$ is a regular lattice in $ \mathbb{Z}^2,\beta>0$ and $h_{x,i}\geq h>0$. The function $\delta$ is the (Kronecker?) delta, that is $$ \delta(a,b) = \begin{cases} 1\, \mbox{ if } a=b, \\\ 0\, \mbox{ otherwise}. \end{cases} $$ What I am wondering, is it possible to bound the absolute value of the covariance function (correlation function in physics?) $$\lvert\langle\delta(\sigma(x),i),\delta(\sigma(y),j)\rangle - \langle\delta(\sigma(x),i)\rangle \langle\delta(\sigma(y),j)\rangle\rvert,$$ by an exponential function if $d(x,y)$ is large? (depending only on $h,\beta$ and $d(x,y)$) If not is the rate of decay known?


title: Why is electrical energy so difficult to store?

text: Does anyone know a general answer to these questions? (I've asked them together because they're all pretty related, it seems.)   * Why is it that we find electrical energy so difficult to store? Do we just find energy difficult to store generally? (...surely not, we can store energy in a block by sending it to the top of a hill.)    * is there something in particular about charge/electricity that makes effective batteries difficult to produce, and, if so, what?   * Is the problem that we're having with storing the energy just an artefact of our use of the energy, or is it difficult to store electrical energy per se?


title: How do different fields interact with each other?

text: Recently I've seen a few talks and lectures about Quantum Field Theory. They explained what a "particle" means in a field, and that a large enough excitement in a certain field can excite another field (effectively creating a new particle), but they didn't mention **how** that interaction happens. Searching around I've heard that the interaction is being transmitted by "virtual particles", but won't these particles require their own field (because a particle is just an specific ripple in a field)? And if those virtual particles do have their own field, then how does that field interact with the 2 other fields that it's mediating? Thank you!


title: How to understand the indeterminate process in thermodynamics?

text: Consider a closed composite system with an internal movable adiabatic wall. If we release the piston, thermodynamics cannot determine the final equilibrium state (the temperature cannot be determined). (About the previous example, you can refer to Callen's _Thermodynamics and an Introduction to Thermostatistics_ 2ed. P53 Problem 2.7-3 and P100 Problem 4.3-1: ![enter image description here](http://i.stack.imgur.com/BtoLr.png) ) How to understand this? Consider a quasi-static process. As the piston moves, the temperature and the pressure of both chambers will change until the pressures coincide. Then there is a well-defined unique final equilibrium state. What is wrong with this argument? Does it mean that thermodynamics is incomplete?


title: What is limiting line in series spectrum?

text: > The wavelength of first line in the Balmer Series is 'whatever(in nm)' . > Calculate the wavelength of the second line and the 'limiting line' in the > Balmer Series. I found this question in an ancient question paper in the library. Now, I have solved the first part by calculating the atomic number from the first relation and then applying it while calculating the wavelengths of the second line in the Balmer series which must mean the line after Balmer (which is paschen). **But for the second part 'what is limiting line', and that's exactly what my question is. Also since the given wavelength is in 'nanometer' what whould be the unit of energy if found.** I certainly don't need any solution for the above noted question( from the question paper), so this question cannot be declared as an 'homework question' due to some weird rules, (ofcourse they are meant for a mass good, but the tag name 'homework' hurts, anyway don't want that to be a topic of discussion here, but please consider it.)


title: Microscopic definition/expression of/for the heat current

text: Often I see the following microscopic definition/expression of/for a heat current due to an external field $$ {\bf j}_Q = 2 \int \frac{\text{d}{\bf k}}{(2\pi)^3} \frac{\hbar {\bf k}}{m} (\epsilon_{{\bf k}} -\mu) f(\epsilon_{{\bf k}}), $$ where $f$ is the solution the the appropriate Boltzmann equation. Note that the factor of two is there for the spin of electrons. I do not really understand where this expression comes from. Especially the factor $(\epsilon_{{\bf k}} -\mu) $ bothers me. Does it tell us that only electrons near the Fermi level contribute to the heat current? Or does it have another meaning?


title: Ambiguity in Asymptotic Perturbative Series and Instantons

text: I know there are a number of questions about the asymptoticity of perturbative series and about instantons on StackExchange (e.g. Instantons and Non Perturbative Amplitudes in Gravity from user566, Instantons and Borel Resummation asked by felix, and How can an asymptotic expansion give an extremely accurate predication, as in QED? asked by yonni). Reading them was helpful but left me with two short questions: 1) What is meant by "ambiguity" in this context? Several posters use the term in alluding to the problems in asymptotic series. Does it have a technical meaning here? 2) How can we see that the instantons would "correct" the series in the full theory? Perhaps the only thing to do is to read these notes ("Instantons and large N" by Marino) (which I plan on doing) but I was wondering if someone could give a quick answer for #1 and perhaps a clever or intuitive way of making #2 plausible.


title: Propagator of Chern-Simons Abelian gauge theory

text: I need to compute the "topologically massive photon" propagator. I've started with : $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\mu}{4}\epsilon^{\mu\nu\lambda}A_\mu\partial_\nu A_\lambda $$ $$ =A_\mu\underbrace{[\frac{1}{2}g^{\mu\lambda}\partial^2+\frac{\mu}{2}\epsilon^{\mu\nu\lambda}\partial_\nu]}_{(\Delta^{-1})^{\mu\lambda}}A_{\lambda} $$ So how can I invert The under braced part which will yield the topologically massive photon propagator?


title: Parity transformation for spinors (pinors) in odd spacetime dimensions

text: What is the transformation law for spinors (pinors) under parity in an odd number of spacetime dimensions? I know how to derive the transformation properties of spinors (pinors) under parity in an even number of spacetime dimensions. Let $$\eta^{ab} = \mathrm{diag} (1, 1\ldots 1, -1, -1 \ldots -1) $$ where there are $p$ entries of $1$ and $q$ entries of $-1$, and $p+q=n$. Let the gamma matrices $\gamma^a$ generate the real Clifford algebra $\mathrm{Cl}(p,q)$, $$ \\{ \gamma^a, \gamma^b \\} = 2 \eta^{ab} $$   The Pin group $\mathrm{Pin}(p,q)$ is defined as the set of invertible elements $S_{\Lambda}$ of $\mathrm{Cl}(p,q)$ that satisfy $$ S_{\Lambda} \gamma^a S_{\Lambda}^{-1} = {\Lambda^a}_b \gamma^b $$ for some element ${\Lambda^a}_b$ of the orthogonal group $\mathrm{O}(p,q)$, and also $S_{\Lambda}S_{\Lambda}^{\tau} = \pm 1$, where the superscript $\tau$ denotes a linear operator that reverses the order of products, e.g. $(\gamma_0 \gamma_1 \gamma_2)^{\tau} = \gamma_2 \gamma_1 \gamma_0$. For each $\Lambda$, there are two solutions for $S_{\Lambda}$ that differ by a minus sign, and the the map that sends these two solutions to $\Lambda$ is a $2-1$, homomorphism from $\mathrm{Pin}(p,q)$ to $\mathrm{O}(p,q)$. A parity transformation in the orthogonal group $\mathrm{O}(p,q)$ that inverts the $i$-th spatial axis is given by $P_i = \mathrm{diag}(1, 1 \ldots 1, -1, 1, 1, \ldots 1)$, with the entry of $-1$ acting on the $i$-th spatial axis. To find a parity transformation on a spinor (pinor), one solves the above equations for $\Lambda = P_i$. In an even number of spacetime dimensions, the solution is $$ S_{P_i} = \pm \gamma_i \gamma_n $$ where $\gamma_n = \prod_{a=0}^{n-1} \gamma_a$. Crucial to make this work is the fact that $\gamma_n$ anti-commutes with all of the gamma matrices $\gamma_a$ in an even number $n$ of spacetime dimensions. However, in an odd number of spacetime dimensions, the operator $\gamma_n$ is proportional to the identity matrix, and thus commutes with everything. And indeed I believe there is no operator in an odd number of spacetime dimensions that anticommutes with all the gamma matrices $\gamma_a$. As a result, I cannot see that there is a solution to the above equations for a parity transformation on spinors (pinors) in odd dimensions. The main reference I have been using is http://arxiv.org/abs/math-ph/0012006. In section 5, page 65, a similar conclusion is reached. Then it is said that the 2-1 homomorphism/covering map from $\mathrm{Pin}(p,q)$ to $\mathrm{O}(p,q)$ given above is not surjective in an odd number of spacetime dimensions, and in particular it does not 'hit' axis reflections in $\mathrm{O}(p,q)$.


