title: Why does a perpendicular force not change speed?

text: I have been taught that a perpendicular force acting on an object will only change the direction of its velocity, not its magnitude. The explanation that was provided to me is that because there no force in the direction of its velocity, there must be no work done and thus change in the object's kinetic energy. However, I have also studied the vectorial form of the equations of motion $$\vec v_f = \vec v_i + \vec a\cdot t$$ So I assume that the velocity after a constant force has acted on it for a period of time is simply the vector sum of $\vec v_i $ and $\vec a \cdot t$, which I just assume is the acceleration vector (determined by the force) scaled up by a factor of the time the force acts.Diagrammatically, I took the sum as ![Vector Sum of initial and change in velocity](http://i.stack.imgur.com/Mo9VJ.png) It is obviously clear from the diagram that the new speed (which is the length of the $\vec v_f$ vector) is bigger than the previous speed. This can be confirmed by Pythagoras's theorem, the magnitude of the $\vec v_f$ vector is $$ |\vec v_f|=\sqrt{|\vec v_i|^2 + |\vec a|^2 \cdot t^2} $$ which is very clearly not equal to $|\vec v_i| $ So, I conclude that the speed of the object has changed even though the force acting on it was perpendicular to its velocity. So my question is, what have I missed? Is my assumption that the final velocity is the vector sum of the initial velocity and the product of acceleration and time incorrect? Or have I made a mistake somewhere else? Thank you in advance for your help.


title: What will happen if a ground state hydrogen atom is placed in a low frequency, but high intensity laser field?

text: Similar questions have been asked before, but this one extends the scope of interpretation and applications. Let us ask the question: What will happen if we have a laser gun that produces a high intensity laser beam, of the order $10^{15} Wm^{-2}$ say, and frequency about $4\times 10^{14}Hz$, and shoot a hydrogen atom in its ground state? The duration of the laser pulse is sufficiently long. Answers in terms of physical arguments will be appreciated.


title: Muon production in particle accelerator

text: PAMELA is a particle accelerator which have two concentric rings, protons are accelerated in the inside ring. At ISIS muons are produced when a 800 MeV proton beam collides with a graphite target(carbon 12) can the same be done with this device? The accelerated protons from inside may be sent to the outside ring which will have slow carbon atoms and the collision will produce pions which will decay into muons. Do pions have any magnetic properties which can help to contain them after production?.


title: The Lorenz gauge in electrodynamics

text: What is the fundamental reason to fix the Lorenz gauge to $0$?


title: Intuitively what's the relationship between forces and connections?

text: In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ where $A$ is the $4$-potential, I've heard it's possible to show that $F$ is the curvature $2$-form on a certain principal bundle. Thinking about that I asked a physicist I know about the other two fundamental forces and he said that they both can also be related with curvature $2$-forms of certain connections. In particular, he said that the structure group of the principal bundle on which we define the connection characterizes the force. I'm just starting to study connections on fiber bundles, so that I yet don't understand the implications of all of this completely but what I'm trying to get is the intuition. What is the intuition on relating forces to connections? Also, why connections on principal bundles? I know that connections can be defined on arbitrary fiber bundles (Ehresmann Connection), so why forces require principal fiber bundles?


From this article on hindustantimes.com - Curiosity sends data about Martian surface: > The resultant flash of glowing plasma is viewed by the system’s 4.3-inch > aperture telescope, which sends the light down an optical fiber to a > spectrometer located in the body of the rover. There, the colours of light > from the flash are recorded and then sent to Earth, enabling scientists to > determine the elemental composition of the vaporized material. How do NASA's Curiosity determine the elemental composition of Mars using spectrometer? Do just examining the colors of light enabled scientist to determine the elemental composition of any object?


title: Having Trouble With The Principle Of Conservation Of Momentum For a Multiparticle System

text: I'am reading John Taylor's Classical Mechanics chapter 1 page 20 where he proves the principle of conservation of momentum which states "If the net external force $F^{ext}$ on an $N$-particle system is zero,the system's total momentum $P$ is constant. I'am not sure if i'am having trouble understanding the usage of summation notation or the entire derivation (which includes the concept) itself.Anyways here it is.A five particle system labelled by $\alpha$ or $\beta = 1,2,...,5$. The particle $\alpha$ is subject to four internal forces,shown by small arrows and denoted $F_{\alpha \beta}$ and the net external force shown by the large arrow and denoted by $F^{ext}_{\alpha}$. ![enter image description here](http://i.stack.imgur.com/5bjnf.jpg) Thus the net force on particle $\alpha$ is $$F_{\alpha} = \sum_{\beta \neq \alpha}F_{\alpha \beta}+F^{ext}_{\alpha}$$ According to Newton's second law,this is the same as the rate of change of $P_{\alpha}$:$$\dot{P}_{\alpha} = \sum_{\beta\neq\alpha}F_{\alpha\beta}+F^{ext}_{\alpha}$$ Then he considers the total momentum of the $N$-particle system,$$P=\sum_{\alpha}P_{\alpha}$$ Differentiating wrt time we get $$\dot{P}=\sum_{\alpha}\dot{P}_{\alpha}$$ or,substituting for $\dot{P}_{\alpha}$,$$\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+\sum_{\alpha}F^{ext}_{\alpha}$$ .The above expression is what is bothering me .I need someone to give me an intuition behind the double summation sign. How come it's $\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+\sum_{\alpha}F^{ext}_{\alpha}$ and not just $\dot{P}=\sum_{\alpha}\sum_{\beta\neq\alpha}F_{\alpha\beta}+F^{ext}_{\alpha}$. He eventually concludes that $$\dot{P}=\sum_{\alpha}F^{ext}_{\alpha}\equiv F^{ext}$$.


title: Can all the systems have a Hamiltonian description?

text: I have heard of mechanical systems that might not have a Hamiltonian dynamics, but I cannot figure out an example that supports it. Please help.


title: Can raw fissile/fusile minerals be triggered by seismic activity?

text: Wikipedia indicates the approximate energy released by an earthquake relative to the destructive power of TNT With reasonably large quakes (> 4.0 on the Richter Scale), the energy dissipated is of the order of Giga-joules. If a seam (Uranium, Thorium, whatever) were caught in plate movement could seismic activity trigger a chain reaction? What if such an earthquake were to hit a waste depot?


title: What does 'energy dissipation' in the description of the Kolmogorov Microscales in fluid dynamics mean?

text: I've asked about microscales before, and thinking more about the problem turned up more basic questions. Consider the length scale of a microscale: $$ \eta = \left(\frac{\nu^3}{\epsilon}\right)^\frac{1}{4}$$ With $\nu$ being the viscosity, and $\epsilon$ the energy dissipation. I understand this to mean the energy transfer from turbulent kinetic flow to thermal energy (at this scale). The unit of $\epsilon$ is Watts. What's the reference volume?


title: Can plasma turn back into gas, solid, or liquid?

text: I wanted to know if, since basic chemistry teaches you that states of matter can be changed, I was wondering particularly about plasma. I know that virtually all of the Sun is plasma, so I was wondering if it's possible to de-ionize the sun, and convert it into a liquid and melt it? I realize this question is laughable, but if the Sun is made out of plasma, and it is converted into a liquid, would there be any immediate benefit in drinking it? Like, what nutritional value would there be in drinking the Sun? I realize this is going off topic to dietary/nutritional concerns, but as physicists can't we all take a step down? Now, I hope you are all aware that this question is meant to be both enjoyable and scientifically credible, so please don't jump the gun and attempt to ionize my question.


title: Green Function for Proca Equation

text: I have tried to find a retarded and advanced Green function for Proca field equation. ## $(\Box - \mu^2)A^{\mu}=J^{\mu}$ where $\mu$ is the mass term. How I did it: first: I made Fourier Transformation for $A^{\mu}$ and its inverse, I got it: ## $Ã^{\mu}(k)=\int d⁴x e^{ikx}A^{\mu}(x)$ ## $A^{\mu}(x)=\int d⁴k e^{-ikx}Ã^{\mu}(k)$ Analogously ## $\bar{j}^{\mu}(k)=\int d⁴x e^{ikx}J^{\mu}(x)$ The Klein–Gordon equation: ## $(k_0^{2} -\vec{k}^{2} - \mu^2)Ã^{\mu}(k)=\bar{j}^{\mu}(k)$ particular solution of the inhomogeneous equation above, is: ## $Ã^{\mu}= \frac{\bar{J}^{\mu}}{k_0^{2} -\vec{k}^{2} - \mu^2}$ The corresponding Green Function: ## $(\Box - \mu^2)G(x)=\delta (x)$ (may I do this?) For $\bar{G}(x)$ we have: ## $\frac{\bar{J}^{\mu}}{k_0^{2} -\vec{k}^{2} - \mu^2} \rightarrow \bar{J}^{\mu}\bar{G}(k)$ using $\omega (\vec{k}) = c \sqrt{\vec{k}^{2} + \mu^{2}}$, whe have: ## $\bar{G}(k)=\frac{c^2}{2 \omega (\vec{k})}[{\frac{1}{ck_{0} - \omega (\vec{k})\pm i0_{+}}}-{\frac{1}{ck_{0} + \omega (\vec{k})\pm i0_{+}}}] $ The choice of $+i0_{+}$ or $-i0_{+}$ gives retarded or advanced Green function: ## $G(x) = \frac{c^2}{(2\pi)^2} \int d^{4}k \frac{e^{-ikx}}{2 \omega (\vec{k})}[{\frac{1}{ck_{0} - \omega (\vec{k})\pm i0_{+}}}-{\frac{1}{ck_{0} + \omega (\vec{k})\pm i0_{+}}}]$ My questions: 1 - Is it right? 2 - There's a another (easy) way to find it?


title: Are length contraction and time dilation physical?

text: Are length contraction and time dilation real physical phenomena, or are they just a measurement issue?


title: Why is lightning considered a plasma?

text: I was wondering why a lightning bolt is coined as "plasma", or a "spark" from an electrical wire/device is as well, yet flares, molten lava, and burning buildings are not(flares are pyrotechnic, exothermic, and very much like flames). I hear arguments saying that plasma is ionized gas, but how can a flare not be ionized at such high temperature(and why isn't ionized, very hot flames considered plasma)? This guy, the top answerer, says that fire is not plasma because it's not ionized: Is fire plasma? Also, some lava exceeds 3,000 Fahreheit. How is that not plasma, and what draws the fine line between plasma and fire itself(not as a state of matter, but as a relation with gas)? Fire is gas, but some say it's plasma if you can see it (check the link the poster added below in the first comment), so when does it become a full-on plasma if it's not normally one?


title: Why do we need an old perturbation theory?

