title: Do instantons support quantum bound states?

text: When one quantizes a scalar in the 1+1 dimensions in the kink background of a double well potential, one finds a spectrum that includes: (1) a zero mode corresponding to the classical particle position, (2) a bound state localized on the kink, and (3) the continuum of states analogous to those found in the trivial background. When we quantize a non-Abelian gauge theory on an instanton background, are there analogs of the bound state (#2 above)? If so, how do they appear perturbatively in my Feynman rules? If not, how can we see that the instanton is really very different from the kink? Thanks!


title: What is the maximum power available from a magnetic field?

text: I just want to validate something I inferred from studying Griffiths (1999). The instantaneous magnetic field $\vec{b}(t)$ at a distance $r$ from a long infinite conductor carrying a current $i(t)$ is $$\vec{b}(t) = \frac{\mu}{2\pi r} \hat{\psi} \cdot i(t)$$ The energy density per unit volume is defined in terms of $\vec{b}(t)$ as $$w_V(t) = \frac{b^2(t)}{2\mu_0}$$ **Question 1:** Does it immediately follow that the power density available per unit volume is $\frac{dw_V(t)}{dt}$? **Question 2:** If I place an inductor in that time-varying magnetic field and connect it to some electrical load, I'll be able to capture that power. Assuming the surface normals $\vec{A}$ of the turns of the inductor coils are perfectly collinear with $\hat{\psi}$, what might be some energy loss mechanisms that prevent me from getting all that power, aside from electrical series resistance in the inductor? This is a self-study question, not a homework.


title: String theory - OPE and primary operators

text: First, a disclaimer: I am new to Physics SE, and I am primarily a mathematician, not a physicist. I apologise in advance for the possibly poor quality of the question, any and thank you for your patience. I am currently trying to understand some basics of String Theory, based on the script by D. Tong, available at: http://www.damtp.cam.ac.uk/user/tong/string.html . I am badly confused about the OPE and some related issues. (For definition, see the mentioned script, pages 69 onwards; I don't quite know enough to know what to mention here). I do understand that a number of "operators" $O(z)$ are supposed to be "inserted" at various points $z$ of the complex plane, and the underlying physics is somehow supposed to be encoded in the singular parts of expressions $O_1(z) O_2(w)$ with $w \simeq z$. It is not quite clear to me how it can be that the singular parts somehow seem to be the only thing that matters, but this is probably too philosophical. I would like for some explanation of the so called primary operators (page 76). Firstly, what is the intuition behind those? Is there some physical entity that they represent? The definition says that $O(z)$ is primary, if it has the OPE with the stress tensor $T(z)$ of the form: $$ T(z)O(w) ~=~ \frac{h}{(z-w)^2}O(w) + \frac{\partial O(w)}{z-w}+\ldots$$ At the same time, it says that this is just saying that the OPE terminates at the second order, so it would sound as if it is _always_ the case that if OPE terminates at the second order, the OPE has this particular form. Is this the case? In particular, it would seem that if $O_1(z)$ is primary with $h_1$ and $O_2(z)$ is primary with $h_2$, then $(O_1+O_2)(z)$ has the pole of order at most $2$, but does not have OPE of this form (or am I getting it wrong?).


title: Path integral with boundary and bulk terms

text: I was wondering if their is a general strategy for computing path integrals with a mix of boundary and bulk integral actions. Do people use divergence theorem to convert the action into bulk integrals, or is there some other trick?


title: Particles as a limit of classical field theory

text: A common academic exercise has been to show that classical mechanics is a limit of quantum mechanics, usually by putting $\hbar \rightarrow 0$. Similarly is it possible to show that a limit to field theory will lead to a particle ?   May be by considering a initial field configuration such that the "lumpiness" of the particle is there, and then as it evolves it maintains that. Clearly such an initial field configuration will continue to evolve as it is in a free field theory as two different modes don't talk to each other, but what about an interacting model ? Are there any instances in the literature which attempts to establish this connection ?


Why do some pressure cookers initially leak steam from the sides and after a while settle down and don't leak? Is it a positive feedback mechanism from the building steam which makes the sealing tighter and vice versa?


title: Relationship between balloon size and pressure

text: I see numerous websites talking about how a small balloon is at higher pressure than a big balloon. This is a fun counter-intuitive factoid... unless it isn't a fact at all. Young-Laplace gives $$T = \Delta p \frac R2 $$ Which means that for a given tension, increasing the radius does decrease the pressure. But why would the tension be constant? A latex balloon is like a 2D spring, right? So the tension in a inflated spherical balloon should be proportional to the surface area, correct? $$T \propto R^2 $$ But this gives $$\Delta p \propto R$$ Which means the bigger the balloon, the higher the internal pressure. Which, again, would be the normal intuition - but it would burst the bubble of some simplified explanations. (har har)


title: Problems with dimensions when solving an ODE

text: I'd like to solve the following differential equation: $$\frac{dQ}{dt}=\frac{k_BT}{m}-\frac{\alpha Q}{m}$$ where $Q$ has units of $\text{m}^2\text{s}^{-1}$, $k_B$ is Boltzmann's constant, $T$ is temperature, $m$ is a mass and $\alpha$ has units $\text{kg s}^{-1}$. Separating variables gives me $$\int\frac{dQ}{\frac{k_BT}{m}-\frac{\alpha Q}{m}}=\int dt$$ Hence $$-\frac{m}{\alpha}\ln\left(\frac{k_BT}{m}-\frac{\alpha Q}{m}\right)=t+C$$ Now, all of a sudden, I have something which does not hold up when I look at the dimensions: taking the logarithm of a dimensionfull quantity does not make any sense. Multiplying by $-\frac{\alpha}{m}$ and exponentiating yields $$\frac{k_BT}{m}-\frac{\alpha Q}{m}=e^{-\frac{\alpha t}{m}-C'}=C''e^{-\frac{\alpha t}{m}}$$ which is again problematic (left hand side has units $\text{m}^2\text{s}^{-2}$ while right hand side is unitless). My question is: what am I missing here? This technique of solving ODE's is widely used, so I must be overlooking something simple. However, my professor was equally stumped as I am.