title: What does the current vs voltage graph of a halogen look like?

text: I have a halogen light with a tungsten filament. It is rated 500w at 130v but I am running it from an inverter in my car that is putting out 110v. I would like to know what the actual power usage of this lamp is at 110v. When I was a Sophomore in high school I had a physics lab where I calculated the temperature of a tungsten filament by measuring the current and then doing some calculation-which I can't pull out of my brain at the moment. I am guessing that I am not the first person to do this and there must be some sort of curve that plots the current vs voltage of a halogen. If anyone knows of such a graph - or better yet some equations I could use to solve for an exact number, I would appreciate it. EDIT: Ok, so I answered my original question, but if someone can give me actual equations instead of the mediocre graphs I found, I would be happy.


title: Speed of sound in air

text: Quick question. I thought that the speed of sound in air was constant, say in the right conditions of pressure and temperature, and humidity... 300 m/s. Now, if I have a sound source that moves towards me at 50 m/s, is the speed of the sound waves 350 m/s when it arrives to me? (this is not about the Doppler effect, which I know about. It's about whether/how to compose the speed of sound and the speed of the source - it seems all the texts I'm seeing assume that the speed is 300 m/s no matter how the source moves)


title: Black hole entropy from collapsed entangled pure light

text: Consider the following scenario, very similar to the one proposed in this question, but this time, the pure quantum radiation used for the black hole collapse, is now being split with down-converter crystals in pairs of entangled photons. So we have two groups of photons, but the overall state is still pure (it is described by an eigenvector, density matrix satisfies $\rho^2 = \rho$) **Brave and Courageous assertion No.1** So the overall quantum state has Von Neumann entropy zero or nearly zero ( _right? wrong? why?_ ) If i were to create a single black hole with the two groups of photons, by the argument explained in the previous question, the entropy of the black hole would be zero or nearly zero, and hence the black hole would be _mysteriously_ extremal, with no measurable charge or angular momentum. But if i take the two groups separately and have them collapse in individual black holes, it would seem to external observers that the Von Neumann entropy of the parts is non-zero, since the partial traces of a pure entangled state will be density matrices of mixed states. Which leads me to my **Brave and Courageous assertion No.2** : So each black hole in the pair would seem to have a non-zero temperature to far-away observers, and they would look classical ( _right? wrong? why?_ ) Assuming both brave and courageous assertions hold true, the question i have is this: > If the two black holes are made to coalesce in a single black hole, what > will be the thermodynamic behaviour of this merged black hole: Would it be a > nearly-zero entropy black hole? or would it radiate as a classical black > hole? if the answer turns out to be: it radiates as a classical black hole, then what happened to the Von Neumann information that is inside the merged black hole and why the formation and merge of black holes changed it so much?


title: Hamiltonian Noether's theorem in classical mechanics

text: How does one think about, and apply, Noether's theorem in the classical mechanical Hamiltonian formalism? From the Lagrangian perspective, Noether's theorem (in 1-D) states that the quantity $$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \frac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} - \left[\sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \frac{d y_j}{dx})} \frac{d y_j }{\partial x} - \mathcal{L}\right]\frac{\partial x^*}{\partial \varepsilon}$$ is conserved if the Lagrangian $\mathcal{L}(x,y_i,y_i')$ is invariant under a continuous one-parameter group of infinitesimal transformations of the form $$T(x,y_i,\varepsilon) = (x^*,y_i^*) = (x^*(x,y_i,\varepsilon),y_i^*(x,y_i,\varepsilon).$$ From the action perspective, Noether's theorem states the equality of the 1-forms: $$\mathcal{L}(x,y_i,y_i')dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*$$ which can be used to determine (additive) symmetries nicely. How do I use this formalism to understand the Hamiltonian Noether theorem in a general context? I'll usually see a claim that $dA/dt = [H,A]$ is the Hamiltonian Noether's theorem, and I can't make sense of this in the context of my description of Noether above. This appears to derive the Poisson brackets as part of Noether from what I've developed above, but I can't make much sense of it to be honest I'm sure the answer is supposed to link the local Lie algebra tangent vector structure to the global Lie group transformation in the Lagrangian, but saying that in words is one thing, in math it's another. Note, these (and other) old stack posts do not answer the question:   * Hamiltonian form of Noether's Theorem   * A kind of Noether's theorem for the Hamiltonian   * Noether's theorem and "translations" of the Hamiltonian function References:   1. Bergmann - Introduction to the Theory of Relativity, appendix


title: Understanding Cherns-Simons-Witten Theory

text: I want to read about Wittens work, on Cherns-Simons theory, and relations to knots and jones polynomials. I am extremely motivated to read his paper: Quantum Field Theory and Jones polynomial. What are the prerequisites in topology, geometry and QFT needed to understand this paper? I have good grasp of basic point-set topology concepts, and have read about manifolds(Nakahara). In QFT, I have just started QED. What do I need to study from each of these? As per my knowledge, I need some knot theory and cohomology(?), and quantum yang-mills theory. Is this correct? Also please provide references.


title: Majorana zero mode in quantum field theory

text: Recently, Majorana zero mode becomes very hot in condensed matter physics. I remember there was a lot of study of fermion zero mode in quantum field theory, where advanced math, such as index theorem, was used. I would like to know if there were previous studies of fermion zero mode where fermions have no $U(1)$ particle number conservation? (Such fermionic zero mode without fermion number conservation corresponds to Majorana zero mode.) Edit: Here, a Majorana zero mode is a zero-energy mode localized at the center of some defect (like vortex or monopole).