text: There are two types of perturbation theory corresponding to explicit lorentz- covariance of amplitudes. The first one is called Rayleigh-Schrodinger perturbation theory. It is based on following ideas. Let's have Hamiltonian in a form $\hat{H} = \hat{H}_{0} +\hat{V}$, where there isn't dependence on time and the second summand is proportional to the small parameter $\lambda$. Also we know solutions of equation $ \hat{H}_{0}\Psi_{n}^{0} = E_{n}^{0}\Psi_{n}^{0}$, where the upper indice zero refers to the proportionality to the zero degree of $\lambda$. Then we can build the full solutions of our Hamiltonian by the expanding solution in a form $\Psi_{n} = \sum c_{mn}\Psi_{m}$, $\hat{H}\Psi_{n}= E_{n}\Psi_{n}$, where $E_{n} = E_{n}^{0} + E_{n}^{1} +...$, $c_{mn} = c_{mn}^{0} + c_{mn}^{1} +... $. After that we can find contributions of a "small" part of the Hamiltonian to the energies and expansion coefficients. For example, $c_{mn}^{1} = \frac{V_{mn}}{E_{n}^{0} - E_{m}^{0}}$, and etc. This result may be used even for relativistic processes (like in QED). In the case when there is dependence on time in Hamiltonian we are not interested in energy of stationary state, so we want to find only the coefficients of expansion. The second type is a theory based on s-matrix formalism which give explicitly lorentz-covariant amplitudes. The first type is little harder and less convenient in compare with the second one. But there is the question: some authors like Weinberg says that we need the first type (he calls it "an old perturbation theory") because sometimes it helps to understand some singularities which are existed in the first type. Can someone make this statement clearer?


title: Effects of microwaves on optical properties of human eye

text: I have read a long series of paragraphs on wiki. Previously I thought that microwaves are not harmful to living beings but Wiki does not claim this explicitly. while doing an experiment in my college laboratory our teacher cautioned us not to see the face of Horn antenna. He said it will damage eyes so I am asking here to be sure about that. > What are the effects of microwaves emitted by Horn antenna upon the optical > properties of human eye(a natural optical device)?


title: The size of the planets

text: What calculations were done to discover the size of all the planets?


title: How does an object falling into a plain Schwarschild black hole appear from near the black hole?

text: I know that when viewed from infinity (or from a very large distance from the black hole event horizon), an object that falls into the black hole will appear to slow down and will become more and more red-shifted as it approaches the event horizon. To the far away observer the object will never be seen to enter past the event horizon since the time dilation at the event horizon approaches $\infty$ as the object approaches it. Conversely, I know that if you are on an object falling into a black hole, you will simply fall past the event horizon and fall faster and faster and will eventually reach the singularity at the center of the black hole in a finite length of time as measured by the observer on the object. However, what happens if the observer is in an orbit around the black hole at say, the last stable orbit for a material object at a distance of $3Rs$ ? **EDIT: (thanks @Ron) Being in orbit or using a rocket engine to hover near the black hole gives a result similar to the observer at infinity: the infalling object will get more and more redshifted as it approaches the horizon but will never be seen to cross the horizon.** **EDIT: So the remaining open question now is:** what happens if there are two observers that are both falling into the black hole with one observer slightly ahead of the other observer by a small distance? What does the second observer see when the first observer crosses the event horizon? How does it change when both observers have crossed the event horizon? **EDIT: (Thanks @Ron) I now understand that the second observer will only see the first object cross the horizon exactly when the second observer himself crosses the horizon. (I think of it as the photons are just sitting there at the horizon waiting for the observer to hit them.) My only remaining question is, does the redshift of the infalling object just increase smoothly and continuously with time from the observer's point of view as they both cross the horizon and as they head for the singularity? What redshift will the observer see when the first object hits the singularity itself?**


title: How strong of magnetic field would noticibly attract a person?

text: There is Iron in blood. Iron is magnetic. Roughly how strong would a magnet have to be to induce a noticeable attraction? It would be nice to know this for several distances. Also, do electromagnets that strong exist?


title: Derivative of the product of operators

text: I'm asked to show that $\frac{d(\hat{A}\hat{B})}{d\lambda} = \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$ With $\lambda$ a continuous parameter Should I use the definition $\frac{d\hat{A}}{d\lambda} = \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon) - \hat{A}(\lambda)}{\epsilon}$ applied to $\hat{A}\hat{B}$ like $\frac{d(\hat{A}\hat{B})}{d\lambda} = \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon)\hat{B}(\lambda + \epsilon) - \hat{A}(\lambda)\hat{B}(\lambda)}{\epsilon}$ and do some algebra to get the RHS of the first equation, or I'm missing something? Thanks for your time Another interesting derivative to pay attention to is: $\frac{d}{d\lambda}\exp(\hat{A}(\lambda) )$?


title: Sign of mass of an anti-particle

text: When deriving the Lagrangian for Spin $\frac{1}{2}$ particles we are naturally led to using $\Psi$ and $\bar{\Psi}$. The Euler-Lagrange equations lead us to two wave equations: \begin{equation} (i\gamma_\mu \partial^\mu - m ) \Psi =0 \end{equation} \begin{equation} (i \gamma_\mu \partial^\mu + m )\bar{\Psi} =0 \end{equation} which differ by a sign in front of the mass term. The same thing happens if we look at the electromagetic coupling of these $\frac{1}{2}$ fields. Again their coupling is different by a sign. This is interpreted as particle and anti- particle having opposite charge. Nevertheless it is unconventional to speak of the anti-particle having negative mass. Why is this the case?


title: My basis set isn't orthonormal?

text: I'm implementing a little QM calculation just for fun and to make sure I understand how it works (calculating the helium ground state energy). My problem is that my basis set doesn't seem to be orthonormal. I'm using the spherical harmonics for the angular part and the Slater type radial function. If I integrate just the angular part $$\int_{\phi = 0}^{2\pi}\int_{\theta = 0}^\pi Y_l^{m*}(\theta, \phi) Y_{l'}^{m'}(\theta, \phi) \sin(\theta)\ d\theta \ d\phi = \delta_{ll'}\delta_{mm'}$$ then the result satisfies the requirement for orthonormality. But if I include the radial Slater component: $$\int_{\phi = 0}^{2\pi} \int_{\theta = 0}^\pi \int_{r = 0}^{\infty} C r^{n-1} e^{-\alpha r} Y_l^{m*}(\theta, \phi) \cdot C r^{n'-1} e^{-\alpha r}Y_{l'}^{m'}(\theta, \phi) \cdot r^2 \sin(\theta)\ dr \ d\theta \ d\phi = \delta_{ll'} \delta_{mm'} \frac{(n + n')!}{\sqrt{(2n)!(2n')!}}$$ with $$C = \left((2\alpha)^n\sqrt{\frac{2\alpha}{(2n)!}}\right)$$ I get some extra factor instead of what should be $\delta_{nn'}$. I've double- checked my references and can't figure out what I'm doing wrong. Thanks!


title: A question about propagator of Maxwell field in different gauge

text: The propagator of Maxwell theory is different, depending on the gauge fixing procedure used. Then why will the S-matrix elements be the same for the same process in different gauges?


title: How Are Galaxies Receding Faster Than Light Visible To Observers?

text: We know that some galaxies are moving away from us faster than the speed of light and we know it by measuring the redshift, but how's that possible? If they're moving away say at 2c, how would the light of the galaxy even reach us? How do we measure "redshift" for something faster than light?


title: Would it matter if the Earth rotated clockwise?

text: In the Futurama episode _"That Darn Katz!"_ they save the world by rotating the Earth backwards saying it shouldn't matter (which direction Earth rotates). If Earth rotated clockwise and remained in it's current counter- clockwise orbit around the Sun, would it affect anything beyond the direction the constellations move across the sky?


title: Why is $|\Psi|^2$ the probability density?

text: I am starting with Quantum Mechanics, learning online. I can't seem to find the reason for $|\Psi|^2$ being the probability density of finding an electron. They've just taken it for granted everywhere. I am learning all this math but I am not able to fully grasp the intuitive idea behind all of it. If anyone could explain this with proper reasoning, I would be grateful.


title: Gravitation within galaxies

text: Do all galaxies radiate gravitational waves? What is the origin of these waves, the origin of the Galactic center? If it exists, do two galaxies warp together due to these waves, when they come closer?


title: Canonical transformation generated by hamiltonian?

text: Someone told me that, in a hamiltonian system, the hamilonian function is the generating function of the canonical transformation given by time translation. However, this statement doesn't make any sense to me. Typically, the generating function is a function of some of the "old" coordinates as well as the new (transformed) ones, but the hamiltonian is a function of **q** and **p**. Please tell me whether the above statement is rubbish, or whether it can indeed be interpreted in a meaningful way.


title: Are there any models for two light Higgs?

text: The LHC results today seem to have two possible Higgs peaks, one at 119 GeV and the other at 125-126 GeV. All the multi-Higgs supersymmetry models I've seen that have multiple Higgs have only one light Higgs, the others being well over 600 GeV. While I think it's unlikely that there really are two distinct peaks, that would certainly be the most interesting result from ATLAS and CMS. Are there any extant models with two light Higgs that are close in energy but distinct?


title: Discretizing the Wave Equation in polar coordinates

text: I want to discretize the wave equation $$\frac{1}{c^2}\frac{\partial^2\psi\left(\vec{r},t\right)}{\partial t^2}=\triangle\psi\left(\vec{r},t\right)$$ in polar coordinates. I find the following relations: $$\vec{r}=r\mathbf{e}_r$$ $$\triangle=\frac{1}{r}\partial_r r \partial_r+\frac{1}{r^2}\partial_\varphi^2$$ Define: $$r=i\Delta r,\qquad\varphi=j\Delta\varphi,\qquad t=n\Delta t$$ Thus: $$\psi\left(r,\varphi,t\right)=\psi\left(i\Delta r,j\Delta \varphi,n\Delta t\right)\equiv\psi_{i,j}^{\left(n\right)}$$ Using the forward difference in $r$ and central differences in $\varphi$ and $t$, I get: $$\psi_{i,j}^{\left(n+1\right)}=2\psi_{i,j}^{\left(n\right)}-\psi_{i,j}^{\left(n-1\right)}+\frac{c^2 \Delta t^2}{\Delta r^2}\left[\psi_{i+2,j}^{\left(n\right)}+\frac{1}{i}\psi_{i+2,j}^{\left(n\right)}-2\psi_{i+1,j}^{\left(n\right)}-\frac{1}{i}\psi_{i+1,j}^{\left(n\right)}+\psi_{i,j}^{\left(n\right)}\right]+\frac{c^2\Delta t^2}{i^2\Delta r^2\Delta \varphi^2}\left[\psi_{i,j+1}^{\left(n\right)}+\psi_{i,j-1}^{\left(n\right)}-2\psi_{i,j}^{\left(n\right)}\right]$$ Now this is divergent at $r=0$, did I something wrong? And how can I set restrictions to the steps $\Delta r$, $\Delta \varphi$ and $\Delta t$?


title: 2 particles collision - deriving the equation for an energy which can become mass of new particles!

text: **Q1:** Lets say a particle with mass $m_1$ and kinetic energy $W_k$ colides with a particle of mass $m_2$ at rest. Can anyone explain why do both particless ($m_1$ and $m_2$) plus an aditionall mass $m$ still exist after the colission? ![enter image description here](http://i.stack.imgur.com/wE0yJ.png) **Q2:** Are particless $m_1$ and $m_2$ stationary? Yes/no? **Q3:** Is the blob of new particles with mass $m$ moving as one with a momentum $\vec{p}$ which is same as momentum $\vec{p}$ before the colision? Yes/no? **Q4:** Is it possible to derive the available energy $W_{av}$ which can be transformed into mass $m$ of new particles? How do we do it? I know that it has to do something with momentum conservation, energy conservation and an invariant $W^2 = p^2c^2 + {W_0}^2$.