title: Calculating laser wavelength/power to cause emission of light in a gas?

text: I've become interested in volumetric displays. In particular I'm looking at a pair of lasers scanning a volume of gas such that the intersection of the lasers would supply enough energy to cause the gas to emit visible light. Since this is the start of my research I would like to investigate a number of gases in order to determine a number of gases which might be suitable for this application. I have, however, forgotten how to determine the amount of energy needed to cause an outer shell electron to emit a photon, and then trying to back-calculate the required wavelength(s) and power requirements of the laser emitters. I'm looking for some information on how to identify the required information for a given gas and how to calculate the energy required to cause visible light emission, and also information on how the wavelength of laser light impacts the ability of the gas to absorb the energy with the intended effect of emitting visible light. Links to introductory texts are always helpful, as are answers which help me get the terminology correct so that I can improve my searches for this information.


title: Is there a cheap way I can do a Bell inequality experiment?

text: My work is almost completely theoretical/computational. But I'd really like to do a Bell inequality experiment just so I can personally see this (the best way to learn something is hands-on, right?) What all do I need do this? I suppose I can use two lasers and polarizing filters, but how exactly do I emit just one photon at a time? I actually feel kind of stupid asking this; I don't even know the basics of how these experiments are performed. Is there something expensive I need? I wonder if my physics department would let an engineering student borrow their resources...


title: Formalities of the variational integral

text: Usually when the variational principle is introduced one starts by defining a Lagrangian density $${\mathscr L}(x,\phi(x),\partial_{\mu}\phi(x))$$ and an action $$S[\phi]=\int_R d(x) {\mathscr L}$$ Then one tries to find the form of $\phi(x)$ that makes the value of $S$ be an extremum (either maximum or minimum). To solve this problem one applies a "small" perturbation to $x$ and $\phi$ to see how the value of $S$ is modified. Thus by "taking the limit" to smaller and smaller perturbations results in an infinitesimal change $\delta S$ which is set to zero. Now there are several ways to implement the "small" perturbation. One is to construct the homotopy between two functions that contains the solution. This means that we have a family of functions $\phi(x;\epsilon)$ parametrized continuously by $\epsilon$. However, one can argue that the homotopy cannot contain all possible variations because it only contains the "intermediate" functions of the deformation of the initial function to the final one. Another way, which is contained in most physics textbooks, is to consider the transformations: $$x^\mu\to x'^\mu=x^\mu+\delta x^\mu$$ $$\phi(x)\to \phi'(x)=\phi(x)+\delta \phi(x)$$ Where both $\delta x^\mu$ and $\delta \phi(x)$ are arbitrary small functions of $x$. Now, the lagrangian density also depends on $\partial_\mu \phi(x)$ so it is straightforward to take another small function, say $\delta \partial_\mu\phi(x)$, and consider $$\partial_\mu \phi(x) \to \partial_\mu \phi'(x) = \partial_\mu \phi(x)+ \delta\partial_\mu \phi(x)$$ However, taking the derivative of the fourth equation gives $$\partial_\mu \phi(x)\to \partial_\mu \phi'(x)=\partial_\mu \phi(x)+\partial_\mu (\delta \phi(x))$$ Then it is tempting to identify $$\delta \partial_\mu\phi(x)=\partial_\mu (\delta \phi(x))$$ But **here is the question** : how can we guarantee that the identification is correct? i.e. that $\partial_\mu (\delta \phi(x))$ is small. For example I can think of a variation of the kind $$\delta \phi(x)=\epsilon \sin\left(\frac{1}{x-a}\right)$$ where $\epsilon$ is a small number and $a$ a constant. Then obviously even when $\delta \phi(x)$ is small, $$\partial_\mu (\delta \phi(x))=-\frac{\epsilon}{(x-a)^2}\cos\left(\frac{1}{x-a}\right)$$ so we cannot guarantee that this variation is small close to $a$.


title: Earth's core electric charge

text: Forgive my limited knowledge, but the Earth's magnetic field is created by the conductive liquid in the core and to the best I can tell organizing itself (which means separating charges?). Would this mean that the core as a whole has an electric charge?


title: Energy transfer by a rotating disk at a constant RPM

text: Does Energy transfer from a rotating disk at a constant RPM into ambient fluid depend on disk material density? Why?


title: Why mostly all the astronomy things are Round in the shape (like, Sun, Moon, Earth, & other planets)?

text: Why mostly all the astronomy things are sphere in the shape (like, Sun, Moon, Earth, & other planets)?


title: Internal/Rotational angular momentum

text: I have some difficulties to understand the relation between the internal and the rotational angular momentum of a rigid body which is also known as König's theorem, so what physical intuition lies behind this equation? (I googled it but I can't find anything, so I will also appreciate any references if possible.) edit: $$H(O/Rg) = OG \times p + H^\star$$ (as vector quantities) $p$ is the moment of momentum , $G$ is the center of mass , $Rg$ is a gallilean reference of frame , $H$ is the angular momentum with respect to a point $O$ fixed in that reference , and $H^\star$ is the angular momentum in the **CM** reference .


title: Representation of the Galileo Group and Central Charges

text: I've arrived at this question because I've been reading Weinberg's Quantum Theory of Fields Volume I, and I'm in the second chapter about relativistic quantum mechanics. Weinberg discusses the representations of the Poincare group, and after deriving the structure constants he gets the structure constants for the Galilean group as well. I have a few questions about this: He introduces these operators $M$ and $W$--where do they come from? (This question may make sense only to those reading Weinberg, I'm not sure, so it's less important). How can we conclude that in the Galilean Algebra \begin{equation} [K_i, P_j] = -iM\delta_{ij} \end{equation} Shouldn't boosts and translations commute? Why is this equal to $M$ instead of total energy? Is it because $M$ is a "central charge"? **What exactly is a central charge?** Does it have to do with the fact that $M$ is in the Lie Algebra, it is a generator of a symmetry transformation, and because it is in the center and commutes, its charge is conserved. But how do we decide what its conserved charge is? What is to prevent us from adding more of these central charges? Also, look at the product of a translation and a boost (K is the generator of boosts): \begin{equation} e^{-i\vec{K}\cdot \vec{v}}e^{-i\vec{P}\cdot \vec{a}}=e^{iM\vec{a}\cdot \vec{v}/2}e^{-i(\vec{K}\cdot \vec{v} + \vec{P}\cdot \vec{a})} \end{equation} Weinberg says that this phase factor (with the M) shows we have a projective representation, "with a superselection rule forbidding the superposition of states of different mass". Where did this phase factor come from? **What are these superselection rules about?**


title: Does Anything Exixts?