title: Inner product of particle-anti-particle spinor components

text: Suppose I have four-component spinors $\Psi$ and $\bar \Psi$ satisfying the Dirac equation with $$\Psi(\vec x) = \int \frac{\textrm{d}^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\vec p}}} \sum_{s = \pm \frac{1}{2}} \left[ a^s_p u^s_p e^{i \vec p \cdot \vec x} + \tilde b^s_p v^s_p e^{-i \vec p \cdot \vec x} \right] $$ $$ \bar \Psi(\vec x) = \int \frac{\textrm{d}^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\vec p}}} \sum_{s = \pm \frac{1}{2}} \left[ \tilde a^s_p \tilde u^s_p e^{-i \vec p \cdot \vec x} + b^s_p \tilde v^s_p e^{-i \vec p \cdot \vec x} \right] \gamma^0$$ with these definitions: $$ \tilde a \equiv a^\dagger \quad ; \quad \gamma^0 = \begin{pmatrix} 0 & 1_2 \\\ 1_2 & 0 \end{pmatrix} \quad ; \quad u^s_p = \begin{pmatrix} \sqrt{p \cdot \sigma} \xi^s \\\ \sqrt{p \cdot \bar \sigma} \xi^s \end{pmatrix} \quad ; \quad v^s_p = \begin{pmatrix} \sqrt{p \cdot \sigma} \eta^s \\\ -\sqrt{p \cdot \bar \sigma} \eta^s \end{pmatrix}$$ $$ \sigma = (1_2, \vec \sigma) \quad ; \quad \bar \sigma = (1_2 , - \vec \sigma) \quad ; \quad p = (p_0 , \vec p)$$ where $\vec \sigma$ is the usual vector of Pauli matrices and $\xi, \eta$ are two-component spinors. The lecturer then goes on to choose an appropriate basis for $\xi$ $$ \xi^1 = \begin{pmatrix}1 \\\ 0\end{pmatrix} \quad ; \quad \xi^2 = \begin{pmatrix} 0 \\\ 1\end{pmatrix}$$ and similarly for $\eta$ such that $\tilde \xi^r \xi^s = \delta^{rs}$ and $\tilde \eta^r \eta^s = \delta^{rs}$. This appears sensible and using this, we can compute various inner products of $u$ and $v$. In the examples discussed in the notes, mixed terms, that is, those containing products of $\xi^r$ and $\eta^s$ never occur (multiplied by 0, hence irrelevant) or cancel out. In an example assignment, however, we are meant to calculate $\bar \psi(-i \gamma^i \partial_i + m) \psi$ which then leads me to terms such as $$ \sum_{s,t} 2 m \tilde a^t_p \tilde b^s_{-p} (p_i \sigma^i) \tilde \xi^t \eta^s$$ which also cancel out in the end, but pose the question in the meantime how $\tilde \xi^t \eta^s$ is actually defined/can be calculated. The identities gained from an appropriate choice of basis for $\eta$ and $\xi$ ‘feel’ wrong here, since, after all, $\xi$ comes from a particle-spinor and $\eta$ comes from an anti-particle spinor. I take it my question could be rephrased as to whether $\xi$ and $\eta$ live in the same space or belong to two different spaces (which would make their inner product an even more thrilling exercise).


title: Gravitational lensing or cloud refraction?

text: **My current understanding of gravitational lensing follows** When a star or other massive body passes between us and another star, the phenomenon generally labeled 'gravitational lensing' occurs. The term 'gravitational lensing' refers to the bending of light rays due to the gravitational influence of the massive body. Since the bends concentrate light for the observer, they act like a magnifying lens. **OTOH** But since I was young I have pondered other explanations for such observations. One thought about what (else) might cause the observed effect is a cloud of gas around the massive body itself. The assumption being that any change in density between the thin gas of interstellar space and the thicker gas surrounding a massive object should form a refractive boundary. In the case of an active star, that gas & might be the heliosphere of the star, and for any massive object that is _not_ creating an heliosphere, it could be due to clouds of gas that the massive object is sucking in. I raised this idea with a physics teacher in high school & he assured me that any possible refraction from clouds of gas had been ruled out in observations of gravitational lensing. I never got to see papers on the subject, so I cannot confirm that (and am not convinced). My questions are:   1. Does anybody know of papers on gravitational lensing that discuss the refraction from clouds of gas around the lensing body?   2. How could an observer/experimenter go about quantifying the likely refractive index of such a gas cloud.


title: Light "diode" and 2nd law of thermodynamics

text: If I had a light "diode" - an object that only allowed light (at least for a range of frequencies) to travel through it in one direction, would this _necessarily_ allow violations of the 2nd Law of Thermodynamics? Or would it be possible to restrict other properties of the device such that it could be okay. Why I think it should violate 2nd law:   1. I can place such devices at temperature T in a bath of thermal radiation at temperature T, such that it forms a box which only lets the light in or out. This would appear to always allow one to create a thermal gradient. Once a thermal gradient is formed, useful energy can be extracted. This seems to allow useful energy to always be extracted from a thermal bath and thus give perpetual motion.   2. Release one such device at temperature T in a bath of thermal radiation at temperature T. Since the photons are absorbed or reflected in one direction, and undisturbed in the other, the device will start moving. It can use a thermal bath to directly generate useful motion. Why I think I'm missing something and such a device could potentially co-exist with the 2nd Law:   1. The above arguments seem valid if the diode was only 'partially' rectifying as well. However there exist sunglasses which largely reflect on one side, but not from the other side.   2. Researchers have succeeded in using meta-materials to create a "blackhole" for microwaves. Can anyone help sort this out?


title: Center Of Mass Troubles

text: I understand the concept of Center Of Mass(com), but I am having a difficult time interpreting the equation of the simplified case of one-dimension. The book I am reading defines the position of the com of a two-particle system to be $x_{com}= \Large\frac{m_1x_1+m_2x_2}{m_1+m_2}$ I'm sorry if this seems like a trivial question, but could someone explain to me the interpretation of this definition? Perhaps even why they defined it to be this way. Here is an excerpt from my textbook: "An ordinary object, such as a baseball bat, contains so many particles (atoms)that we can best treat it as a continuous distribution of matter. The “particles” then become differential mass elements $dm$, the sums of Eq. 9-5 become integrals, and the coordinates of the center of mass are defined as (9-9)...Evaluating these integrals for most common objects (such as a television set or a moose) would be difficult, so here we consider only uniform objects. Such objects have uniform density,or mass per unit volume; that is, the density $\rho$ (Greek letter rho) is the same for any given element of an object as for the whole object. From Eq. 1-8, we can write $\rho = \large\frac {dm}{dV} = \frac {m}{V}$" What does the author mean by, "the 'particles' then become differential mass elements $dm$? Is $\rho = \Large\frac{dm}{dV}$ the derivative of the density function? If so, how would I interpret that? Furthermore, if it is indeed the derivative of the density function, why is it also equal to the original function of density?


title: Spin of 125 GeV Higgs boson

text: Can someone please explain to me why (according to decay of Higgs boson into 2 photons) Higgs boson cannot have spin $S=1$?


title: How can a fly be in the space of a roofless car, get out of the car space and get back in while the car is driving at the speed of 80mph?

text: I was driving in a cabriolet car at the speed of 80mph, while a fly flew in and out of the frame of the car. My question is, How can the fly leave the frame and get back in? Is it possible that it flies at 80mph?


title: Non reciprocal light propagation

text: In search for some explanation in why magneto-optical materials (like the one used in the Faraday rotator and, consequently, in the "optical diode") act in such a "strange" way, I saw that this kind of materials break the time- reversal symmetry and a different kind of reciprocity is applicable, the Rayleigh-Carson one. What I cant figure out is how the interaction of the the light with the material can rotate de polarization vector depending on the direction? How can I visualizate it in a microscopic level (using Quantum Mechanics)?


title: Circular motion and centrifugal force

text: Assuming a race car drives around in a circle of radius r, center (0,0), linear velocity v, and ignoring centrifugal forces and friction I can calculate the position at any time, t. Angular velocity $$\omega = \frac{v}{r}$$ degrees turned through (assume at time t = 0 car is at position (r,0)): $$\theta = \omega t$$ So the positional (x,y) coordinates, p, at time t, are $$p(t) =(r\cos(\theta) , r\sin(\theta ))$$ Please correct me if I am wrong up to this point. Now I want to include slipping, so that the car will be thrust outwards by the centrifugal (?) force, with friction force countering it. Supposing the centrifugal force is stronger, then there will be a resultant force in the direction tangential to the linear velocity at time t. Suppose this force is $F$, then I want to find the modified position of the car at time t, as it won't be travelling in a circle but, I think, will be travelling in an increasing spiral (this is theoretical, I haven't taken into account spinning around its axis or crashing into a barrier). So this is where I am going to use very naive maths/physics to try and find the modified position. Assume the car has mass $m$. Then the acceleration outward from the centre of the circle will be $$a = \frac{F}{m}$$ Now I'm going to assume acceleration is constant to make my life simpler, and I can fudge a value for outward velocity u, $$u = \int_{0}^{t} a dt = at$$ and more dodgy physics, I can calculate the position vector by multiplying by t, assuming constant velocity outwards, $$\text{position} = (u t \cos(\theta),u t \sin(\theta))$$ Remembering at time t, the angle is $\theta$ I can see that the new position, let's say $q(t)$, is given by $$q(t) = p(t)+(u t\cos(\theta), u t\sin(\theta)) =p(t)+ (\frac{Ft}{m} t\cos(\theta),\frac{Ft}{m}t\sin(\theta))$$ One problem I have noted, just looking at this is that $q(t) \tilde{}t^{2}$ since to get the position from acceleration I multiply by t twice, which seems too much, the car will spiral too fast. I am very skeptical this is correct. Please correct me where I have gone wrong. Edit: Also I just pulled the value $F$ from thin air. Could you explain how I could calculate it based on the cars velocity, mass, radius of circle, coefficient of friction of surface or any other variables?