title: Why does the higher of two medical IV bags empties first?

text: I'm a nurse, so my physics background is about that of a lay person. Please explain on that level. Ordinarily an Intravenous (IV) fluid bag is one liter. An IV piggyback is usually 2500ml. usually the fluid in each is the same 0.45% NaCl. The piggyback may also get medication mixed in with it. such as a Gm of Rocephin. But the effect is the same without it. The main one has a small gauge tube that is spiked into the port at the bottom of the bag that continues to the patient's arm, infused into the vein with, commonly a 18 Ga catheter. That tube will have a port, there the same gauge tube can be connected to the piggy back bag. The Main bag is on a pole, about 1.5 meters above the infusion site. the piggyback is placed just higher than the main bag. The accepted explanation for why the piggyback flows and runs out before the main one starts (intentional) is gravity. To say the higher one has higher gravity. I tend to disbelieve this explanation. Does anyone know what the real reason for this effect is? assume all settings are equal and there is no pump, and is gravity fed.


title: Quantum annealing computing

text: What is Quantum Annealing and quantum annealing computing and what are its advantages and disadvantages with respect to quantum circuit quantum computing/computers?


title: Gravity as a gauge theory

text: Currently, (classical) gravity (General Relativity) is NOT a gauge theory (at least in the sense of a Yang-Mills theory). **Why should "classical" gravity be some (non-trivial or "special" or extended) gauge theory? Should quantum gravity be a gauge theory?** Remark: There are some contradictory claims in the literature to this issue. To what extent gravity is "a gauge" theory?Obviously, GR is not a YM theory. So, why do some people say that gravity "IS" a gauge theory? I found this question relevant, e.g., then we handle GR in the Einstein-Cartan theory or any other theory beyond GR, like teleparallel theories or higher-derivative gravitational theories. So I think it could be useful to discuss the "gauge flavor" of gravity here.


title: Can Gravity be described in terms of velocities and rotations of local Minkowski space?

text: In the paper _The River Model of Black Holes_ :   Am.J.Phys.76:519-532,2008, Andrew J. S. Hamilton, Jason P. Lisle   http://arxiv.org/abs/gr-qc/0411060   The authors give a way of describing the action of a rotating (and or charged) black hole through a collection of local Minkowski frames, that is, as a sort of collection of preferred reference frames, or more accurately, by the tidal effects arising from the movement from one frame to another. Each frame is defined by a "river field" $\omega_{km}$ (See around equation 74). This field is composed as follows:   $\omega_{0m} = \beta_m$ is the "velocity" of the river, while   $\mu^i = 0.5 \epsilon^{ikm}\omega_{km}$ gives the "twist" of the river. Then the motion of objects due to the black hole can be calculated from the tidal change $\delta\omega^k_m$ which is a local infinitesimal Lorentz transformation. My question is this: Can this description of a black hole be used to describe general relativity? Note that there is an obvious limitation: since this is based on a flat background metric you can't get worm holes and the like, but I mean, subject to the requirement of trivial topology, can every GR situation be described by a "river field"?


title: Is there any reason to major in physics if you're certain you will never do a Ph.D?

text: > **Possible Duplicate:**   >  What sort of jobs do physics graduates with B.Sc/M.Sc get? I've been told over and over again..."I would not major in physics unless you are looking to get a Ph.D." What is it that physics majors do with B.S./M.S. degrees? I'm currently studying engineering physics (though most of my courses lean toward engineering theory and not pure physics, really) (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.) Thanks, I'd greatly appreciate some responses


title: What is the electric field generated by a spinning magnet?

text: Consider a cylinder of permanently magnetized material, with uniform magnetization pointing along the cylindrical symmetry axis (the $z$-direction). The magnet is rotating about its cylindrical symmetry axis with angular velocity $\omega$. What electric field does the rotating magnet generate? Backstory: Moving permanent magnets generally generate an electric field, even in cases where $d\vec{M}/d t = 0$. In the case of uniform motion, this electric field is straightforward to determine using a Lorentz boost. I'm interested in cases where the simple Lorentz boost does not work. EDIT:   As perceived by some of the answers, I am not specifically interested in a cylinder. If your solution is for a ring, a sphere, or pretty much any nontrivial cylindrically symmetric object rotating about its cylindrical symmetry axis, I'm interested, as long as $d\vec{M}/d t = 0$. Landau and Lifshitz describe a similar, interesting case where the rotating magnet is also a conductor. I'm interested in the case where the rotating object is not a conductor. Unipolar induction is very interesting, but again, involves a rotating conductor, which I am not asking about.


title: Final-state baryons in $p \bar p$ collisions in Pythia

text: I'm trying to simulate inclusive $\chi_c$ production in $p \bar p$ collisions at very low energies (~ 5.5 GeV) using Pythia8 event generator. Leaving aside problems bounded with applicability of parton model at such low energies, I have encountered the problem that Pythia8 produces baryons ($p \bar p$ mostly, but also neutrons) in the final state in all events (I've tried to simulate more then 100k events). On the other hand, the total baryon number is zero, and it would seem that also $\pi$ mesons or whatever can be produced instead of baryons (at least in some cases). Since baryons stay in the final state, there is no enough phase volume for significant $p_T$ of $\chi$ meson (at least ~100 MeV) or even it can not be produced because of the nonzero momentum of the remnants. I know that, baryons should stay in final state in some cases at low energies and for that reason some $k$-factor (~0.5) is usually introduced in order to take into account suppression of the inclusive cross section. However, Pythia8 leaves baryons in all events. So, my question is: what are the physical reasons why baryons should always be present in the final state, or may be it is only feature of the Pythia model? In the last case, are there some instruments to suppress baryon production in Pythia?


title: Newton's second law?

text: $F=ma$. A car strikes a wall at 60 mph. Its acceleration is zero at the time. The force of the car against the wall or vice versa is? To look at the car the force is not zero. Please explain.


title: Local minima in Ising model in a Monte Carlo simulation

text: Is there any way to check whether in a Monte Carlo simulation using Ising model is stuck in any (false) local minima of energy or not, particularly in 3D system ?


title: Raman vs. Brillouin Scattering

text: In my class notes, I have the two types of inelastic scattering described as follows. (Note that the "difference in energy" here is the difference between the incident frequency of the light and the emitted radiation) > _Brillouin scattering_ : The difference in energy generates acoustic phonons > > _Raman scattering_ : The difference in energy generates a vibrational > oscillation From what I understood, phonons are vibrations of atoms in a molecule. So what's the difference here? Is there a better way to describe the difference?


title: What's a geometric explanation for exponential-falloff fundamental forces?

text: Gravity and electromagnetism are inverse-square laws. This makes geometric sense -- if you build a spherical shell around a lamp then a shell with twice the radius has four times the surface area and hence a quarter of the intensity per unit area. The strong interaction has a nontrivial distance-strength relation with an exponential term. Is there an intuitive geometrical explanation for this?


title: Why is the decibel scale logarithmic?

text: Could someone explain in simple terms (let's say, limited to a high school calculus vocabulary) why decibels are measured on a logarithmic scale? (This isn't homework, just good old fashioned curiousity.)


title: Non-uniform field pressure on fluid

text: I have been studying fluid mechanics and currently I was trying to understand Euler's equation for fluid flow. For that purpose, I was following this webpage. My problem is in understanding Eq.(12). The author states that that is the contribution due to an external $\underline{\text{uniform}}$ (gravitational) field, $g$. What I don't understand is why does that definition only applies to an uniform field. What would the contribution due to a, say, central field be? Can anyone help me?


title: Where does the energy for repulsion in charges come from?

text: If I bring together 2 objects with the same charge, they repel. But even after the repulsion has taken place, there is no loss of charge in the objects. Where does the energy come from then?


title: Work Done in an Isobaric Process

text: I am given the information that an air parcel undergoes isobaric heating from 0° C to 20° C, and that's all I'm given. I have to determine the work done by the parcel on its surroundings. I know that $dW = pdV$, and that $pV = RT$ ($V$ is the specific volume), but I don't know how to go about solving this in terms of the change in temperature and not the change in volume.


title: Is the number of rays projected by a source of light finite?

text: Take a source of light which gives out infinite number of rays, each ray with finite number of photons and each photon with a finite amount of energy, Then, Aren't the number of photons become infinite and hence the energy in the beam of light becomes infinite? If this is wrong, then how's this finite?


title: $\nabla$ and non-locality in simple relativistic model of quantum mechanics

text: In Wavefunction in quantum mechanics and locality, wavefunction is constrained by $H = \sqrt{m^2 - \hbar^2 \nabla^2} $, and taylor-expanding $H$ results in: $$ H = \dots = m\sqrt{1 - \hbar^2/m^2 \cdot \nabla^2} = m(1-\dots) $$ While the person who asked this question accepted the answer, I was not able to understand fully. Why would $\nabla^{200}$ in the taylor-expansion of $H$ be so problematic (200 can be replaced by any arbitrary number) - resulting in non-locality? Isn't this just some exponentiation of dot product of gradient?


title: Proof of equality of the integral and differential form of Maxwell's equation

text: Just curious, can anyone show how the integral and differential form of Maxwell's equation is equivalent? (While it is conceptually obvious, I am thinking rigorous mathematical proof may be useful in some occasions..)


title: Contraction of indices

text: We use contraction of indices method to manipulate Tensors. However, I cannot relate that manipulation visually. We can change covariant tensor to contravariant tensor and vice versa by contracting indices with the metic tensor. Covariant tensor represent gradient and contravariant tensor basically represent tangent vector. So, using contraction we are doing something awesome but how exactly can I can see this visually. What exactly does contraction do geometrically?


title: Cause of sea ice freezing during an upwell event

text: One of the Episodes of BBC's documentary Frozen Planet describes an upwell phenomenon in the Antarctic Ocean. I am trying to understand the cause of seawater freezing when this upwell occurs. Liquid sea water can remain liquid below salt water's one-atmosphere freezing temperature when that salt water is maintained under pressure at deep ocean depths. The high pressure conditions deep in the ocean apparently prevent the salt water from freezing. According to the documentary, shifting ocean currents occasionally bring a deep mass of liquid antarctic sea water from near the bottom of the ocean all the way up to the surface very quickly. And much of that mass of liquid sea water quickly freezes due to the change in pressure. Pressure is reduces when the water moves from the bottom of the Antarctic Ocean and arrives near the surface. If temperature remains constant but the state changes from liquid to solid, I assume this state change is caused by the pressure allowing the sea water to expand and form a crystalline lattice - ice. Someone watching the BBC documentary with me believes the ice formed in such an upwell event is caused by the molecular movement of the water molecules slowing down with the reduction in pressure, and she argues the slowing of movement brought on by reduction in pressure causes the ice to form. She asserts the reduced molecular movement is the cause, not the physical expansion of the overall volume of the water as it rises. My understanding is that the extreme high-pressure water simply doesn't have enough physical space to expand into a volume which will allow the molecules to form a lattice, and this has little or nothing to do with the molecular movement. What is the correct way to understand this phenomenon?