text: I Mean that Does Anything In The Universe Exists By Means Of Quantum Mechanics...I Heard This From Someone But i wanna be sure that it is really true or what?


title: Nonextensive statistical mechanics

text: I know that the Tsallis($S_q$) entropy is called nonextensive information measure in the sense that if $P$ and $Q$ are two probability distributions then $S_q(P\times Q)=S_q(P)+S_q(Q)+(1-q)S_q(P)S_q(Q)$. My question is what is meant by nonextensive statistical physics? What is its connection with Tsallis entropy maximisation?


title: Quantum circuit, two control not gates

text: Consider the quantum circuit in the picture below: ![Figure 1](http://i.stack.imgur.com/s80aW.jpg) We have a Hadamard gate followed by a CNOT gate, this puts the 2nd & 3rd state in the bell state $\beta_{00}=\frac{1}{\sqrt2}(\mid 00\rangle +\mid 11\rangle$ This occurs because the CNOT gate flips the target qubit iff the control qubit is $\mid 1 \rangle$. The two qubits are now entangled. When we proceed to the second CNOT gate we only care about what state the second wire is in? Would someone be able to help explain how we can compute how the top wire acts in this situation? Or if someone would be able to explain in general how we treat qubits entangled such as this if we want to say something about them separately? At the end of the circuit we have two detectors $x$ and $y$, these will collapse wave function, two two lower qubits will carry on unchanged in their bell state, how would we describe the middle wire being measured by $x$?


title: Invariance and forms of the Lagrangian

text: I have been doing Landau and due to its concise style been facing a few problems. I hope you can help me out here somehow. 1)Does the "homogeneity of space and time" essentially talk about the invariance of the Lagrangian with co-ordinate system or time invariance? Is this the basis of Landau's proof of the Law of Inertia? 2) The Lagrangian for a closed system of particles is described by adding a function of the co-ordinates. This essentially means that the same trajectory satisfies the extremal principle.(I am a bit confused about this though). So why the function added is not of time too? 3) On the subject of " the Lagrangian being defined only to within an additive total time derivative of any function of co-ordinates and time", is it true that the function can be of velocity too ( Though this would make the entire exercise useless, since we would know everything)? The disadvantage of trying to teach oneself physics without anyone to refer to is that I am not completely sure that what I am saying makes sense.


title: What is the activity of a planet in a double star system?

text: If a planet is directly between two stars _y_ distance apart (so it's _y/2_ from each planet), with velocity _x_ such that _x_ is sufficient to continue orbiting at that distance ( _y/2_ ), which does it orbit, and how does it continue?


title: What is a chiral field?

text: I have not found a clear definition of this. A teacher told me that it was a field having some constrains but that is not very convincing for me. He told me also that some examples could be skyrme model, sigma model and sine-gordon model. Is this right? Could you give me a complete definition of chiral fields and some examples? Please I also would very grateful if you could refer me some book or paper about this topic.


title: Superconducting gap, temperature dependence: how to calculate this integral?

text: Tinkham (page 63) states that the temperature dependence of the gap energy of a superconductor $\Delta(T)$ can be calculated using the following integral: ![enter image description here](http://i.stack.imgur.com/d3lip.png) How can this actually be carried out? I am not sure how to approach this problem or re-arrange the equation for finding $\Delta(T)$ numerically. A URL containing a little more inforamtion about this Eqn: http://katzgraber.org/teaching/ss07/files/burgener.pdf (slide 35)


title: Polarization Measurement

text: Assume having a laser beam which is horizontal linear polarized. As one measure the sqrt(intensity) transmitted through a rotatable linear polarizer its pattern corresponds to a cosine. Plotting this in a polar coordinate system results in the so called "polarization ellipse". But the result is far away from being a (more or less) horizontal line. But what does the polarization ellipse tell me? My problem is that from the polar plot one may interpret that the original beam has E-Field components also in non-horizontal direction. But this is (ideally) not true. But only the polarizator has (excepted for the vertical position) components in horizontal direction leading to a transmitted field.


title: Clocks in special relativity

text: One book on special relativity says: > _Any observer at rest relative to his own timepiece will see that other > clocks moving with respect to him run fast - the greater their speed, the > faster they are._ Other book says: > _Observers measure any clock to run slow if it moves relative to them._ Don't they contradict each other? If yes - who's right? If no - why are they both right? I assume it is a very newbie question, but relativity is one of the topics where you recheck every statement again and again, so I want to be sure.


title: What is geostationary orbit radius?

text: I'm asking this apparently "general reference" question for the simple reason: I was unable to find whether the quoted everywhere "35,786 kilometers (22,236 mi) above the Earth's equator" means "radius" or "altitude above equator." (yes, for it to be geostationary it must be located above the equator, but I'm really not sure if the number includes Earth radius or not). With Earth radius at Equator equal 6,378 km that's a considerable difference. So, is the orbital radius 35,786km, and altitude 29,390 km or is the altitude 35,768 and radius 42,164 km?


title: Does quantum manybody theory with 2nd quantization completely equivalent to 1st quantization formulation?