title: if a force is 1 newton metre, what is it at 2 meters?

text: If I have a force, say 24 kg/cm what would that equate to at 2cm? I would like to know the formulae for calculating this. For example. If a motor can hold an object of 24kg at 1cm from its pivot point, what is it cable of holding at 5cm? And how is it calculated?


title: How to get the gradient potential in polar coordinate

text: In polar coordinate, $$\nabla U = \frac{\partial U}{\partial r}\hat{\mathbf{r}} + \frac{1}{r}\frac{\partial U}{\partial \theta}\hat{\mathbf{\theta}} .$$ Can anyone show me how to get this result?


title: Fitting of exponential data gives me a constant function

text: I conducted an experiment, where a capacitor is discharged over a (big) resistor for $t$ seconds and then the remaining charge is measured with a ballistic galvanometer. The data I have is:               t       phi     -----------     1.00    65     1.75    60     1.75    61     2.28    56     3.13    52     3.94    48     5.16    44     5.85    40     13.35   20      When I plot this data (gnuplot here) in a semilog plot, I get this: ![](http://wstaw.org/m/2012/10/18/latex-edit1.png) As far as I can tell, those points lie on a straight line, which would then be an exponential function. And that is what I would expect as well, since this should be $$Q(t) = Q_0 \exp\left(\frac{-t}{RC}\right)$$ However, when I fit it using this gnuplot snippet, I get fit parameters that make no sense to me.               f1(x) = a * exp(-x/g);     fit f1(x) "test.dat" using 1:2:3 via a,g      Yields:               Final set of parameters            Asymptotic Standard Error     =======================            ==========================          a               = 49.5555          +/- 7.647        (15.43%)     g               = -3.43204e+07     +/- 3.281e+13    (9.561e+07%)      I tried this same thing in Octave and Grace as well, they all give me this constant function. How can I get a meaningful fit for this, other than trying to fit it by hand?


title: How much of himself a person can see in the mirror?

text: > A man who is $6$ ft tall is standing in front of a plane mirror that is $2$ > ft in length. The mirror is placed lengthwise with its bottom edge $4$ ft > above the floor on a wall that is $5$ ft away(Assume that his eyes are right > at the top of his head).   > 1) How much of his image (i.e. what length of himself) can the man see?   > 2) How much of the $8$ ft-tall tree behind him, $10$ ft away from the wall, > can he see?   > 3) When the man moves to stand next to the tree, with the mirror staying in > place, how much of himself can he now see? How much of the tree can he now > see? ![enter image description here](http://i.stack.imgur.com/ItDe9.jpg) **Attempt** :1) I know about the $2:1$ rule so since the mirror is $2$ ft long he must be able to see the upper four ft of his body. Does the distance from the mirror matter though?   2) He should not be able to see the tree because he himself is covering it. I am not sure about that. Hints please.   3)The answers will be the same as in two previous questions. Again I am really not sure about that. An intuitive explanation or hint would be great. Thanks.


title: What's the average position of oscillating particles in a box with periodic boundary conditions?

text: Imagine an open box repeating itself in a way that a if a particle crossing one of the box boundary is "teleported" on the opposite boundary (typical periodic boundary position in 3D). Now put a hundred of oscillating particles inside. Each particle position is identified in a uniq fashion through time. **I want to get the average position of each of these particles.** **EDIT:** Thanks for your input. But I think I did not explain myself correctly. Please forgive me for that. That's why I removed most of my wordy description and created the following picture as a 1D reduction of my problem instead. ![1D simplification](http://i.stack.imgur.com/McuWL.png) _**Both pictures:_**   * This is a 1D projection of the 3D problem.   * The axis is infinite.   * The `blue` line represents the _1D unit cell_ repeated by translation along the axis without overlapping.   * The `red` dot represents the position of one unique and identified particle.    * The `green` dots represents the equivalent particles to the `red` one obtained by translation. These are really equivalent to the `red` one: same nature, velocity, force field, etc.   * As an input, I only have particle positions inside of the _1D unit cell_ (the `blue` line). _**Top picture:_**   * The `red` particle is present in the unit cell for each time step.   * I take the particle position $x(t)$ and average it over $t$ to get $\bar{x} = \frac{1}{4}$ of the unit cell in this case (`black arrow`). _**Bottom picture:_**   * The `red` particle is **crossing** the boundary of the _unit cell_.   * However, $x(t)$ is only defined in the _unit cell_ , therefore in this case at:      * $t=0$, the position of the `green` particle (on the right) is taken into account ;     * $t=1$, the position of the `red` particle (on the left) is taken into account     * _and so on_ ;   * If the average $\bar{x}$ is calculated without much care, it is false since this average is done between two different particles (`red arrow` instead of `black arrow`). I developed a small algorithm that tries to correct this assuming that $x(t+\Delta t) - x(t) < \frac{\Delta x}{2}$, with $\Delta x$, the size of the _unit cell_. However this algorithm is quite slow. I know I can improve the algorithm itself and my implementation of it. But I also know that this is a trivial (and inefficient) way to do it. I also hate to reinvent the wheel again and again... Do you know any smart implementation or algorithm able to calculate this kind of "periodic" average?


title: What is the physical meaning of the action in Lagrangian mechanics?

text: The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian. I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical meaning of action.


title: Definition of quasi-static assumption

text: I'm reading this paper, where atop pg 3, the author mentions the "quasi- static" assumption for earth displacements which leads to the mechanical deformation of a fluid-solid system to be governed by: $$\nabla \sigma + \rho_h g = 0$$. What exactly does "quasi-static" mean in this context?


title: Can the current in a semiconductor be independent of mobility of charge carriers?

text: Can any device be made in such a way that the current through it is independent of carrier mobility? In a usual semiconductor or a device which is made of semiconductor material the current due to the charge carriers (both thermally generated and carriers due to doping) is a function of mobility which in turn is a function of temperature, doping levels, electric field applied. My question is: Is there any device which can be made in a way that the conduction through it is independent of the carrier mobility?