title: Did really everything begin with a state with very low entropy?

text: As emphasized by Penrose many years ago, cosmology can only make sense if the world started in a state of exceptionally low entropy. The low entropy starting point is the ultimate reason that the universe has an arrow of time, without which the second law would not make sense. However, there is no universally accepted explanation of how the universe got into such a special state. Are there some observations that would really tell us that the early universe was with small entropy? Is this claim really consistent with our theories?


title: If something has zero resistance, does it have infinite amperage?

text: If amps = volts / ohms, and ohms is 0, then what is x volts / 0 ohms?


title: Increase of threshold current by temperature in a laser diode

text: I read somewhere that if we increase the temperature of the material in a laser diode the threshold current for lasing also increases. Can someone explain to me why is this happening? What is the mechanism inside the diode behind this increase?


title: Why doesn't Lenz's law predict the behavior of a rod on springs in a magnetic field?

text: This comes from one of the free response questions from the AP Physics B 2014 exam. > ![diagram of rod suspended by springs in magnetic > field](http://i.stack.imgur.com/i3Qo2.png) > > 5\. (15 points) A conducting rod of mass $m$ and length $L$ hangs at rest > from two identical conducting springs, each with spring constant $k$, as > shown in the figure at left above. The upper ends of the springs are fixed > at points $P$ and $Q$, and the rod is in a uniform magnetic field > $\mathbf{B}$ directed into the page. A battery is then connected between > points $P$ and $Q$, as shown in the figure at right above, resulting in a > current $I$ in the rod. The rod is displaced downward, eventually reaching a > new equilibrium position with the springs stretched an additional distance > $\Delta y$. > > (a) Which point, $P$ or $Q$, is connected to the positive terminal of the > battery? Justify your answer. The answer is Q. But I put P. I said there will be a change and increase in flux. This means, by Lenz's Law, that there will be an induced magnetic field in the opposite direction. Using the hand rule, you point your thumb in the direction of the magnetic field, and CURL your fingers towards the direction of current, which makes it counter-clockwise. Thus, it can be seen that P is the right answer. Why is my answer wrong? My other classmates are telling me the answer is Q, why?


title: QED Commutation Relations Implications

text: In Brian Hatfield's book on QFT and Strings there is the following quote: > In particular $$ [A_i (x,t), E_j(y,t)] = -i \delta_{ij}\delta(x-y) $$ > implies that $$ [A_i(x,t),\nabla \cdot E(y,t)] = -i\partial _i > \delta(x-y).$$ I'm not sure how to get between those lines. If I take the partial of the fist line I get $$ [\partial_j A_i(x,t),E_j(y,t)] +[A_i(x,t),\partial_jE_j(y,t)] = -i\partial_i \delta(x-y) $$ So perhaps my question turns into: "Why is $[\partial_j A_i(x,t),E_j(y,t)] = 0$ ?" Thanks.


title: Relationship between Polarizability and Conductivity

text: I've seen in the literature the relation: $\sigma (q,\omega) = \frac{i e^2 \omega}{q^2}\chi(q,\omega)$ where $\sigma$ is the conductivity and $\chi$ the polarizability. However my attempt to derive it leaves me without the factor $q^2$. I have \begin{equation*}J_{int} = \sigma E_{tot}\end{equation*} \begin{equation*}P = \epsilon_0 \chi E_{tot}\end{equation*} \begin{equation*}\rho_{int} = -\nabla \cdot P = -iq\epsilon_0 \chi E_{tot}\end{equation*} in Fourier space. The continuity equation reads as: \begin{equation*}iqJ_{int} = i\omega \rho_{int} = iq\sigma E_{tot}\end{equation*} Relating the two through $\rho_{int}$ and cancelling $E_{tot}$ yields: \begin{equation*}\sigma = -i\omega\epsilon_0 \chi\end{equation*} Forgetting the difference between the units, where is the missing $q^2$? Is there an improper Fourier transform?


title: Is force a "real thing", or a tool for explaining changes in measurable phenomena?

text: My physics text says that force is "an interaction between two bodies or a body and its environment." When an object undergoes acceleration we explain it with a force. But we don't measure force, we measure the change in acceleration. Is force just a helpful construct to be a placeholder in the equation F = ma? Is force defined as the quantity mass times acceleration, or is it really "an interaction"?


title: Calculating the mechanical power of a water pump

text: Say I want to pump water from one container to another. The water levels are 3 meters apart, and I want to pump 10 litres per hour. I figure the mechanical power necessary, assuming no losses, is: $$ \require{cancel} \dfrac{10\cancel{l}}{\cancel{h}} \dfrac{kg}{\cancel{l}} \dfrac{\cancel{h}}{3600s} = \dfrac{0.0028kg}{s} $$ $$ \dfrac{0.0028kg}{s} \dfrac{3m}{1} \dfrac{9.8m}{s^2} = \dfrac{0.082kg\cdot m^2}{s\cdot s^2} $$ $$ \dfrac{0.082\cancel{kg}\cdot \cancel{m^2}}{\cancel{s}\cdot \cancel{s^2}} \dfrac{\cancel{J} \cdot \cancel{s^2}}{\cancel{kg} \cdot \cancel{m^2}} \dfrac{W\cdot \cancel{s}}{\cancel{J}} = 0.082W $$ But, I know from practical experience that real centrifugal pumps that can work at a 3m head are big and certainly require orders of magnitude more electrical power. What explains the difference? Intuitively, I figure this must be because the pump must exert some force to balance the force of gravity from pushing water backwards through the pump, siphoning the water back to the lower container, then exert yet more force to accomplish what was desired, pumping to the higher container.   1. How is this force calculated mathematically?   2. Assuming an ideal electric centrifugal pump, can we establish the electrical power required by the pump, given the difference in heights between the containers?   3. Does this apply to all pumps, or just centrifugal pumps?


title: mathematical explanation for UV divergences and $ \delta ^{(n)}(0) $

text: is there any mathematical explanation for the UV divergences ?? i have read that in the framework of Epstein-Glser theory :D these UV divergences appear from the product of distributions anyone does the numbers (divergences) of the form $ \delta ^{(n)}(0) $ have a meaning within the renormalization framework ?? since $ \delta ^{(n)}(0)=i^{n} \int_{-\infty}^{\infty}dx x^{n} $


title: Can a complex lens be represented by wavefront aberrations?

text: Usually complex lenses are described by their geometric design - usually a sequence of spherical cap sufraces and materials. The lens behavior, ie. how it transfers rays, can be evaluated by ray tracing. On the other hand wavefront aberrations can be computed for a particular lens. Are they enough to represent the ray transfer behavior of a lens? Is it true that wavefront aberrations describe the way a beam of collinear rays are transferred though the lens? Is the list of wavefront aberrations valid only for a single direction of collinear rays or for any direction? Thanks! PS: I'm not a physicist but a computer graphics programmer, so excuse me in case the questions seem trivial for an optician :).


title: In 't Hooft beable models, do measurements keep states classical?

text: This is a questions on 't Hooft's beable models (see here: Discreteness and Determinism in Superstrings?) for quantum mechanics, and the goal is to understand to what extent these succeed in reproducing quantum mechanics. To be precise, I will say an "'t Hooft beable model" consists of the following:   * A very large classical cellular automaton, whose states form a basis of a Hilbert space.   * A state which is imagined to be one of these basis elements.   * A unitary quantum time evolution operator which, for a series of discrete times, reproduces the cellular automaton evolution rules. 't Hooft's main argument (which is interesting and true) is that it is possible to reexpress many quantum systems in this form. The question is whether this rewriting automatically then allows you to consider the quantum system as classical. The classical probabilisitic theory of a cellular automaton necessarily consists of data which is a probability distribution $\rho$ on CA states evolving according to two separate rules:   * Time evolution: $\rho'(B') = \rho(B)$, where prime means "next time step" and B is the automaton state. You can extend this to a probabilistic diffusion process without difficulty.   * Probabilistic reduction: if a bit of information becomes available to an observer through an experiment, the CA states are reduced to those compatible with the observation. I should define probabilistic reduction—it's Bayes' rule: given an observation that we see produces a result $x$, but we don't know the exact value $x$, we know a the probability $p(x)$ that the result is $x$, the probabilistic reduction is $$ \rho'(B) = C \rho(B) p(x(B)), $$ where $x(B)$ is the value of $x$ which would be produced if the automaton state is $B$, and $C$ is a normalization constant. This process is the reason that classical probability theory is distinguished over and above any other system—one can always interpret the Bayes' reduction process as reducing ignorance of hidden variables. The bits of information that become available to a macroscopic observer internal to the CA through experiment are not microscopic CA values, but horrendously nonlocal and horrendously complex functions of gigantic chunks of the CA. Under certain circumstances, the probabilistic reduction plus the measurement process could conceivably approximately mimic quantum mechanics, I don't see a proof otherwise. But the devil is in the details. In 't Hooft models, you also have two processes:   * Time evolution: $\psi \rightarrow U \psi$.   * Measurement reduction: the measurement of an observable corresponding to some subsystem at intermediate times, which, as in standard quantum mechanics, reduces the wavefunction by a projection. The first process, time evolution, is guaranteed to keep you not superposed in the global variables, since this is just a permutation in 't Hooft's formulation, that's the whole point. But I have seen no convincing argument that the second process, learning a bit of information through quantum measurement, corresponds to learning something about the classical state and reducing the CA probabilistic state according to Bayes' rule. Since 't Hooft's models are completely precise and calculable (this is the great virtue of his formulation), this can be asked precisely: is the reduction of the wavefunction in response to learning a bit of information about the CA state through an internal observation always mathematically equivalent to a Bayes reduction of the global wavefunction? I will point out that if the answer is no, the 't Hooft models are not doing classical automata, they are doing quantum mechanics in a different basis. If the answer is yes, then the 't Hooft models could be completely rewritable as proper activities on the probability distribution $\rho$, rather than on quantum superposition states.


title: Average acceleration when more than two different velocities occur

text: > Suppose a car travels at 5m/s north for 5 seconds, it then turn east and > travel at 7m/s for 10 seconds, finally it turns north east and travel at 10 > m/s for 20 seconds. What is the average acceleration over these 35 seconds? I know acceleration average is the change in velocity over the change in time. If there was just two vectors for velocity, then I would subtract those two vectorally. However, I don't know how to find the average acceleration from 3 or more vectors. Do I just subtract these 3 vectors vectorally and divide that the result by 35 seconds?


title: Fourier integral form of the delta function?

text: I'm learning basic maths for physicist and was wondering what do we use the Dirac delta function for? What is the difference between "the Fourier integral form" and the usual way of expressing the delta function? I know how to use it from elementary quantum mechanics classes so for instance $\delta _{ij}$ is equal to one if $i=j$ and zero if not.