text: by equivalent I mean:   1. the Hilbert spaces are isomorphic;   2. the operators can therefore be mapped 1:1


title: Topological phase transitions - breaking of continuous translational invariance

text: I'm relatively new to the theoretical side of physics. I have a question about topology, continuous symmetry breaking and phase transitions. Your help is much appreciated! Ok so I have an infinite array of metallic nanoparticles with a two atom basis, which can equivalently be thought of as two equivalent but independent sub-lattices (think of graphene). Anyway, I start with a configuration such that the lattice is a honeycomb structure (with three nearest neighbours), I then deform the lattice by scaling the position vector of the second basis particle, which can be seen in the figure as 0.9, 0.858, 0.856 and 0.8 times the original second basis particle vector. I have generated plots which show if the resulting dispersion of the plasmons is gapless (cream) or gapped (orange) for arbitrary dipole orientation (am modelling the LSPs as dipoles). Ok now I am going to say a paragraph and I would like you to tell me if I'm talking rubbish or not... "At some value between $\vec{d}_3 = 0.9 \vec{e}_3$ and $\vec{d}_3 = 0.858 \vec{e}_3$ there is a topological change which can be seen in the breaking of the continuity of gapless states around the points $(\frac{\pi}{2},\frac{5\pi}{12})$ and $(\frac{\pi}{2},\frac{8\pi}{12})$. There is another topological change that occurs at a value between $\vec{d}_3 = 0.858 \vec{e}_3$ and $\vec{d}_3 = 0.856 \vec{e}_3$ where initially there is a continuum of gapless states about the lines $\theta = 0$ and $\theta = \pi$, then subsequently a continuum of gapped states about the same lines. At these values of $\vec{d}_3$ where a continuous translational symmetry has been broken, it is reasonable to assume this is accompanied by some form of phase transition." ![enter image description here](http://i.stack.imgur.com/tbKc4.png)


title: Relativity Problem Given Rest Length and Proper Time

text: > A relativistic train with a rest length of 500.0 m takes 780 ns to pass a > stationary observer standing on the train platform, as measured by the > stationary observer. > > (a) What is the speed of the train? (Hint: Remember to account for the > Lorentz contraction of the spaceship.) I've been at it for a couple hours and I just don't see how I can use the Lorentz formulas. If S is the inertial frame with respect to the observer and S' is the inertial frame with respect to the train, then the front of the train and an observer line up at t = t' = 0 and x = x' = 0, Then, in the S' frame, the front of the train sees itself at $x' = 500 m$ when $t = 780 \times 10^{-9} seconds$ The correct answer is: $$2.72 \times 10^8 \frac{m}{s} = 0.907c$$ But I don't see where to head from where I set up. Any help is much appreciated at this point.


title: What's the second terminal in corona discharge?

text: I don't get corona discharge. There's an electrode (say naked high-voltage wire) and a current develops from that wire and ionizes gas (air) on its way. But where does this current go? What's the destination of that discharge?


title: What is the minisuperspace Lagrangian for gravity plus a scalar field?

text: In this paper by Sean Carroll and Grant Remmen, in equation (11) they write a Lagrangian of the form $$\boxed{\mathcal{L}=3a\left(k-\dot{a}^2\right)+a^3\left[\frac{1}{2}\dot\phi^2-V(\phi)\right]}$$ for gravity plus a scalar field with potential $V(\phi)$, here given $N=1$ and $M_{\text{Pl}}=1$. So how is this Lagrangian obtained? I'm trying $$\mathcal{L}=\mathcal{L}_{\text{gravity}}+\mathcal{L}_\text{homogeneous SF}$$ with $\mathcal{L}=\frac{\widehat{\mathcal{L}}}{\sqrt{|g|}}$, so that it would be (with -+++ signature) $$\mathcal{L}=\frac{1}{2}R-\frac{1}{2}g^{00}\dot\phi-V(\phi)$$ where I've taken $\partial_i\phi=0$ for the scalar field $\phi$ is homogeneous. Now the Ricci scalar for FRW is $$R=\frac{6}{a^2}(k+\dot{a}^2+a\ddot{a})$$ so that $$\mathcal{L}=\frac{3}{a^2}(k+\dot{a}^2+a\ddot{a})+\frac{1}{2}\dot\phi^2-V(\phi)$$ and using the acceleration equation for the scale factor, $$\ddot{a}=-\frac{a}{6}(\rho+3p)=\frac{a}{3}\left[V(\phi)-\dot\phi^2\right]$$ where I used $\rho=\frac{1}{2}\dot\phi^2+V(\phi)$ and $p=\frac{1}{2}\dot\phi^2-V(\phi)$. Thus $$\boxed{\mathcal{L}=\frac{3}{a^2}\left(k+\dot{a}^2\right)-\frac{1}{2}\dot\phi^2}$$ and it doesn't seem to fit, even if I multiply by an $a^3$ factor, which seems to be done to get the expression. What is going on here?


title: Creation of matter in the big bang

text: I appreciate your patience to my neophyte question. I am working on my dissertation in philosophy (which has nothing or little to do with physics) about the "problem of naming." Briefly what I am arguing is that when we name something, we stop it from being anything or everything else. It is a phenomenological question and has a lot to do with language as an object. My question for you is that, is it true that all matter was somehow formed in the Big Bang or in those famous three minutes following? I think I understand that helium and hydrogen were formed and are they then to be considered the basis of all matter today? A friend said to me a long time ago that we are made of the same atoms that were present at the Big Bang; could this possibly be true? (And how wonderful if it is...) Thanks in advance.


title: Poynting vector and Rindler flux under time inversion

text: This question is about some reply by John Baez on sci.physics.research the post is this: https://groups.google.com/d/msg/sci.physics.research/F6x5GkFt0ic/fxsfuNl9d8gJ the article he is talking about is this one: http://xxx.lanl.gov/abs/gr- qc/0106075 the question itself is pretty simple. In that reply, which i'll quote for completeness, reads:               Unruh says that Haisch and Rueda's calculations are wrong, and     that a correct calculation shows a uniformly accelerating observer     zipping through the vacuum state of a quantized electromagnetic      field on Minkowski spacetime sees a *perfectly thermalized* bath      of photons.          In particular, this means such an observer will see no "Rindler flux" -      i.e., the expectation value of the Poynting vector is zero.  Or in     less fancy language: there will be, on average, no net flux of momentum      in the photons seen by the accelerating observer.          He gives a very simple argument showing that the expectation value      of the Poynting vector *must* be zero: the whole situation is      time-symmetric, and time reversal flips the direction of the      Poynting vector!        Now, i don't understand at all that last symmetry argument. Yes, time inversion will reverse direction of real vectors, so? an accelerated particle has a definite acceleration vector, that is reversed under time inversion, and is not zero. Why would any Poynting vector be zero?