title: FWHM in resonance amplitude square derivation

text: Consider a linear harmonic oscillator subject to a periodic force: $$ \ddot x + 2 \beta \dot x + \omega _0 ^2x = f_0\cos \omega t$$ The solution tends to: $$A \cos (\omega t - \delta)$$ where: $$A^2=\dfrac{f _0 ^2}{(\omega _0 ^2-\omega ^2)^2+4\beta ^2\omega ^2}$$ The maximum of this function is reached when: $$\omega ^2 = \omega_0^2-2\beta ^2$$ which is: $$A^2 _{max}=\dfrac{f_0 ^2}{4\beta ^2\omega_0 ^2-4\beta^4}\approx \dfrac{f_0 ^2}{4 \beta ^2\omega_0 ^2}$$ if $\beta$ is small compared to $\omega _0$. **I need to find theFWHM of $A^2$**. I put: $$\dfrac {A_{max}^2}{2}=\dfrac {f_0 ^2}{8 \beta ^2\omega_0 ^2}=^!\dfrac{f _0 ^2}{(\omega _0 ^2-\omega ^2)^2+4\beta ^2\omega ^2}$$ $$\iff (\omega _0 ^2-\omega ^2)^2+4\beta ^2\omega ^2=8\beta^2\omega_0^2$$ My book gives the hint "Is it ok to consider $\omega + \omega _0 \approx 2\omega _0$ or $\omega \approx \omega _0$, it is _not_ ok to consider $\omega - \omega _0 \approx 0$". With this hint it's not hard to find the solution: $$( \omega - \omega _0)^2(\omega + \omega_0)^2+4\beta^2\omega^2\approx 4\omega _0 ^2[(\omega-\omega_0)^2+\beta^2]$$ $$\therefore (\omega-\omega_0)^2\approx\beta^2\iff\omega=\omega_0 \pm\beta $$ solved. Now my question is: it's pretty clear that the approximation $\omega -\omega _0 \approx 0$ wouldn't work. It is also clear to me that it is not contradictory to assume $\omega \approx \omega _0$ , for we use this approximation for the other term $\beta ^2 \omega ^2 $. _Is there a more precise way to see this (i.e. that the approximation $\Delta \omega \approx 0 $ is not correct)??_


title: relating spinor and fundamental representation for $E_8$

text: While proving a very important relation which is satisfied both by $SO(32)$ AND $E_8$, which makes it possible to factorize the anomaly into two parts. The relation is $Tr(F^6)=\frac{1}{48}TrF^2TrF^4-\frac{1}{14400}(TrF^2)^3$, where trace is in adjoint representation. I am able to prove this relation but while doing so, I have some identities which relates the spinor representation $128$ of $SO(16)$ to fundamental representation of $SO(16)$ which I must show but this is not working out. The simplest one being $TrF^2=16trF^2$, where $Tr$ is in spinor representation $128$ and $tr$ is in fundamental representation. There are other relations showing the equality between $TrF^4$ and $tr(F^2)^2$ and $trF^4$. I am aware that spinor representation would be $\sigma_{ij}$ which is $128$ dimensional. While trying to prove these identities, I have noticed that if $F^2$ in the fundamental representation is diagonal with only two elements -1 and -1 and if $\sigma_{ij}^2=\frac{-I}{4}$ where $I$ is $128$ dimensional identity matrix then I can get the result. But I can not convince myself why it should be true. Any details would be appreciated of how to prove it. The identities can be found in GSW chapter 13 last section (VOL.2).


title: How you call the constant $\alpha$ within the heat equation in general and in terms of electromagnetism?

text: The heat equation or diffusion equation does contain a constant $\alpha$. $$\frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$$ **How is it called?** I'm interested in a general name which can be used for different circumstances apart from heat transfer and diffusion of fluids. For example in electromagnetism this constant is related to the skindepth $\delta$ via: $$\alpha = \frac{\sqrt{2j}}{\delta} = \sqrt{j\omega \kappa \mu}$$ If there is no name, what would you suggest?


title: How to define the order parameter of the q-state Potts model?

text: The order parameter of Ising model can be defined as $m=\frac{N_1-N_2}{N}$, if $N$ is the total number of lattice points, $N_1$ and $N_2$ is the number of lattice points spin up and down respectively, $N=N_1+N_2$. But I am not able to write down the order parameter of the q-state Potts model. Any help will be appreciated!


title: Torque on wire summarized with magnetic moment

text: The magnetic moment of a current-carrying wire loop $L$ is $$ \boldsymbol\mu = \frac I2\oint_L\mathbf{r} \times \mathrm{d}\mathbf{r} $$ so the torque it experiences under a uniform magnetic field $\mathbf{B}$ is \begin{equation} \boldsymbol\tau = \boldsymbol\mu \times \mathbf{B}\tag{$\ast$} \end{equation} But if I calculate from first principles, a small wire element $\mathrm{d}\mathbf{r}$ experiences a force $I(\mathrm{d}\mathbf{r} \times \mathbf{B})$. Thus \begin{align} \mathrm{d}\boldsymbol\tau &= \mathbf{r} \times I(\mathrm{d}\mathbf{r} \times \mathbf{B})\\\ \implies \boldsymbol\tau &= I\oint_L \mathbf{r} \times (\mathrm{d}\mathbf{r} \times \mathbf{B}) \end{align} which I have no idea how to reconcile with $(\ast)$. What I'm really trying to learn from this exercise is how to "summarize" a complicated integral be defining a convenient quantity such as magnetic moment. Here's my attempt at converting one formula into the other using triple product expansion: \begin{align} \frac{\boldsymbol\tau}{I} &= \oint_L \mathbf{r} \times (\mathrm{d}\mathbf{r} \times \mathbf{B})\\\ &= \oint_L (\mathbf{r} \cdot \mathbf{B})\mathrm{d}\mathbf{r} - (\mathbf{r} \cdot \mathrm{d}\mathbf{r})\mathbf{B}\\\ &= \underbrace{\oint_L (\mathbf{r} \cdot \mathbf{B})\mathrm{d}\mathbf{r}}_X - \underbrace{\mathbf{B}\oint_L\mathbf{r} \cdot \mathrm{d}\mathbf{r}}_Y \end{align} $Y$ is zero because of the curl theorem (the vector field $\mathbf{r}$ has zero curl), but I have no idea how to proceed for $X$.


title: Naive questions on the concept of effective Lagrangian and equations of motion?

text: Let us consider a LC circuit containing an electric dipole moment, the **quantum** system (electric field $E$ coupled with a dipole moment) can be described by the path integral $$Z=\int DEDxe^{i\int dtL},$$ where the total Lagrangian $$L=\frac{1}{2g}(\dot{E}^2-\omega _{LC}^2E^2)+\frac{m}{2}\dot{x}^2-\frac{m\omega _{0}^2}{2}x^2+exE.$$ After integrating out the dipole $x$, we obtain an **effective Lagrangian** $L_{eff}$ for the electric field $$L_{eff}=\frac{1}{2g}(\dot{E}^2-\omega _{LC}^2E^2)+\frac{e^2}{2m}E(\partial_t^2+\omega _{0}^2)^{-1}E.$$ On the other hand, from the **classical point of view** , by solving the total Lagrangian $L$, we can obtain a **4th** order equation of motion for the electric field $$[\partial_t^4+(\omega _{0}^2+\omega _{LC}^2)\partial_t^2+\omega _{0}^2\omega _{LC}^2-\frac{e^2g}{m}]E=0.\tag{a}$$ My questions are:   1. Can the second term in $L_{eff}$ be written as a function of $\dot{E}$ and $E$?   2. Can we derive an 'Euler-Lagrangian' equation from the **effective Lagrangian** $L_{eff}$? If yes, is this equation the same as the above **4th** order equation of motion $(a)$ of the classical system?   3. Can we construct another **effective Lagrangian** from the **classical dynamics** which gives rise to Eq.$(a)$? Is the concept of **effective Lagrangian** ONLY meaningful for the **quantum system**?


title: bulk model of the oceanic mixed layer

text: Can anyone explain where equation 7 in this paper comes from? They mention a paper by Kraus and Turner, but by looking through that paper I cannot see how I would end up at this equation.