title: Impulse & Momentum

text: please could someone check this MIT video (http://www.youtube.com/watch?v=Lkuo6nZ6nZM) at 26mins 31secs. He says that if you threw a tomato on a bathroom scale then you would get a certain force indicated by the scales. He then says if you then use a bouncy ball instead of a tomato that the momentum change would be 2mv. He then says that the force on the scales would read double. This is where I am confused. Earlier in the video he had two balls of equal mass but one was elastic and one was inelastic. He said the momentum change of the elastic ball was 2mv but the force exerted on the floor was the same for both balls because the inelastic ball, although having half the momentum, had half the impact time giving rise to the same force. Therefore, he seems (from my point of view) to have contradicted himself with the tomato-scale situation. Wouldn't the bathroom scale read the same force because although the bouncy ball will have a momentum change of 2mv, the collision time will be twice as long as the impact time for the tomato and the scales?


title: Solving differential equations without approximations?

text: In physics, many problems start with a mathematical relationship of the physical phenomenon at hand, and then, in many occasion, always only leave whatever in the first order to get a nice and solvable differential equation. Then there may be terms of higher order considered later, by as far as I know, it rares goes to the third order or above. My question is, are there any extensive analysis about how physical problems would behave when we no longer do any approximations, by keeping _all_ orders intact?


title: pressure in fluid mechanics of incompressible liquid

text: Most liquids can be approximated to be incompressible, since the Mach-number is much smaller than 1. That means that the density variations are **negligible** and from the relation between pressure p and density ρ, $$ p=c_s^2 \rho $$ we see that the pressure in constant as well. Now, say that I look at a pipe with the following geometry: ![enter image description here](http://i.stack.imgur.com/TYaJD.gif) From Bernoulli's equation we get that the pressure and velocity will be different between the large-radius part of the pipe and the small-radius part. How does this varying pressure conform with the constant pressure/density obtained from the equation of state?


title: Soft Bremsstrahlung: why $\hat{k}\cdot\mathbf{v}= \mathbf{v}'\cdot\mathbf{v}$?

text: On page 181 in Peskin & Schroeder they say that we consider the integral (intensity) $$\tag{1}\mathcal{I}(\mathbf{v},\mathbf{v}') = \int\frac{\mathrm{d}\Omega_\hat{k}}{4\pi}\,\frac{2(1-\mathbf{v}\cdot\mathbf{v}')}{(1-\hat{k}\cdot\mathbf{v})(1-\hat{k}\cdot\mathbf{v}')}-\frac{m^2/E^2}{(1-\hat{k}\cdot\mathbf{v}')^2}-\frac{m^2/E^2}{(1-\hat{k}\cdot\mathbf{v})^2}$$ in the extreme relativistic limit (ERL). Then they say that in this limit most of the radiated energy comes from the two peaks in the first term of $(1)$. Is this because in the ERL one can take the mass $m$ to be zero: $m=0 ~(\text{ERL})$ so only the first term in $(1)$ remains? The next question is what I really want an explanation for: They claim that in (ERL) we break up the integral into a piece for each peak, let $\theta=0$ along the peak in each case. Integrate over a small region around $\theta=0$, as follows: $$\tag{2}\mathcal{I}(\mathbf{v},\mathbf{v}') \approx \int_{\hat{k}\cdot\mathbf{v}= \mathbf{v}'\cdot\mathbf{v}}^{\cos\theta=1}\mathrm{d}\cos\theta\,\frac{(1-\mathbf{v}\cdot\mathbf{v}')}{(1-v\cos\theta)(1-\mathbf{v}\cdot\mathbf{v}')} \\\\[1cm] +\int_{\hat{k}\cdot\mathbf{v}'= \mathbf{v}'\cdot\mathbf{v}}^{\cos\theta=1}\mathrm{d}\cos\theta\,\frac{(1-\mathbf{v}\cdot\mathbf{v}')}{(1-v'\cos\theta)(1-\mathbf{v}\cdot\mathbf{v}')}. $$ Then they claim that the lower limit are really not that important, but in any case: my question is where the lower limits comes from and how about the replacement in the denominator of the integrand, in other words: **How does one go from $(1)$ to $(2)$?** I should add that $\mathbf{v}, \mathbf{v}'$ are the particle velocity before and after interaction. I think one must have access to the book to understand the question unfortunately, other than that, I just want to understand where the lower limits of the integral comes from. **NOTE:** PS are working in a frame where $p^0=p^{'0}=E$ which (according to them) implies $$k^\mu=(k,\mathbf{k}),~~p^\mu=E(1,\mathbf{v}),~~p^{'\mu}=E(1,\mathbf{v'}) $$ where (I guess) $k=||\mathbf{k}||. $ Then for instance $(k_\mu p^\mu)^2$ becomes $(Ek)^2\left(1-\frac{\mathbf{k}}{k}\cdot\mathbf{v}\right)^2$ which is (I assume) one of the denominators (up so some factors) in $(1)$. So I guess the correct notation in $(1)$ should be $$\tag{3}\mathcal{I}(\mathbf{v},\mathbf{v}') =\int \dots-\frac{m^2/E^2}{\left(1-\hat{\mathbf{k}}\cdot\mathbf{v}'\right)^2}-\frac{m^2/E^2}{\left(1-\hat{\mathbf{k}}\cdot\mathbf{v}\right)^2}. $$ Overall, bad notation is used IMO on the pages near 181 in PS.


title: Why is $\vert \phi \vert ^2$ infinite in QFT?

text: I've read here¹ that for a scalar field $\phi$, the square $\vert \phi \vert ^2$ is infinite (which gives an infinite contribution to mass), more precisely: > the square of the field – a quantity which diverges in QFT as necessary > consequence of the commutation rules of the theory and unitarity. It seems that this is related to the fact that $\phi$ is a distribution and the square would be the correlation function. > if we compute in quantum field theory a correlation function like $\langle > A(x)A(y)\rangle$ and let $x\to y$, we find a divergent quantity. Is there a simple way to see why is this true, with little knowloedge of QFT?   1. Inertial Mass and Vacuum Fluctuations in Quantum Field Theory, Giovanni Modanese, PACS: 03.20.+i; 03.50.-k; 03.65.-w; 03.70.+k 95.30 Sf


title: Does physics need an epsilon-delta definition for the concept of limit?

text: This question has been inspired by a question asked on Mathoverflow about "effectiveness of (epsilon, delta) definition". Most mathematicians have a strong opinion about the need of such definitions: They are a must for mathematics as a discipline and accordingly, they are a must for whom studies mathematics. But, It happens (like the current term) that I do teach calculus to physics students where I cannot come up with a straightforward decision about the use of such definitions. The question is: To what extent are they also a "must" for a person who studies physics?


title: Normalized Projection Operator

text: What is meant by normalized projection operator? What is its physical meaning in quantum mechanics? I am pretty confused regarding the physical interpretation of both projection operator and normalized projection operator.


title: What is the interface tension between ordered and disordered phases of the Potts model?

text: I read in these papers(1,2) the concept of interface tension. I can't understand its definition. I can hardly imagine there is some tension in a model. Any help will be appreciated.


title: What is the pressure at the top of a sealed tube being drawn out of water?

text: Is the atmospheric pressure in a closed container the same as that of the surroundings (1 bar at sea level)? Consider a tube with both ends open with one end dipped into water (like a pipette in chem lab). Now if we close the other end with a thumb and draw it out of the water, the water level in the tube will be higher than the surrounding water level. If we say it is because the atmosphere that pushes up on the water in the tube is same as that of remaining air in tube pushing down on the water, won't the water fall out due to its own weight as the upward and downward pressure is balanced? I would like an explanation of the whole process which compares the weight of the water with up and down pressure by the atmosphere.


title: Circuits and Power

text: "The diagram below represents a simple circuit composed of 5 identical light bulbs and 2 flashlight batteries. Which bulb (or bulbs) would you expect to be the brightest? ![enter image description here](http://i.stack.imgur.com/QT2O4.png) a) V only b) V and W only c) V and Z only d) V, W, and Z only e) all five bulbs are the same brightness" The solutions say the answer is d, but I'm wondering if someone can explain why. I know X and Y have the same brightness because they're in parallel, but I don't understand why it's lower than the others. (I'd prefer an intuitive answer rather than one that makes reference to formulas)


title: Entropy change relation to the number of lost bits

text: Can we use entropy change value to define the (perhaps fractional) number of bits of information that are lost by and, or, xor gates?


title: Differential squared vs. differential of squared

text: Why it is said that $$\frac{dx^2}{dt^2}=\upsilon^2$$ I can only understand the following one: $$\left (\frac{dx}{dt} \right)^2=\upsilon^2$$ **Edit** : Excerpt from Landau's Mechanics: ![enter image description here](http://i.stack.imgur.com/R24Yp.png) Execrpt from Shourt Course of Theoretical Physics by the same authors (Landau and Lifshits): ![enter image description here](http://i.stack.imgur.com/sPr9X.png) So it seems to be a universally accepted notation in the old days (both books are from 70s).


title: Irreducible Representations Of Lorentz Group

text: In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as $P^{\mu} |p,\sigma\rangle = p^{\mu} |p,\sigma\rangle $, where $\sigma$ is any other label. Then he shows that, for a Lorentz Transformation : $$P^{\mu}U(\Lambda)|p,\sigma\rangle = \Lambda^{\mu}_{\rho} p^{\rho}U(\Lambda)|p,\sigma\rangle $$ Therefore: $$U(\Lambda)|p,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,p)|\Lambda p,\sigma'\rangle.$$ Then he wants to find $C$ in irreducible representations of the inhomogeneous Lorentz group. For any $m$ he chooses a $k$ such that $k^{\mu}k_{\mu} = - m^2$. Then defines express $p$'s with mass m, according to $p^{\mu} = L^{\mu}_{v}(p)k^v$. Then he defines $$|p,\sigma\rangle = N(p)U(L(p))|k,\sigma\rangle$$ (where $N(p)$ are normalization constants). I didn't understand this last statement. Is $\sigma$ an eigenvalue of the corresponding operator, or just a label? I mean, if $J |k,\sigma \rangle = \sigma |k,\sigma\rangle $ then is it true, $J |p,\sigma\rangle = \sigma |p,\sigma\rangle$. If so how can we say that if $$U(\Lambda)|k,\sigma\rangle = \sum_{\sigma'} C_{\sigma' \sigma}(\Lambda,k)|\Lambda k=p,\sigma'\rangle$$ Thanks for any help. First pages of these notes on General Relativity from Lorentz Invariance are very similar to Weinberg's book.


title: The energy spectrum in quantum field theory

text: One of the key elements of any quantum mechanical system is the spectrum of the Hamiltonian. But what about in quantum field theory? It seems as if nobody ever discusses the spectrum of a system at all -- or have I missed something? Just what role does the spectrum of any given potential play in field theory? To put it another way: does knowledge of a spectrum in QM help to understand the analogous problem in QFT?


title: How do head mounted displays simulate infinite focus?