title: For a fixed power input, is it better to hit something with a light rod or a heavy rod?

text: My friends and I got into an argument about whether it was more damaging to hit an object with a heavy rod or a light rod. The sides of the argument go something like: "If you swing the light rod, you'll be able to swing it much faster and therefore do the most damage!" "Yes, but if it's too light there won't be enough mass behind it and it won't damage as much!" I have been toying around with this question for a few months and I have yet to come up with a satisfactory answer. I'm setting up the problem like this: **Person A** is swinging a rigid rod of mass $m$ and fixed length $L$ at **Object B** located at a fixed distance $d$ away. Imagine **Person A** is swinging the rod much like a baseball player would swing to hit a ball. **Person A** has a fixed power $P$ to input to the rod over distance $d$ before making impact with **Object B**. Given input power $P$, rod length $L$, and swing distance $d$, what is the optimal mass $m$ of the rod to maximize the damage of **Object B**? For simplicity, I'm modeling the power application to be over a linear rather than angular distance, and I'm modeling **Object B** as a spring with spring constant $k$ attached to a fixed wall. Effects of gravity should be ignored. Also, for now ignore the constraints of human strength and assume we can swing anything as heavy as we like.


title: Speed of light in a vacuum

text: I see many references to the speed of light in a vacuum implying that it is only truly a constant measurement in a vacuum. I can live with that, but what kind of vacuum? Are we still talking about the kind of vacuum under a bell-jar with all the air pumped out or the kind of vacuum you could find in interplanetary space, or outside the heliosphere, or between stars, or between galaxies? Each has a lower energy content and therefore a lesser effect on the light. We know that light is noticeably bent by massive bodies, I guess it is also slightly bent by **any** kind of energy in its proximity, even other light (or am I wrong). Should we be talking about the speed of light in a dark, intergalactic vacuum? If so, what would be the difference? What subtle adjustments are we failing to make by measuring this speed at the bottom of a deep gravity well here on the earth's surface? How accurate is our measurement and could we have it sufficiently wrong to affect some other ideas we may have about the universe?


title: Do greenhouse gasses make the world habitable?

text: An ageing gent says he does not accept this statement:               It's a fact that greenhouse gases, predominantly water vapour, keep the Earth      from being hostile and cold; that CO2 albeit in trace quantities is also a      greenhouse gas and therefore helps keep the planet warm...      I know no more about why he rejects this statement - maybe it is that water vapour alone would do the job. The gent in question has a first class degree and a doctorate in physics. Is there a debate within the physics community over this or am I misunderstanding?


title: Integration of constant: $\int dp = \Delta p$ in impulse formula

text: In University Physics, it has something like: $$\int \sum F dt = \int \frac{dp}{dt} dt = \int dp = \underbrace{p_2 - p_1}_{\Delta p?}$$ But I thought $\int dp = p$? Though my maths is really rusty ... $p$ refers to momentum, $F$ is force


title: Is time a Scalar or a Vector?

text: In Wikipedia it's said that time is a scalar quantity. But its hard to understand that how? As stated that we consider only the magnitude of time then its a scalar. But on basis of time we define yesterday, today and tomorrow then what it will be?


title: Special Theory of relativity on electromagnetic waves

text: Since time slows down and length **contracts** , when we travel _almost_ at speed of light, if the **speed** of light (or EM waves) remains same and the **wavelength** of light remains same, do we measure the wavelength more than it actually is? Does it mean, we can "see" the ultraviolet light, if we can contract our length by an order of 100..?


title: Curious relation between the dependance in ℏ of Planck units and units dimensions

text: Looking at Planck units, there seems to be a curious rule between the dependance in $\hbar$ of a Planck unit and the unit dimensions of the corresponding physical quantity. Let the dimensions of the physical quantity be ($Q$ being the electric charge unit and $\Theta$ being the temperature unit): $$ L^l M^m T^t Q^q \Theta^\theta.$$ Then, if : $$l + m + t + q + \theta = 0$$ the Planck unit does not depend on $\hbar$. This seems to work for all base Planck units, and, consequently, for all derived Planck units. Is it just chance, or is there a more fundamental reason?


title: White Dwarf radius

text: So I've been reading this about white dwarves, and various other sites about white dwarves. In all of them, they say that we can find the radius of a white dwarf by minimizing its total energy. I know that for almost all physics problems we do this, but why is it so? Does it have something to do with variational calculus or something else entirely...? Grateful for any help!


title: Physical interpretation of an intermittency definition

text: A random function $v(t)$ is said to be intermittent at small scales of its "Flatness" $F$, given as $$ F(\Omega) = \frac{\langle (v_{\Omega}^{>}(t))^4\rangle}{\langle v_{\Omega}^{>}(t))^2\rangle} = \frac{\langle v_{\Omega}^{>}(t)v_{\Omega}^{>}(t))v_{\Omega}^{>}(t))v_{\Omega}^{>}(t))\rangle}{\langle v_{\Omega}^{>}(t)v_{\Omega}^{>}(t)\rangle \langle v_{\Omega}^{>}(t)v_{\Omega}^{>}(t)\rangle} $$ diverges as the high pass filter velocity $\Omega \rightarrow \infty$. $v(t)$ which can for example be the velocity in the fluid, is decomposed into its Fourier components $$ v(t) = \int_{R^3}d\omega \, e^{i\omega t}\hat{v}_{\omega} $$ and $v_{\Omega}^{>}(t)$ is the high frequency part of for example the velocity in the fluid $$ v_{\Omega}^{>}(t) = \int_{|\omega| > \Omega} d\omega \, e^{i\omega t}\hat{v}_{\omega} $$ I am struggling hard to understant the physical meaning of this definition and nead some help in this: First of all, why is $F$ called "flatness", it should be a measure of flatness of what? What does it mean for $F$ to diverge in the high frequency (UV) limit? Looking at the expression for $F$, I got the impression that it could mean that higher order correlations in time ("n-point functions") start to dominate in case of intermittency and more "local" 2-point interactions, which are responsible to maintain a scale invariant inertial subrange (?), become negligible such that the turbulent system starts to deviate from a Kolmogorov inertial subrange for example. In addition, other measures of intermittency can be defined involving higher order correlations in length $l$, such as the so called hyper-flattness defined as $F_6 (l) = S_6 (l)/(S_2(l)^3)$ etc... Does this mean that one could more generally say that for a turbulent system that shows intermittency, the Wick theorem can not be applied to calculate higher order n-point functions from 2-point functions? I am finally interested in understanding intermittency from a quantum field theory point of view, which is unfortunatelly not the point of view of the book I have taken these definitions from ...