title: Adding 3 electron spins

text: I've learned how to add two 1/2-spins, which you can do with C-G-coefficients. There are 4 states (one singlet, three triplet states). States are symmetric or antisymmetric and the quantum numbers needed are total spin and total z-component. But how do you add three 1/2-spins? It should yield 8 different eigenstates. Which quantum numbers do you need to characterise the 8 states? It is not as easy as using C-G-coefficients and the usual quantum numbers as for the total momentum the doubly degenerate 1/2 state and quadruple degenerate 3/2 state can describe only 6 or the 8 states. You will need an additional quantum number for the degeneracy. So how do you get the result? (I actually tried out myself with a large 8x8 matrix. The total spin 1/2 is each doubly degenerate. For the additional quantum number I chose the cyclic permutation. Spin 1/2 states are neither symmetric nor antisymmetric. But what is the usual way to derive this?)


title: Lenses (refractor) or mirrors (reflector) telescope?

text: What differentiates, in terms of practical quality, not technical implementation, a refractor from a reflector telescope? Why would one prefer a refractor over a reflector, when reflectors come with such large diameters at a smaller price?


title: What is the penetration length of static electric field into conducting metals?

text: How large is the penetration length for static electric field into good conductors? I have two versions: (1) few atomic spacings $$a\sim n_{e}^{-1/3},$$ and (2) Debye length computed by Fermi energy $\varepsilon_{F}$ (not the temperature) $$\lambda_{D}=\sqrt{\varepsilon_{F}/4\pi e^2 n_{e}}.$$ First variant is preferable since $$n_{e}\lambda_{D}^3<1.$$


title: Use of escort distribution in nonextensive stat. mech

text: In some of the articles which I read recently, I happen to see the following statement. > In Nonextensive statistical physics, it is inappropriate to use the original > distribution $P=(p_i)$ instead of the escort distribution $P'=(p_i')$ where > $p_i'=\frac{p_i^q}{\sum_j p_j^q}$ in calculating the expectation values of > physical quantities. Can someone clarify this point?


title: Why is Dirac Lagrangian in Curved Spacetime Weyl Invariant?

text: Are there any references on the Weyl invariance of the Dirac Lagragian in general spacetime?


title: What does "single inclusive" mean exactly?

text: I thought I knew what single inclusive scattering was, but today when I went to look up a definition to check my memory, I couldn't find one. A Google search yielded no shortage of papers that use the term, but no definition or explanation in the first several pages of results. I'm asking here so I'll have something to look up if/when I forget again: what specifically does "single inclusive" mean in particle phenomenology?


title: Schwarzschild black hole geometry in Novikov coordinates

text: As I am a layman in gravitation I would really appreciate any suggestions on how to approach and tackle the following problem: > Choose a time-symmetric 3-geometry for the initial geometry of Schwarzschild > black hole and continue the foliation of spacetime in free fall coordinates. > Show that this results in Novikov metric. I believe this is the general relativistic initial value problem within the ADM formulation. I read that the curvature of spacetime can be integrated analytically only for some highly symmetric and simple cases, otherwise approximations and numerics have to be applied. My idea is to solve standard 3+1 decomposition (foliation) of Einstein field equations (ADM equations). I would start with initial data for the Schwarzschild spacetime at the moment of time symmetry and with gauge fixing for lapse and shift $\alpha=1$ and $\beta_i=0$, which would attach coordinates to free fall observers (also called _geodesic slicing_ ). In [1, p.535] the time-symmetric 4-geometry is defined as one which has a spacelike hypersurface with extrinsic curvature 0. This is thus one constraint for the curvature tensor of initial 3-geometry: $K_{ij}=0$. [1]: Misner, C.W, Thorne, K. S., Wheeler, J.A., _Gravitation_ , 1973.


title: Primordial flunctuation gave rise to cosmic structures?

text: I'm not a physicist, not even a physics student. I'm just reading Lawrence Krauss's book _A Universe From Nothing_ and I got stuck understanding a concept. In his book, Lawrence says: > Quantum fluctuations, which otherwise would have been completely invisible, > get frozen by inflation and emerge afterward as density fluctuations that > produce everything we can see. I'm a bit confused here, didn't big bang already created all the raw materials needed to produce "everything we can see"? How can quantum fluctuation after inflation affect stars/galaxies/clusters formation?


title: Position with respect to time with friction

text: I'm interested in equations of motion when friction is present for a little graphical side project I am working on. I'm really rusty with physics, so I apologize if this is a basic question or doesn't make sense... Consider an object on a plane in three dimensions. It has initial position $ \vec{x_0} $ on the plane and initial velocity $ \vec{v_0} $ tangential to the plane. Assume it has acceleration (i.e. downhill gravity), $ \vec{a_G} $, which is constant and tangential to the plane. Now assume it has an additional acceleration (from friction, which I'm modeling as an acceleration), $\vec{a_F}(t) $, with constant magnitude $ k $ but variable direction, opposite that of velocity. Is there a function $ \vec{x}(t) $ that gives position with respect to time? Something in the vein of the $$ x(t) = x_0 + v_0*t + 0.5*a*t^2 $$ that I learned in high school?


title: Does water have surface tension in a vacuum?

text: I could be totally wrong here but I was thinking about water surface and what creates that. My thought is it is the thin mixture of water and air separating the two. This mixture creates the boundary between water and air that has the property of surface tension. So does the surface of water in a vacuum act in the same way? Is there a surface with surface tension when there is no air to make the mixture?


title: Spatial light modulation for mode multiplexing, how does it work?

text: If a laser light becomes spatial phase modulated (by phase only spatial light modulator) what happens to the intensity distribution and frequency spectrum of the laser light ? I know that this technology is used to perform mode multiplexing. But how affects the spatial phase modification the mode field distribution (e.g. transform a SMF mode to a MMF mode)? I suppose the Fourier Transformation can help here? But how exactly ?


title: How does the period of an hourglass depend on the grain size?

text: Suppose I have an hourglass that takes 1 full hour on average to drain. The grains of sand are, say, $1 \pm 0.1\ {\rm mm}$ in diameter. If I replace this with very finely-grained sand $0.1 \pm 0.01\ {\rm mm}$ in diameter but keep the hourglass otherwise the same, how long a time will the "hourglass" now measure? Does this depend on the size of the funnel, or should all one-hour hourglasses change in roughly the same way? Bonus: is the new hourglass more or less precise? (precision defined at $\sigma_t /t$, with $t$ the time to drain)


title: Why is the Coulomb taken as the unit positve test charge, though $1 C$ charge is a high value?

text: Why is the Coulomb taken as the unit positve charge, though $1 C$ charge is a high value?


title: Standard Deviation in Particle Physics

text: I'm familiar with sigma, and how its usually calculated and used, but would like to know how it's applied to particle physics. I recall reading that the discovery of the Higgs would only be credible unless it meets a 5 sigma criterion. What are the particle physics factors/numbers that would go into a sigma formula for a credible Higgs discovery?


title: Varying electric field?

text: Maybe a silly question, but anyway: We always hear about "electromagnetic induction", that is when a time varying magnetic flux induces an electric field. Does the same thing happen when there is a varying _electric_ field? Does a time varying electric field result in an induced magnetic field?


title: How would one best dissipate or absorb shockwaves and pressure caused by an explosion within a confined space

text: I am trying to find a way to limit the effect that detonation of an explosive gas, introduced into a confined space would have when detonated. Can one introduce some form of panelized material or construction that would work to minimize the effect of the initial explosion and shockwave resulting from such?


title: What happens if we place a hydraulic platform under rocket exhaust

text: Imagine I place a hydraulic platform mechanism under a rocket where its engine exhaust is placed.This hydraulic platform is fixed on the ground. During propulsion we actuate/lift platform to certain height (assume 100meters) with a pump. We can do some maths and find the initial velocity of the rocket and with same velocity we will lift the platform. Will it double the velocity of rocket? Will there be any issues apart from the platform design?