text: First, when I say infinite focus, I am using the term from photography, perhaps incorrectly. Secondly, I believe am not talking about convergence, but about focus. If I am not mistaken, with stereoscopic displays, only the convergence is manipulated, but if the monitor is 4 feet away, your eyes remain focused at 4 feet. With head mounted displays, the display is extremely close to your eyes. It is very difficult to focus on your hand if it is only a couple inches from your face, and makes your eyes hurt a great deal. Somehow head mounted displays overcome this. I've always thought it was just through the use of a projection off of a reflective surface, just to add a few more inches to the distance from the image source to your eyes. I.e. the bulky part above your forehead projecting forward, and then reflecting off of the mirrors in front of your eyes. So I believed that head mounted displays would still involve you focusing on something at a fixed distance, that simulated focusing on something 5" or so from your eyes. Now I read about this HMD Oculus Rift, which claims that you are always focused on the "distance" which I assume is the same as infinity focus in photography. I wonder how they achieve this. I feel like I have a pretty good handle on how focusing works, and why something appears blurry when not in focus, i.e. when the light from some point that enters our pupil, but is not converging on a single point on our retina. It just boggles my mind though, how something so close to your face would simulate having to focus on the distance. I understand how it would be able to simulate this for convergence, but not for focus. The real reason I find this interesting, is thinking about multifocal displays. I.e. something that can dynamically display some regions in different depths of field: http://www.opticsinfobase.org/abstract.cfm?uri=FiO-2009-FTuM3 But baby steps: An HMD on the other hand is relatively simpler in that it deals with a single focal plane, and I'm really curious: **How is an HMD able to make it appear that the focal plane is far in the distance?**


title: Why do the formulas for planck's units use h instead of hbar?

text: On the Answers.com page on Planck length, I see two almost-same formulas for the Planck length that differ only by the use of h and hbar. However, the constants are the same, and my calculator gives the correct answer for hbar instead of h, so the first use of h was probably intended to mean hbar. Why doesn't the Oxford Dictionary (and my textbook!) use hbar instead? UPDATE: the equation (from Oxford dictionary?) I was talking about that uses h: ![planck stuff](http://i.stack.imgur.com/2PaMT.jpg) and the equation from Wikipedia that uses hbar, but gives the same constant for the Planck length: ![wiki planck stuff](http://i.stack.imgur.com/S4oVI.png)


title: Cancellation of the quadratic divergence in QCD

text: I am currently reading about QCD in QFT-Peskin&Schroeder. When calculating 1-loop diagrams for QCD and using dimensional regularization, the 3-vertex boson loop, 4-vertex boson loop and ghost loop yield the following divergent terms which contain $\Gamma(1-\frac{d}{2})$ and $\Gamma(2-\frac{d}{2})$. The coefficient for $\Gamma(1-d/2)g^{\mu\nu}q^2x(1-x)$ is $(1-\frac{d}{2})(2-d)$. At around $d=2$, this term vanishes clearly. However, for physical dimentions, i.e. $d\rightarrow4$, this is not equal to $0$. The book also says that "the pole at $d=2$ is canceled, $\Gamma(1-\frac{d}{2})$ becomes $\Gamma(2-\frac{d}{2})$". Can anyone help me understand this? Or is there a better explanation why the cancellation of the divergent quantities here happen to


title: Where do electrons in electricity come from?

text: Where do the electrons come from when an electric generator is making electricity? Is from the air? Would a generator work in a vacuum? Electrons have mass so where would they be pulled from if there is nothing?


title: Are background independence and diffeomorphism invariance the same?

text: What is the difference between background independence, which loop quantum gravity formalism claims to have, and diffeomorphism invariance?


title: Reversed freezing point

text: Helium-3's Phase diagram shows that at the right temperature and pressure combination, the solid region dips _downward_ as temperature increases. That means that you can _heat it up and it will freeze_. What is going on?


title: Do electromagnetic fields interact with air pressure?

text: I would like to know whether electric and magnetic fields interact with air pressure. Do changes in magnitude or direction of such fields increase or decrease the air pressure in a given volume of air? Does it Does it affect the air flow? For example, could it change the flow from laminar to turbulent?


title: How to define tensor contraction without referring to summation?

text: The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation convention. My question is: Can we define tensor contraction without referring to basis, components, and summation, but just the definition of tensor as a multilinear function? If components are inescapable, why bother define tensors as multilinear functions?


title: Topple a mass standing still

text: Consider the following situation: ![enter image description here](http://i.stack.imgur.com/pH3N4.jpg) This is taken from an example from my book with a solution. The box weight is $W$ and the coefficient of friction between the floor and the object is $\mu$. What is the minimal magnitude of the horizontal force $F$ so that the box would topple? The answer in the book states this: We get the 2 following: $$F_{friction}=F \\\N=W$$ Now find $x$ by moments: $$ \text{(1) } x\cdot N=x\cdot W=\frac{b}{2}\cdot W-h\cdot F$$ so that $$x=\frac{b}{2}-h\cdot \frac{h}{W}$$ and now equate $x$ to zero and find $F$: $$F_{min}=\frac{b}{2h}\cdot W .\text { Answer.}$$ I didn't understand the whole process written in $\text{(1)}$ at which most of the question is solved. **EDIT:** This $F_{min}$ is correct for most geometric bodies i.e cone, cylinder, prism and so.


title: what does "mode" means in Mode-Stirred Reverberation Chamber?

text: Recently I have been studying about properties of Mode-Stirred Reverberation Chamber (MSRC). I have run through some of the research papers about the MSRC, but none of them talks about the meaning of mode. What does the "mode" mean in the term MSRC?


title: Junction conditions in GR including electromagnetism

text: I have recently learned about the Israel junction conditions in GR (as explained in for example Gravitation by MTW). I then tried to generalize it when including Electromagnetism, ie matching two spacetimes over a junction but when there is some non-zero electromagnetic content in, but I ran into difficulties. Does anyone know where this has been studied in the literature? Basically the problem I ran into was that the Einstein equations are schematically on the form G=F^2+T Here T is a matter stress energy tensor for some delta function matter at the junction, F is the electromagnetic field tensor and G is the Einstein tensor. When deriving the Israel junction conditions we would integrate over a small slice over the junction and it turns out that the Einsten tensor has a delta function which comes from a discontinuity in the extrinsic curvature so we obtain relations between the jump of the extrinsic curvature and the stress energy tensor of the shell. However, no such delta function can appear in F^2 if we assume a discontinuity in eg the four potential. Differentiating A then gives a delta function in E and/or B but F^2 is then a delta function squared and contributes nothing. If we argue that it is really B and E that are the physical quantities of interest and they can not possess a delta function, then F^2 would contribute zero to the junction integral since there are not derivatives wrt E or B. Thus the Israel junction conditions should be unchanged by adding electromagnetism, but I dont think this is the right answer...?


title: Is there a theory which treats particles as classical point singularities?

text: Is there a published theory that looks at all matter as occupying no space and only being felt because of its gravitational pull? We've been taught in school that matter has mass and occupies space. I was just wondering if anyone tried to look at matter as having no physical reality...


title: Path of Orbital Bodies

text: I am trying to figure out how to parametrize the path of a body under the influence of gravity from another body, but I am stuck. I have looked at the Wikipedia page on Kepler orbits, but it is rather hard to follow, and doesn't seem to ever actually give the final formula for the position WRT time. (It should somewhere near the end of the "Mathematical solution of the differential equation (1) above" section, right before the "Some additional formulae" section, right?) I have a number of equations I've collected that describe the path of an orbital body, but I have been unable to figure out how to get $\overset{\rightharpoonup}{r}(t)$ directly in terms of $t$. I use$\theta(t)$ is to mean the polar angle of $\overset{\rightharpoonup}{r}(t)$, and $k$ to mean the whole "product of masses and gravitational constant" that keeps showing up everywhere ($k = m_1 \times m_2 \times g$). $$\begin{matrix} \text{Newtonian Gravity:} && \frac{\mathrm{d}^2 \overset{\rightharpoonup}{r}(t)}{\mathrm{d}t^2} = \frac{\mathrm{d} \overset{\rightharpoonup}{v}(t)}{\mathrm{d}t} = \frac{k \cdot \hat{r}(t)}{ \overset{\rightharpoonup}{r}(t){}^2} \; \\\ \text{Kepler's 1st Law (conics):} && \overset{\rightharpoonup}{r}(t)=\frac{\hat{r}(t) \cdot l}{1 + e\cos{\theta(t)}} \text{(} l \text{ and } e \text{ are scalar constants)} \\\ \text{Kepler's 2nd Law (sweep):} && \int_{\theta(t_0)}^{\theta(t_0+\Delta t)} \overset{\rightharpoonup}{r}(t){}^2 \mathrm{d}\theta(t) = c \ \text{ constant WRT } \Delta t \\\ \text{Potential Energy:} && \mathrm{d} \left \| \overset{\rightharpoonup}{v}(t) \right \| = \mathrm{d}\sqrt{\frac{k}{ \left \| \overset{\rightharpoonup}{r}(t) \right\|}} \text{(did I get this one right?)} \end{matrix}$$ I am somewhat lost here, could someone please tell me what I need to do, where to look online, etc, to find the information I need? **What is the correct parametrization for $\overset{\rightharpoonup}{r}(t)$?**


title: Hodge star operator on curvature?

text: I've a question regarding the Hodge star operator. I'm completely new to the notion of exterior derivatives and wedge products. I had to teach it to myself over the past couple of days, so I hope my question isn't trivial. I've found the following formulas on the internet, which seem to match the definitions of the two books (Carroll and Baez & Muniain) that I own. For a general $p$-form on a $n$-dimensional manifold: \begin{equation} v=\frac{1}{p!} v_{i_1 \ldots i_p} \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_p} \end{equation} the Hodge operator is defined to act on the basis of the $p$-form as follows: \begin{equation} *\left( \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_p} \right) = \frac{1}{q!} \tilde{ \varepsilon}_{j_1,\ldots,j_q}^{i_1,\ldots,i_p} \mathrm{d} x^{j_1} \wedge \cdots \wedge \mathrm{d} x^{j_q} \end{equation} where $q=n-p$ and $\tilde{ \varepsilon}$ is the Levi-Civita tensor. Up until here everything is fine, I managed to do some exercises and get the right answers. However, actually trying to calculate the curvature does cause some problems with me. To give a bit of background. I'm working with a curvature in a Yang-Mills theory in spherical coordinates $(r, \theta, \varphi)$. Using gauge transformation, I've gotten rid of time-dependence, $r$ dependence and $\theta$ dependence. Therefore, the curvature is given by: \begin{equation} F = \partial_\theta A_{ \varphi} \; \mathrm{d}\theta \wedge \mathrm{d} \varphi \end{equation} Applying the Hodge operator according to the formula above gives: \begin{equation} * \left(\mathrm{d} \theta \wedge \mathrm{d} \varphi\right) = \frac{1}{(3-2)!} \tilde \varepsilon^{\theta \varphi}_r \mathrm{d}r=\mathrm{d}r \end{equation} such that: \begin{equation} *F = (\partial_\theta A_{ \varphi}) \mathrm{d} r \end{equation} However, three different sources give a different formula. Specifically they give: \begin{equation} *F = (\partial_\theta A_{ \varphi}) \frac{1}{r^2 \sin \theta} \mathrm{d} r \end{equation} It is not clear to me where they get this from. Something is being mentioned about the fact that the natural volume form is $\sqrt{g} \; \mathrm{d} r \wedge \mathrm{d} \varphi \wedge \mathrm{d} \theta$ with $\sqrt{g}=r^2 \sin \theta$, which I agree with. However, I do not understand why that term is incorporated in the Hodge operator. Boaz and Muniain define the Hodge operator as: \begin{equation} \omega \wedge * \mu = \langle \omega , \mu \rangle \mathrm{vol} \end{equation} But I don't see how that formula is applicable to calculating the Hodge operator on the curvature. Could anybody tell me where I am going wrong or provide me a source where they explain this?