Is there a connection between the fluctuation-dissipation theorem and the Green–Kubo relations? I have a hard time finding out if there is a relation and what it is, because the fluctuation-dissipation theorem always seems to be stated in another way and for specific cases. Both formula seem to be a statement about how a macroscopic property is determined as an integral over Greens function'esque objects. What is the exact relation and the hierarchy of the two concepts?


title: What is the Lie algebra of the Galilean group and what is the structure of it?

text: I read Freeman Dyson's article _Missed Opportunities_ , in which he talked about the mathematical attractiveness of the Lorenz group compared to the Galilean group. I am reading _Florian Scheck_ 's book on special relativity, and I feel very confused because he emphasized that though $P_{G}$ is an `affine bundle', it is nevertheless not trivial. To illustrate this he argued that since the general form of Galilean transformation is $$x'=Rx+vt+a, t'=t+s, R\in SO(3)$$ He claimed that if $v\not=0$, then the projection of the spacial coordinate is no longer invariant for the rest frame and moving frame. Does this just mean $x'\not=x?$ I am confused with his language. A related question is what is the structure of $P_{G}$ afterall? The wikipedia article is sadly incomplete; I do not want to know what the central extension of $P_{G}$; I want to understand how to describe it mathematically (like a matrix group or as a fibre bundle). What is its topological structure? What is its group structure? What makes it so complicated as Dyson claimed? The representations of $P_{G}$ are not clear to me, either as the indices mixed up everywhere. Overall I feel very confused because I believe Galiean transformations should be easy to understand. A strategy of unwind this ambiguity is to work with lower dimensions. The $P_{G}$ for one dimensional space should only have the form $$x'=x+vt+a,t'=t+s$$ and is only 3 dimensional as a Lie group. Group structure operates by $$(v_{2},a_{2},s_{2})(v_{1},a_{2},s_{1})=(v_{1}+v_{2},a_{1}+a_{2}+v_{2}s_{1},s_{1}+s_{2})$$ But I am not sure what kind of group this is. Is this $\mathbb{R}^{2}\rtimes \mathbb{R}^{1}$ with $$\phi_{s{1}}(v_{2},a_{2})=(v_{2},a_{2}+v_{2}s_{1})?$$ and how this generalizes to higher dimensions?


title: Why is the mass of even the upper limit of the possible Higgs boson is less than the top quark?

text: On 7 March 2012, the DØ and CDF collaborations announced that, after analyzing the full data set from the Tevatron accelerator, they found excesses in their data that might be interpreted as coming from a Higgs boson with a mass in the region of 115 to **135** GeV. But the top quark has a mass of 172.9 ±1.5 GeV which is about the same mass as an atom of tungsten.   1. why is the mass of even the upper limit of possible Higgs boson less than the top quark!? $$135 \frac {\mathrm{GeV}}{c^2} < \left( 172.9 \pm 1.5 \right) \frac {\mathrm{GeV}}{c^2}$$


title: Can human push earth away from the sun?

text: It's said the earth is approaching the sun. Can we push it away little by little? Although we can emit rocket towards the sun, but it's too expensive to make a rocket: ![step1](http://i48.tinypic.com/2hqb5w8.png) So I think maybe we can recycle the rocket, at first, the rocket is send out with a very fast speed, so give back a big force to the earth. And then, we let the rocket slowly comes back to the earth, and re-send it again: ![step2](http://i47.tinypic.com/wgtjjl.png) Well, this solution sounds a little inefficient. Still I don't know if it works. Better ideas? **EDIT** Umm.. It seems the earth is leaving the sun. That's ok. Then, the question is how can we push the earth towards the sun, i.e., correct the orbit by human force? Even if the sun turns into a red giant someday, we may finally find a way to put the earth at a safe position far enough from heat.


title: Density of universe during recombination

text: How can I calculate the density of the Universe at recombination?


title: Do any naked-eye stars have planets

text: Are there any known exoplanets around naked-eye stars? I know that Fomalhaut has a planet, and Vega has a dust belt that may be a protoplanetary disk or even analogous to our Kuiper belt. What else is out there orbiting naked-eye stars?


title: Why is Avogadro's law always true?

text: Why is Avogadro's law always true? How and why do equal volumes of gases at equal pressure and temperature contain equal number of molecules? I know it is a fundamental principle in chemistry but I wonder how it works.


title: Increasing resistance, by heat?

text: I was wondering the magnitude of effect to a conductor's resistance when heated by a few degrees. Can this be controlled way of changing conductor's resistance, or fixing in a system?


title: Centripetial Force in Rotating Platform and relation between Period and radius

text: The experiment setup looks (roughly) like: ![enter image description here](http://i.stack.imgur.com/7Ynpe.png) First, a weight (W2), is hung to simulate centripetal force when the platform is not rotating. Then its removed (W2 is now hanging to one side as shown in the bottom image) and we adjust the speed of rotation until W1 is aligned again. Then we measure the time taken for 20 revolutions. This experiment is repeated for varying radius (the distance between the 2 vertical columns). Then I'm asked to plot a graph of radius vs Period^2. But I don't get why Period^2? Then when the experiment is repeated for constant radius, but varying W2, I'm asked to plot a graph from radius to Period^(-2), why powers of 2 and -2?