title: Interpretation of decay associated difference spetrum

text: We measured transient absorption spectra and we got $\Delta A(\lambda,t)=A(\lambda,t)-A(\lambda,0)$. If we use DAS (decay associated spectra) for interpretation, we assume that our data have this dependance: $\sum_{i}A_i(\lambda)e^{-t/\tau_i}$, where $\tau_i$ is a lifetime of a component $i$. Decay associated spectra are then $A_i(\lambda)$. If we focus on one of the $i$ spectra ($A(\lambda)$), is there any easy way how to interpretate it? I am confused with mixing decacy associated spectra together with transient absorption.


title: How to prove that identical particles are attracted or repelled in a given spin-s interaction theory?

text: Let's assume that we have integer spin interaction theory (EM field, linearized gravity, arbitrary gauge spin s theory). How to prove the consequence that in interaction theory with spin $s = 2n$ two identical particles (equal spins, masses, all charges etc) are attracted (like gravity), while in $s = 2n + 1$ theory they are repelled (like in EM interactions)?


title: Time Reversal, CPT, spin-statistics, mass gap and chirality of Euclidean fermion field theory

text: In Minkowski space even-dim (say $d+1$ D) spacetime dimension, we can write fermion-field theory as the Lagrangian: $$ \mathcal{L}=\bar{\psi} (i\not \partial-m)\psi+ \bar{\psi} \phi_1 \psi+\bar{\psi} (i \gamma^5)\phi_2 \psi $$ with Yukawa coupling to scalar term (with $\phi_1$) and pseudoscalar term (with $\phi_2$). Here $\bar{\psi}=\psi^\dagger\gamma^0$. This $\mathcal{L}$ is: (i) Hermitian (ii) invariant under Lorentz group SO($d,1$). * * * In the Euclidean fermion field theory, there are many attempts in the literature to find an analytic continuation from Minkowski to Euclidean field theory, and to satisfy the conditions of: (i) Hermitian (ii) invariant under rotational group SO($d+1$) for example, in Steven Weinberg Vol II, $$ \mathcal{L}_E= \psi^\dagger(i\not \partial_E-m)\psi $$ or this PRL paper, $$ \mathcal{L}_E= \psi^\dagger i \gamma^E_5(i\not \partial_E-m)\psi $$ in the case of this PRL paper, $\gamma^E_0=i \gamma_5$, $\gamma^E_5=-i \gamma_0$, and $\gamma^E_j=\gamma^E_j$ for $j=1,2,\dots$ The clifford algebra: ${\gamma_\mu,\gamma_\nu}=2(+,-,-,-,\dots)$, then $$ {\gamma^E_\mu,\gamma^E_\nu}=2(-,-,-,-,\dots) $$ with $x_E=(t_E,x_1,x_2,x_3,\dots)$ * * * I wonder in the Euclidean fermion field theory: **(Q1) Time Reversal and CPT:** how does the Time Reversal $T$ work in Euclidean theory, say $T_E$? Is time reversal $T_E$ the same as parity operator $P$ then? **Is $T_E$ unitary or anti-unitary?** And whether there is CPT theorem still? If partity $P$ and time reversal $T_E$ operators are (almost) the same meaning in Euclidean space, then how **CPT theorem in Euclidean field theory version** becomes? **(Q2) spin and spin-statistics:** In Minkowski signature, although spin state does not change under parity $P$, but the spin state changes under time reversal $T$ - spin flips direction, (see, for example, Peskin and Schroeder QFT p.68), say in 4D (d+1=4), $$ \xi^s \to (-i \sigma_2)(\xi^s)^* \equiv \xi^{-s} $$ Flip spin twice and measuring the spin eigenvalue by $(\vec{n} \cdot \vec{\sigma})$, $$ (\vec{n} \cdot \vec{\sigma}) \xi^s = s \xi^s \to \\\ (\vec{n} \cdot \vec{\sigma}) (-i \sigma_2 )\big((-i \sigma_2 )(\xi^s)^*\big)^* = (\vec{n} \cdot \vec{\sigma}) (\mathbf{(-1)}\xi^s) = s (\mathbf{(-1)}\xi^s) $$ will give the **same eigenvalue** of $s$, but the statefunction picks up a **(-1) sign**. This is related to spin-statistics theorem of fermions. In Euclidean signature, whether there is a defined notion of spin? Does spin state change under parity $P$ and time reversal $T_E$? Is there any kind of spin-statistics theorem in Euclidean signature? **(Q3) pseudoscalar:** In Minkowski, we have a Hermitian pseudoscalar term $\mathcal{L}_{M,pseudo}=\bar{\psi} (i \gamma^5) \psi$, with: $$ P \mathcal{L}_{M,pseudo} \psi P= -\mathcal{L}_{M,pseudo} $$ $$ T \mathcal{L}_{M,pseudo} \psi T= -\mathcal{L}_{M,pseudo} $$ $$ C \mathcal{L}_{M,pseudo} C= +\mathcal{L}_{M,pseudo} $$ In Euclidean space, is there a peudoscalar term $\mathcal{L}_{E,pseudo}$ in the version of this Euclidean theory: $\mathcal{L}_E=\psi^\dagger i \gamma^E_5(i\not \partial_E-m)\psi$, such that $\mathcal{L}_{E,pseudo}$ is Hermitian and SO(d+1) rotational invariance, and with $$ P \mathcal{L}_{E,pseudo} \psi P= -\mathcal{L}_{E,pseudo} $$ $$ T_E \mathcal{L}_{E,pseudo} \psi T_E= -\mathcal{L}_{E,pseudo} $$ $$ C \mathcal{L}_{E,pseudo} C= +\mathcal{L}_{E,pseudo} $$ what is $\mathcal{L}_{E,pseudo}=?$ in the case of **2D (d+1=2) and 4D (d+1=4)**? **(Q4) mass gap and chirality:** In Minkowski theory, the mass gap is due to the term: $$ \mathcal{L}_{M,mass}=m \bar{\psi}\psi= m (\psi_L^\dagger \psi_R+\psi_R^\dagger \psi_L) $$ where $\psi_L,\psi_R$ interpreted as massless limit of Left-, Right-handed Weyl spinors. In Euclidean theory, however, in Steven Weinberg QFT II's theory, the mass term is written as $m \psi^\dagger \psi$, does that mean $$ \mathcal{L}_{E,mass}=m \psi^\dagger \psi=m (\psi_L^\dagger \psi_L+\psi_R^\dagger \psi_R) $$ the mass gap is generated from a forward scattering term( $L\to L, R\to R$)? (This is weird!) Or does the projection $P_L, P_R$ and chirality is defined differently in Euclidean theory? How does this projection $P_L, P_R$ reflect the group structure of $$ SO(4) \simeq SU(2) \times SU(2)/Z_2 \text{ in the case of 4D} (d+1=4)? $$


title: Is the graviton hypothetical?

text: Wikipedia lists the graviton as a hypothetical particle. I wonder whether graviton is indeed hypothetical or does its existence directly follow from modern physics? Does observation of gravitational waves amount to the discovery of graviton or could there be gravitational waves not composed of gravitons?


title: Speed of light in different parts of the universe

text: Do we have enough evidence to claim that speed of light remains the same at 'c' even at the most distant parts of the visible universe?