title: Fourier transform of two pulses of light

text: I have laser beam path that fires two pulses of light in a gaussian distribution, so the intensity graph over time is two identical gaussians separated by a distance $t_0$. In other words, a gaussian convolved with two $\delta$ functions equidistant from the origin. When I take the Fourier transform of this, I get a graph with beats, but I'm having trouble figuring out what this graph means physically. When we feed this beam through a spectrometer, we are trying to find out if we should see fringes, and if so, what their frequency is. Any help would be appreciated.


title: Virial theorem and variational method: a question

text: I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ (using the variational method). After having included the Darwin correction $$ H_D = D\delta(r) $$ with $D = \frac{\alpha^2\pi Z}{2}$, into the Hamiltonian $$ H_0 = -\frac12\nabla^2-\frac{Z}{r} $$ I calculated $$\langle\psi(\beta)|-\frac{\nabla^2}{2}|\psi(\beta)\rangle = \langle \psi(\beta)|T|\psi(\beta)\rangle = \frac{\beta^2}{2}$$ then I thought to use the virial theorem to calculate the part $\langle V\rangle $: $$ \langle \psi(\beta)|-\frac{Z}{r}|\psi(\beta)\rangle = \langle \psi(\beta)|V|\psi(\beta)\rangle = -2\langle \psi(\beta)|T|\psi(\beta)\rangle = -\beta^2 $$ with the Darwin part left to be easily calculated. Looking at the solutions that my professor wrote, I've just found a different result: > $$\langle V\rangle = \langle \psi(\beta)|-\frac{Z}{r}|\psi(\beta)\rangle = > \frac{Z}{\beta}\langle \psi(\beta)|-\frac{\beta}{r}|\psi(\beta)\rangle = > -2\frac{Z}{\beta}\frac{\beta^2}{2} = -\beta Z$$ Why's that? Is this somehow related to the fact that I'm going to use the variational method later? Please, help me understand.


title: Are orbits reversible in general relativity?

text: It seems if I reverse velocities then things begin orbiting backwards, at least in classical mechanics. From here: > Every orbit and trajectory outside atmospheres is in principle reversible, > i.e., in the space-time function the time is reversed. The velocities are > reversed and the accelerations are the same, including those due to rocket > bursts. Thus if a rocket burst is in the direction of the velocity, in the > reversed case it is opposite to the velocity. Of course in the case of > rocket bursts there is no full reversal of events, both ways the same > delta-v is used and the same mass ratio applies. What's up when I put relativistic effects into the mix? So for example I watch a super light test particle orbiting a black hole in a highly precessing flower shaped orbit. Then I put a bouncy wall into it's path that's at rest from my viewpoint when the particle hits it, so the particle bounces back reversing it's velocity from my viewpoint. Would it begin running its orbits backwards? The actual reason I'm asking this, because I want to know whether I can use backwards ray-tracing to render a black hole.


title: What is the near point for the eye

text: So I am a teaching assistant for an introductory physics class. One of the problems on this weeks workshops is: Where is the near point (far point) of an eye for which a contact lens with power of +2.75 (-0.83) diopter is prescribed. So I know that the power is the inverse of the focal length, so I can find f in the thin lens equation. I'm pretty sure the near point is the image of what someone with the lens would see? So would I plug in 20/25 cm in for object distance (which is roughly near point of the eye) and then solve fir image distance? And then for far point, I figure I probably just need to take object to infinity and then the image will actually be the power (in m)?


title: Is that true that neutron activity on Earth in 2013 is unusual?

text: I have just read curious statement on this website: http://www.trinitas.ru/rus/doc/0016/001d/00162241.htm (unfortunately this is in Russian, but I will try to translate) "В заключение можно отметить, что всплески потоков частиц, не связанных с солнечными вспышками, по данным нейтронных мониторов впервые стали наблюдаться в 2013г." "In conclusion we can say that splashes (on charts) that are not connected to the solar flashes, according to the neutron monitors are observed for the first time ever started in 2013." The website also links to http://www.bartol.udel.edu/gp/neutronm/ as to the source of data used for analysis. Can anyone please confirm or reject this statement? Te authors suggest that there is some sort of increased underground neutron activity, that is detected by the monitors. And that previously nothing (on that scale) like that was ever detected, in other words it was thought that most neutrons come from solar flashes and now many of neutrons come from somewhere else.


title: Do two magnets stacked on top of each other repel/attract stronger than just one magnet?

text: In designing a switch, I made it such that it "springs" back via two neodymium magnets (one in button repelling one in switch). I've found the magnets are too weak and don't spring back. I've resorted to putting a spring in the prototype for now and it works fine. I have to produce 1000s of these. It's cheaper to use springs, but I'd much prefer use magnets. I can get more magnets (double my order) to hopefully see a stronger repulsive force, but I'm not sure if magnetic fields work like this. Do two magnets together repel more than just one? Here's some text to illustrate: **S** = switch magnet, **B** = button magnet **[-S+] [+B-]** (repulsion is too weak) **[-S+-S+] [+B-+B-]** (proposed change) Does the repulsion stack? Is it a linear increase? I need to know because I can either use springs (guaranteed to work), or more magnets (what I want to do but unsure if it works).


title: Why do we expect our theories to be independent of cutoffs?

text: **Final edit** : I think I pretty much understand now (touch wood)! But there's one thing I don't get. What's the physical reason for expecting the correlation functions to be independent of the cutoff? I.e. why couldn't we just plump for one "master Lagrangian" at the Planck scale and only do our integration up to that point?   * Perhaps it has something to do with low energy experiments not being influenced by Planck scale physics.   * Maybe it's because there isn't any fundamental scale, i.e. that $\Lambda$ must be arbitrary in a QFT approximation, for some reason. I'll award the bounty to anyone who can explain this final conundrum! Cheers! $$***$$ Apologies if this question is too philosophical and vague! I've been thinking about QFTs and continuum mechanics, and reading about their interpretation as effective theories. In these theories we have natural cutoffs at high momentum (small scales). We make the assumption ($\star$) that the large scale physics is decoupled from the small-scale. Therefore we hope that our predictions are independent of the cutoff (after some renormalization if necessary). Why is the assumption ($\star$) so reasonable? I guess it seems observationally correct, which is powerful empirical evidence. But could it not be the case that the small scale physics had ramifications for larger scale observations? In other words, would it be reasonable to expect that the predictions of a TOE might depend on some (Planck scale) cutoff? This question may be completely trivial, or simply ridiculous. Sorry if so! I'm just trying to get a real feel for the landscape. **Edit** : I'd like to understand this physically from the purely QFT perspective, without resorting to analogy with statistical physics. It might help if I rephrase my question as follows. In the Wilsonian treatment of renormalization we get a flow of Lagrangians as the energy scale $\Lambda$ changes. For a renormalizable theory we assume that there's a bare Lagrangian independent of $\Lambda$ in the limit $\Lambda \to \infty$. We calculate with this quantity, by splitting it into physical terms and counterterms. I think these counterterms come from moving down the group flow, but I'm not quite sure... But **why do we care about (and calculate with) the bare Lagrangian** , rather than one at some prescribed (high) energy scale (say the Planck scale)? I don't really understand the point of there existing a $\Lambda\to \infty$ limit.


title: Power factor for impedance

text: I need to compute the power factor for two ideal impedances, a capacitor: $$Z_C=\frac{-j}{\omega C} $$ and a inductor: $$Z_L=j\omega L $$ I know that the power factor is defined as a term $\cos(\phi)$, but I don't know how to compute it in terms of $Z_C$ and $Z_L$. Moreover I need to discuss wether $\cos(\phi)$ is a function of $\omega$, the frequency of the AC circuit. Any hint or reference will be appreciated. I'm translating those terms into english so I hope that my question doesn't sound wierd or is even well defined.


title: Possibility of low-radius orbit?

text: Please bear with me, as I'm not in the field of physics, this question may seem a bit simple. This question is concerning stable circular orbits around celestial bodies. I know the equation relevant to my question is given by the equality of gravitational to centripetal force: $$G\frac{mM}{r^2}=\frac{mv^2}{r}$$ where $m$ is the mass of the orbiting body, $M$ the mass of the celestial body, $G$ the gravitational constant, $r$ the distance between the centers of mass of both bodies and $v$ the tangential velocity.   1. Assuming a celestial body without an atmosphere; When expecting the above formula, it seems that one, should be able to orbit at any   given radius, however is this true in reality?   2. Assuming a celestial body with an atmosphere; Will drag be the main reason why one cannot orbit at heights inside the atmosphere?


title: Wavefunction as a combination of two stationary states - how to find those states?

text: Lets say we have a particle in a infinite square well which has a wavefunction like this ($A$ is some constant and $d$ is the width of the well): \begin{align} A\left[ \sin \left(\frac{2 \pi x}{d}\right) + \frac{1}{3}\sin \left(\frac{3 \pi x}{d}\right)\right] \end{align} How do we know from which stationary states this wavefunction is composed of? I would say that it is a combination of 1st and 2nd **excited state** , but the anwser in the book is 1. and 3. **excited state**. How do we find this out?


title: Question about relative speeds on a different "scales of perception"

text: I was kind of puzzled yesterday when thinking about this.   1. If we observe an object moving away from us with 10m/s we would say the object is moving away from us really slowly   2. But if we (humans, making an observation) would be the size of a proton, previously mentioned 10m/s would appear enormously faster than in the first case What puzzles me is this. If we say two protons move away one of another with relative speed 0.999c, then those (second case) two-man-protons would say the relative speed between them is much greater than 0.999c. Does this make any sense?


title: Is it true that no one, not even Planck himself, knows how Planck's Constant was arrived at?

text: Planck's Constant is said to be one of the smallest constants used in Physics. Some even say it is one of the imponderables – something that we know exists and is true but cannot be explained yet. I remember reading somewhere that Max Planck somehow discovered this constant but was unable to explain how he did so. While experts may dismiss it as a concept of quantum mechanics, a branch of science that is still under intense research, a common man might see it as nothing short of a mystery.


title: Logarithmic discretization in Anderson´s model

text: Is there some motivation for the construction of Ladder operator that compound the recursive halmitonian of the Anderson model for numerical renormalization contained is this paper?