title: What is the spatial derivative of strain tensor?

text: I'm looking for a metric that contains enough information of a 3D solid's deformation: stretch, bend and twist, etc. Stretch, bend and twist may be ambiguous for a 3D solid, since unlike a rod, it doesn't have a curve direction. Let's assume that we've found three directions, along which the deformations are independent and the combined deformation of the three directions is the total deformation of the solid. (Do the three directions even exist? I'm still looking for some references on this.) Strain tensor only describes stretch at a material point, because it's the same order as the gradient of displacement. It needs the strain at neighbor positions to measure other deformation behavior. I'm thinking of taking the spatial derivative of the strain tensor and the resulting 3-order tensor may contains the information I want. My question is: what does every entry of the "strain gradient" mean? Is it that some of them measure stretching, some measure bending and some others measure twisting? Does anyone know any books that may cover this topic? Thanks!


title: Coverage of Quantum Electrodynamics (QED) in introductory Quantum Field Theory (QFT) books

text: Which QFT books also cover QED? I am not very familiar with QED, so I am looking for QFT books which cover QED too (I know they cover Quantum Chromodynamics (QCD).).


title: What is the difference between "kinematics" and "dynamics"?

text: I have noticed that authors in the literature sometimes divide characteristics of some phenomenon into "kinematics" and "dynamics". I first encountered this in Jackson's E&M book, where, in section 7.3 of the third edition, he writes, on the reflection and refraction of waves at a plane interface: >   1. Kinematic properties: (a) Angle of reflection equals angle of incidence > (b) Snell's law >   2. Dynamic properties (a) Intensities of reflected and refracted radiation > (b) Phase changes and polarization > But this is by no means the only example. A quick Google search reveals "dynamic and kinematic viscosity," "kinematic and dynamic performance," "fully dynamic and kinematic voronoi diagrams," "kinematic and reduced-dynamic precise orbit determination," and many other occurrences of this distinction. **What is the real distinction between _kinematics_ and _dynamics_?**


title: Event Horizon violability?

text: Is the "event horizon" of a black hole potentially violable? Black holes are commonly described as being unidirectional (matter / energy goes in, but doesn't come out), but is the event horizon of a black hole potentially deformable? That is, in the case of two black holes colliding (let's say), is there not a point when the effective gravitational field between the two black holes "cancel out", thereby causing a potential "escape path" for light traveling perpendicular to the axis the two black holes are on? In a more complicated case, consider two black holes of similar mass orbiting each other in a stable configuration; since no matter or energy actually pass the event horizon of a black hole, would the area between the black holes be a potential zone for escape of energy? (I'm assuming minimal frame dragging effects here.) And since the black holes would need to rotate at a very high rate of speed to avoid actually colliding, would that not restrict the emission of energy to be along the axis of rotation? A bit of clarification here; the primary question here is whether or not the gravitational force vectors of the black holes can properly cancel each other out at an appropriate point between them. If that's the case, then (in a static system) it can be easily determined that there is a plane through that point perpendicular to the axis along which the two black holes exist where the gravitational force vectors are deeply reduced (not zero, of course, except at the precise point where the masses perfectly balance; however, reduced significantly based upon the inverse distance from said axis). In an (again static) two black hole system, orbiting material could enter a "figure eight" orbit; much of this material would pass through this "nodal point" (or very nearby it). The high speeds and volumes of the material would indicate a large number of collisions; some resultant (high-energy) ejecta from these collisions would result in some (small) quantity of ejecta traveling along the plane of reduced gravitational force. Now consider the (more dynamically stable) case of stably orbiting black holes; the plane of reduced gravitational force is itself rotating (about the axis of rotation of the black holes); this turns the "escape plane" (not entirely just a plane, it having some (small) quantity of thickness) into a line (again, not entirely just a line, but a very thin cone). The primary point is that that the cancellation of force vectors effectively makes for a "dimple" in the event horizon of each of the black holes; that is, the existence of another strong gravitational attractor in close proximity deforms the event horizon of a black hole. The question is, is there anything which invalidates this analysis? As mentioned before, frame dragging is a concern; is there anything else which would be a concern? (I suspect the frame dragging concern would actually be resolved by the mutual forced frame dragging, i.e. the holes' spins would "synchronize" because of the mutual frame drag effect.) If this is a possibility, then does this not imply another means of black hole mass "evaporation" other than Hawking radiation? (And, in fact, wouldn't it imply the type of "polar jet" that has been observed?)


title: "finite" QFTs and short-distance singularities and vanishing beta functions

text: I am not sure that I can frame this question coherently enough - it springs from various things in QFT that I have recently been thinking and reading about. May be these thoughts are mis-directed but it would still help to know why they are if they are so! I would like to hear some discussions about these.   * I guess there are QFTs which are "exact" or "finite" as in they don't require a regulator or cut-off for their partition function to be defined (..and I guess one can evaluate the partition function exactly..) I guess the really non-trivial "finite" QFTs would be ones which are so even on non-compact spacetimes. Are there such?    * I guess there exists QFTs which have a non-perturbatively 0 beta function. (like $\cal {N} =4$ SYM?)  Are the above two properties related? Like does being "finite" imply it has an exactly $0$ beta function or conversely? (..seems no..see below..)   * I guess CFTs (..or any QFT sitting at a zero of its beta-function -"critical" ?..) sort of by definition have a 0 beta function but they do have non-trivial OPEs coming from short-distance singularities. This is somehow not intuitive because one would have naively thought that large momentum is like short distance and hence if the theory requires no regulator and hence has no large momentum divergence then it should also correspondingly not have any short-distance singularity. But this seems to be wrong - hence I guess one is led to thinking that having an exactly 0 beta function has nothing to do with the theory being finite.  Its not clear to me that there is any direct relationship between having short-distance singularities and whether or not momentum space integrals diverge (..which should possibly imply that the partition function also diverges..)(..Like Yuji Tachikawa in the comments points out the simple case that even free boson theory has short-distance singularity but since there are no loop processes in it I guess it doesn't make sense to ask whether momentum space integrals converge but I guess its partition function is not always well-defined..) Like on page 441, Weinberg in his volume 1 of his QFT books, says in italics that _"renormalization of masses and fields has nothing directly to do with the presence of infinities and would be necessary even in a theory in which all momentum space integrals were convergent"_ To may be summarize my query - is one saying that there are conceptually different multiple sources of infinity in a QFT like,   1. divergence of the momentum space integrals   2. short distance singularity   3. divergent partition functions    4. coupling constants being sent to infinity by the beta function (.I thought of also adding the phenomenon of Landau pole in the above list but I guess that is not so fundamental a property and is only an indication of the failure of the perturbation technique..thought I may be wrong..) So is there a way to think of these "different" infinities as cause and effect of one another? Or is it possible that any combination of these can show up in some QFT? And/How are these related to the property of the beta-function being non- perturbatively 0 or not? (..except for the "by definition" case that for non- perturbatively 0 beta-function (4) can't happen..)