title: Evaluation of expectation values

text: I will denote operators with hats. Suppose we got an operator of the form $i[\hat p, \tan^{-1}(e^{\hat x})]$ and we want to calculate the amplitude for a transition from a state $|p_i\rangle$ to the same state $|p_i\rangle$, as might happen in elastic scattering problems. Below are two evaluations which look like they are correct but take to two different results. Denoting $\hat f=\tan^{-1}(e^{\hat x})$, 1) \begin{equation} \begin{array}{lcl} \langle p_i|i[\hat p,\hat f]|p_i\rangle&=& i\langle p_i|(\hat p \hat f-\hat f\hat p)|p_i\rangle =i\langle p_i|(p_i \hat f-\hat fp_i)|p_i\rangle\\\ &=&i\,p_i\langle p_i|(\hat f-\hat f)|p_i\rangle=0 \end{array} \end{equation} 2) \begin{equation} \begin{array}{lcl} \langle p_i|i[\hat p, \hat f]|p_i\rangle&=&i \langle p_i|\left(-i\hbar\frac{\partial \hat f}{\partial x}\right)|p_i\rangle =\hbar\langle p_i| \frac{e^{-\hat x}}{1+e^{-2\hat x}}|p_i\rangle\\\ &=&\hbar\langle p_i| \int dx |x\rangle \langle x| \frac{e^{-\hat x}}{1+e^{-2\hat x}}|p_i\rangle =\hbar\int dx \frac{e^{- x}}{1+e^{-2 x}} \langle p_i| x\rangle \langle x |p_i\rangle\\\ &=&\hbar\int dx \frac{e^{- x}}{1+e^{- 2x}} \frac{e^{\frac{i}{\hbar}p_i x}}{\sqrt{2\pi \hbar}}\frac{e^{-\frac{i}{\hbar}p_i x}}{\sqrt{2\pi \hbar}} =\frac{1}{2\pi}\int_{-\infty}^{+\infty} dx \frac{e^{- x}}{1+e^{- 2x}}\\\ &=&\frac{1}{2\pi}\,\frac{\pi}{2}=\frac{1}{4} \end{array} \end{equation} The integral in the evaluation 2) can be found in Jeffrey, Eq. (11), Sec. 15.3.1 . I chose the form of $\hat f$ to bring the wanted results, but the question is general. Many other simpler forms can be employed to bring similar discordant results. Moreover, the results may be generalized saying that: from the first evaluation the result is always zero, from the second evaluation the result is always $\propto f\big|_{x=+\infty}-f\big|_{x=-\infty}$. What do I do wrong?


title: SUSY as the only way to unify bosons and fermions

text: Is SUSY really the only known approach to "merge"/unify bosons and fermions in a common framework? BONUS question: If SUSY does exist at high energy, it seems unnatural and "not simple" in the sense it could imply a non-minimal model (or at least a very special MSSM from the current LHC data). Are these "facts" good or bad things for both theory and phenomenology?


title: Macro-Level Explanation of Fluid Heat Transfer

text: Here's something I've been struggling with for a while. Say you have a hot fluid and cold surface (or the reverse, so long as there is a temperature difference) where the fluid flows over the surface. I know from heat transfer class that the heat transfer rate goes as the heat transfer coefficient, which _increases_ with relative velocity. Looking at the equations, this makes sense. What I'm struggling with is a qualitative understanding on the macro level. To help demonstate my issue, here's another example There are two long slabs each of a different temperature. They are drawn across each other at some velocity. The total heat transfer which occurs while the slabs are in contact _increases_ as velocity _decreases_. This is the opposite of what would happen if one slab were a fluid. Now we could change one slab to a fluid, then increase its viscosity until it behaves like a solid, so to me, my conclusion that they behave differently must be non-physical. Why? Some ascii graphics for the second example:               +---------------------+     |    slab 1: v = x    |                                          FRAME 1     +---------------------+  +-----------------------+                              |      slab 2: v = 0    |                              +-----------------------+                           +---------------------+               |    slab 1: v = x    |                                FRAME 2               +--------------+------+----------------+                              |      slab 2: v = 0    |                              +-----------------------+                            +---------------------+                       |    slab 1: v = x    |                        FRAME 3                       +------+--------------+--------+                              |      slab 2: v = 0    |                              +-----------------------+


title: Work done in pulling a rope from a boat

text: > _A boy sitting on a boat pulls on a rope with a constant force $F$ over a > duration of time $t$. The other end of the rope is either tied to a bridge > or to another freely floating boat of equal mass. Does the boy do more work > in the case of the bridge, or in the case of the other freely floating > boat?_ Here's my attempt: Work is defined as $\int F dx$. Since $F$ remains constant for both cases, we only need to analyze the difference in displacement. This is where I am confused. In the case of the bridge, only the boy is moving (the bridge is fixed). In the second case, the boy is moving towards the other boat, and the other boat is also moving towards the other boy. The relative speed of approach would seem to be greater in second case (exactly twice), and since $x = \int v dt$, the boy does more work in the second case. Is my analysis sound? Is there a better way to deduce the result?


title: Why is the second order perturbative correction to the ground state energy always down?

text: What is the physical/deeper reason for the second order shift of the ground state energy in time independent perturbation theory to be always down? I know that it follows from the formula quite straightforwardly, but I could not find a deeper reason for it even if if I took up examples like the linear Stark effect. Would you please tell me why?


title: The conduction electrons in metals is a thermal phenomenon?

text: When applying an external electric field in a metal at absolute zero, there is electrical current? There must be thermal fluctuations in the electron's band to be occurs current?


title: Flaw in the radius of a spherical electron model (classical electron radius)

text: My textbook asks me to derive an equation for the potential energy ($U$) of sphere ($r_0$) filled with an electric charge of uniform density ($\rho$), expressed in terms of the total charge $Q$. The equation can be derived by treating the potential energy ($U$) as the total work it would require to build the sphere and computing an integral [which came from the standard charge through a sphere formula (if you feel the need for a diagram or intermediate mathematical steps let me know)]: ($Work$) = $\int_0^r ((4\pi\rho)^2r^4\cfrac13)~dr$ which means that: ($Q$) = $4\pi\cfrac13r_0^3\rho$ , and from the relationship between potential energy and Work, we obtain: $$U = \frac{3Q^2}{5r_0}$$ The next question (based off of the equation we just derived) asks us to set our answer equal to $mc^2$, and attempt to calculate the proper radius of an electron (the value we receive is, I now know is the "classical radius" of the electron, or the "Lorentz radius" $\approx$($2.8 *10^{-15}\;\rm m$). Finally we arrive at our problem : It asks us to explain why this theory does not adequately analyze the proper radius of an electron, or to find out what the flaw in this theory is. I really do not know, I wouldn't have used this method in the first place. Any thoughts here? Is looking at an electron as a sphere inncorrect? I actually attempted to research this matter on my own, and could not ascertain the reason that the classical electron's radius is a poor theory, only that it does not work well in quantum mechanics.


title: What is the physical structure of light?

text: Conceptually, I understand what light is, but I don't know what a photon would "look like" if it could be frozen in space/time. For instance, the notion of amplitude seems to be absent when discussing light, even tho it is drawn as orthogonal sine waves. Is the sine wave just a way to represent the periodic change in field strength, or do the fields occupy a volume such as would be generated by rotating a sine wave about its axis? Does a light quantum have length, or is it only an instantaneous value at a point in space (and then how can it be red- or blue-shifted)? How does the magnetic field component satisfy the requirement that all field lines be closed? Does light even behave like this on a discrete level?


title: Could symmetric molecular stretching cause a physical object to become amorphous?

text: I understand very little of this, but my google-foo has yielded me next to nothing, I thought I might then just ask. My real question here is if it might be possible through the use of vibration to cause a solid object (rock) to become amorphous/polymorphous by causing the bonds between particles within a molecule to weaken. If the bonds that made up each molecule were weakened and the particles that it were made up of were somehow dissipated or transferred temporarily, it makes sense to me that the object might become polymorphic and manipulatable with ease. Possible? Yes/no would be helpful and then how/why. Thanks in advance! _Edited to ask a single more specific question_