title: Calculating the magnetic field due to a rotating ring using Biot–Savart law

text: I wish to calculate the magnetic field due to a charged ring rotating about an axis perpendicular to the plane of the ring and passing through a point on its circumference $(P)$. **I arrive at something unusual**. ![Ring](http://i.stack.imgur.com/2uxU4.png) > Here's what i did > > Let the radius of the ring be $R$, let it rotate with an angular velocity > $\omega$. > > According to Biot-Savart law, the magnetic field due to a moving charge is > $\dfrac{\mu_o}{4\pi}\dfrac{dq(\vec{v} \times \hat{r})}{|\vec{r}|^2}$. Let us > consider a charge $dq$. Let $\angle POQ$ be $\theta$ and let the angle > subtended by the small element $dq$ be $d\theta$. Through some geometry, the > distance $PQ$ (let us call it $x$) is found to be $2R\sin > \left(\dfrac{\theta}{2}\right)$ > > We now see that all such elements move in circles with radius $x=2R\sin > \left(\dfrac{\theta}{2}\right)$. > > So the magnetic field due to all such elements will be pointing out of the > plane. We also note that the velocity and $x$ are always perpendicular. > > Therefore, the magnetic field due to that element at point $P$ > $$|\vec{dB}|=\dfrac{\mu_o}{4\pi}\dfrac{dq \ > v}{x^2}=\dfrac{\mu_o}{4\pi}\dfrac{\omega \ dq}{x}$$ > > Substituting $x=2R\sin \left(\dfrac{\theta}{2}\right)$ and > $dq=\dfrac{Qd\theta}{2\pi}$, > $$\color{blue}{|\vec{B}|=\dfrac{\mu_oQ\omega}{16\pi^2R}\displaystyle\int_{0}^{2\pi} > \csc \left(\dfrac{\theta}{2}\right) \ d\theta}$$ But i don't think the last integral converges. **Note** that this integral can be evaluated if the values of the integrating limits are anything other than $0$ and $2\pi$, i.e., if it is not a full ring. **Why is this happening? What does this physically mean**?


title: A common definition of a scalar

text: Some dictionaries define a scalar as follows: > A quantity, such as mass, length, or speed, that is completely specified by > its magnitude and has no direction. -- The Free Dictionary However, it is my impression that in many contexts scalars can be signed, in which case their magnitude (their absolute value) does not specify its value. This definition is even used on a test question here. Is it true that this definition is inaccurate?


title: Why does one get an illusion as moon following him?

text: When you run or ride bike at night if you observe the moon you feel like he moves along with you as the same speed you are going. Why?


title: Observer looking out through the window of a train

text: When travelling by a train, it seems that the nearby objects move in the opposite direction(which I can explain) but the distant objects appear stationary. I can't explain this. By the concept of relative velocity shouldn't all objects appear to move in opposite direction?


title: Sixth Power Law

text: The Law:   * The size of a solid material that can be carried by a (water) stream is proportional to the sixth power of its velocity. I have just discovered that I read this law long time ago (it has a name starting with N or P or something), and I was required to construct a convincing model that naturally lead to the sixth power. Can anyone give me an interesting or simple model?


title: Why Can We Observe Space Curvature / Warping At All?

text: I don't understand why we are able to see and measure curvature / warping of space at all. Space as I understand it determines distances between objects, so if space were "compressed" or warped, shouldn't distances be compressed or warped the same way (like crumpling up a sheet of paper) ? Then, however, our units of measure and frames of reference should be compressed likewise so that there should not be any visible changes to our cognition. This would also rule out warp drives unless they form some wormhole in hyper- space (analogous to two points touching each other on the crumpled sheet of paper)... What am I missing ?


title: Why is the dephasing time T2* gaussian?

text: In MRI dephasing is characterized by $T_2^*$. $T_2^*$ is measured by performing a Ramsey experiment and is defined as the 1/e decay time of the measured oscillation. I have read that when there is a slow spin bath the decay will be a Gaussian but cannot find any good sources describing why. Could someone explain me why the decay has a gaussian ($e^{-(t/\tau)^2} $)profile?


title: Regarding the possibility of Closed Timelike Curves

text: I've been looking a lot at Closed Timelike Curves, and how if a theory allows for these curves it doesn't respect causality. I understand that about the curves themselves (Grandfather Paradox), but can't seem to fathom how a theory would allow for such structures, since they seem to be "geometrically" impossible in a spacetime. To my understanding, CTC are simply worldlines that loop back on themselves, and are therefore closed. The problem comes in when I actually try to picture a CLOSED worldline: If I start at a point in Minkowski Spacespacetime and draw any closed curve, I end up _always_ having a portion of it be spacelike, and therefore the curve is never _fully_ timelike. Meaning It's impossible to draw a CTC. So my question is, how can a theory allow for such worldlines, since the fundamental principles behind the geometry of spacetime simply prohibit it?


title: How can a fruit be used as a battery cell?

text: Fresh fruits can be used as batteries for glowing bulbs, but how this is possible? I mean how can electric charges flow through fruits? Do they contain chemicals like cells?


title: How is the set of displacement operators best called?

text: Displacement operators $\hat D(x,p), \ \ x,p\in\mathbb{R},$ follow a composition rule $$D(x,p) D(x',p') = \exp\frac{i(px'-xp')}2 D(x+x',p+p').$$ Because of the extraneous phase factor, the set of all displacements is not a group. I could take a set of all displacements multiplied by all possible phase factors and I believe that would be isomorphic to $U(1) \rtimes \mathbb{R}^2$, but if there was an algebraic name for the kind of object that the set of displacements alone forms, that would suit my purpose better.


title: Total energy of a body following circular motion

text: I learned that when a body rotates, its total energy is, $$energy=\left(\frac12\right)mv^2 + \left(\frac12\right)I\omega^2 $$ However, if an astronomical object is orbiting around the earth, is there any rotational kinetic energy? or is it purely translational?


title: What does an analog voice transmission look like in the visible spectrum?

text: Analog radio signals are transmitted using light in the radio area of the spectrum. If it was transmitted using the visible spectrum instead (using a visible light emitting device instead of a radio antenna, obviously), what would it look like? Would it look like a solid light, because the vibrations would be imperceptibly fast? Or would one be able to visually tell when a voice started and stopped talking? Would you be able to see the difference between high and low pitches? Loud and quiet? Multiple tones and a single tone? Has anyone ever done this (e.g. transmitted analog audio over fiber optic or something like that)? **Edit** : another way to think of this question. You know how you can take an x-ray video recording and then create a visual-light video with it (examples)? Imagine you do the same thing with a radio-wave video camera pointed at a radio tower broadcasting analog audio.


title: Calculating electric motor's weight lifting ability knowing power. What else do i need to know?

text: So, my company works with electric motors, and my boss wanted me to calculate how much weight could be lifted by a motor rated at 750W. I know that i can't just calculate the work done by it without any further specification of characteristics, but what i what to know is a couple of formulas that could get me there using measurable values, for example, i can measure current consumed when lifting a 10kg weight. Also, I'm open to suggestions, if you think there is an easier way to get where I want to, I am open to new ideas. Thanks,


title: How can net output of photons in laser occur?

text: The way I see it, in both spontaneous and stimulated emission, one atom being de-excited would result in one photon being produced. So if we excite $10^{23}$ atoms, only $10^{23}$ photons would be produced when they de-excite. How can there be a net output of photons then? How is it that lasers can be sustained for a long time at high intensities? Also, my notes state that when a system is in equilibrium, the rate at which atoms are excited is equal to the rate at which the atoms are de-excited. What is meant by equilibrium in this context?


title: Best way to solve P-V-T Question

text: Typically in an exam or problem set we have a problem that's like this: Determine the specified property at the indicated state of water. Locate the state on a sketch of the T-v and P-v diagram. p= 3 bar, T= 240°C. Find v, in $m^3/kg$. I believe I can figure out $v,u,h$ when it ask to find this information, however I'm having trouble figuring out what's the best way to look at the steam table and figure out how to draw the correct information. For reference I've provided what the answer to this problem is. Here is the solution: ![Problem Set Answer](http://i.stack.imgur.com/XDuc3.png) Thanks,


title: What is the spectrum of energies for the potential $ a^{x} $?

text: Given a certain potential $ a^{x} $ with positive non-zero 'a' are there a discrete spectrum of energy state for the Schrodinger equation $$ \frac{- \hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}f(x)+a^{x}f(x)=E_{n}f(x)$$ Is there an example of this potential in physics? EDIT: what would happen if we put instead $ a^{|x|} $ so the potential is EVEN and tends to infinity as $ |x| \to \infty $


title: What is the likelihood of a long-period comet impact?

text: I always see lots of information about asteroid impacts, but very little is said about comet impacts. As I see, a long period comet impact is much more dangerous because a) we wouldn't have time to prepare for it, and b) even small bodies would have high kinetic energy, equivalent to large asteroids. On the flip side, I think that a long period comet would have trouble hitting the Earth, as it wouldn't have time to interact with our planet as an asteroid does. It would just hit us if, by chance, it was crossing the ecliptic plane at 1 AU from the Sun and we happened to be at the same place as well. So, my question boils down to the following:   * Do long period comets cross the ecliptic plane at a preferential distance from the sun (or: what is the chance of them crossing at approximately 1 AU)?   * What is the area of the hitting plane, versus the cross section of our planet?   * Does the assumption that a comet wouldn't have time to interact with us holds?


title: What is non-thermal plasma?

text: I read about non-thermal plasma, but I still have some questions:   1. The ions and neutral particles are not in thermal equilibrium with the electron, does that mean that the overall temperature is low like in neon signs?   2. Does the non-thermal plasma have relatively low ionization percentages, like 2 or 3%? Or higher ionizations can be achieved?    3. Is the super cold plasma the only way to get a non-thermal plasma?


title: EmDrive Cavity modes

text: The EMDrive described in patent application GB 2493361 is a simple section of a sphere surrounded by a superconducting cooling system. I have attempted to solve the oscillation modes possible and find spherical Bessel functions for radial directions and Legendre polynomials for "theta". When I apply boundary conditions, $E_r$ must be zero at the side walls so $P_n^0$(cos($\theta_w$)) = 0 can only happen for specific angles. If the wall is not at the right angle, $E_r$ must be zero every where. Same is true for $E_\phi$, but that has zeros for $P_n^1$(cos($\theta_w$)) so it would be a different mode. The claim in the patent is that putting RF into the cavity creates a force - but it does not say in what direction! The EmDrive web site talks about group velocity, but I don't see how that makes sense for a standing wave in a cavity. What am I missing? I can find $B$ from $\nabla\times E$), but is there an easy way to find energy density directly from the $E$ field solutions? Once I know energy density, finding forces should be straight forward. What I expect is the net total force to be zero, but it would be nice to prove it.


title: Why does $\nabla \to ik$ when you Fourier transform?

text: I am reading a text that describes the scattering of light by a particle with dielectric constant $\epsilon$ After a bit of maths starting from Maxwell's equations they obtain: $$\nabla (\nabla \cdot E(r)) - \nabla^2E(r)=\mu_0\omega^2\epsilon(r)\cdot E(r)$$ then say "Fourier transforming with respect to $r$ gives (replace $\nabla$ by $ik$):" $$[k^2 \hat{I}-kk]\cdot E(k) = \mu_0 \omega^2 \int \epsilon(r) \cdot E(r) \exp(-ik \cdot r) \,\mathrm{d}r$$ I don't understand why all $\nabla$ have turned into $ik$? Is there any way to visualize why this can be done?


title: How can I apply to the differential form of Gauss' Law?

text: I'm trying to learn Maxwell's equations but I got stuck. I couldn't understand the usage of the differential form of the Gauss' law. How can it be applied to questions? For example, let's say there is a plate sizes of 10 cm X 5 cm and has a charge of 1 C. To calculate the electric field on the point **_a_** , can I use ∇⋅E=ρ/ϵ0? (The distance between plate and point **_a_** is much smaller than the size of the plate)