title: Functional relations for Kochen-Specker proofs

text: Many proofs of the Kochen-Specker theorem use some form of the following argument (from Mermin's "Simple Unified Form for the major No-Hidden-Variables Theorems" ) > [I]f some functional relation $$ f(A,B,C,\ldots)=0 $$ holds as an operator > identity among the observables of a mutually commuting set, then since the > results of the simultaneous measurements of $A,B,C,\ldots$ will be one of > the sets $a,b,c,\ldots$ of simultaneous eigenvalues of $A,B,C,\ldots$, the > results of those measurements must also satisfy $$ f(a,b,c,\ldots)=0 $$ Parity-type contradictions (e.g., $1=-1$ or $0=1$) are then seen to arise when $a,b,c\ldots$ are assigned values independently of the context in which they are measured. The only explicit forms of $f$ that I have seen are either (i) $A+B+C+\ldots$ or (ii) $(A)(B)(C)\ldots$ (see e.g., "Generalized Kochen- Specker Theorem" by Asher Peres, where both forms are used). My question, then, is: are there examples of parity-type proofs where $f$ is, necessarily, not of the above forms (i) or (ii)? For example one could consider $A+(B)(C)\ldots$ etc. Ideally, I'm looking for explicit examples where $f$ is spelled out, but I would also be interested in arguments where a different kind of $f$ is implicitly used.


title: How do you know which way to choose the limits of an integral?

text: I am reading http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S4 In the beginning of equation 13.18, in which Mr. Feynman calculates the potential energy of an object outside a spherical shell, it says: $$W=\frac{Gm'2 \pi a \mu}{R} \int \limits_{R+a}^{R-a} \, dr$$ The result of the integral should be: $$W=\frac{Gm'2 \pi a \mu}{R} \bigl((R-a)-(R+a)\bigr)$$ $(R-a)-(R+a)$ is $-2a$ and so $$W=-\frac{Gm'4 \pi a^2 \mu}{R}$$ $$W=-\frac{Gm'm}{R}$$ **How does Mr. Feynman know how to choose the limits of the integral?** If I chose the limits from $R-a$ to $R+a$, I would get in the end: $W=\frac{Gm'm}{R}$. The only difference is the algebraic sign at the beginning. I don't know why you have to choose the beginning of the integral further away from P ($R+a$) and end closer to P ($R-a$). Is there any rule that states that, or why did Mr. Feynman choose the limits of the integral the way he did? Did he think: I've got to get a negative result, so I choose the limits from $R+a$ to $R-a$?


title: Electric field between two conducting plates of different potential?

text: not sure if this question's been asked before, though I couldn't see one in my brief search on here... Well, the problem I am trying to solve is that I want to determine the electric field strength between two parallel rectangular- shaped plates with different voltages on each plate. For example, rectangular plate 1 has a voltage of +180V and rectangular plate 2 (parallel to plate 1) has a voltage of -5V. What equation(s) could we use to determine the field strength at any point between these two plates? The distance of separation of the plates is finite, say 8.6mm.


title: Heat & thermodynamics question based on heat loss

text: ![enter image description here](http://i.stack.imgur.com/FkbmQ.jpg) A Sphere A is placed on a smooth table.Another sphere B is suspended as shown in the figure.Both the spheres are identical in all respects.Equal quantity of heat is supplied to both spheres.All kinds of heat loss are neglected.The final temperatures of A & B are T1 & T2 respectively,then   * T1>T2   * T1 = T2   * T1    * None of these Please give reasons for your answer.


title: How to calculate the exit velocity of a coil gun projectile?

text: First off, what quantities need to be factored in? Voltage and current through the coils, the magnetism of the projectile, the magnetic fields, etc.? Next, how would you calculate the speed of the projectile as it exits the coil gun? Is there an equation? Also, an equation on the force exerted on the projectile would be appreciated.


title: Why is 55-60 MPH optimal for gas mileage of a passenger car?

text: My driver's education teacher back in high school said 55 MPH is optimal for gas mileage of a passenger car. Just last week, I read an article in a magazine saying 60 MPH is optimal. These numbers are pretty close, so there's some validity in the statement. What's the physics explanation for this 55-60 MPH sweet spot?


title: Where does the energy go, when light is blocked by polarisation

text: I've been looking around about LCD monitors, and how they polarise light. When a pixel needs to be black, the light is "twisted" so that it can't go through the polarising sheet in front. What happens to this light? Does it relfect back into the screen? Surely that would mean the screen would get quite hot if you leave it on a black image for a while? (but I've never noticed that happen). What happens to the energy (from the electricity used to create the light) when it's blocked by polarisation.


title: Spinning wheel on rotating stool - why am I also affected even when the wheel axis is horizontal?

text: So I can point the axis of spin in one horizontal direction - and then move it in another direction (so it stays horizontal - like the picture below). And that still changes the direction of spin (on the rotating stool). Why does that happen? I already know the full explanation in the case that the wheel axis is vertical and the wheel is flipped over. ![enter image description here](http://i.stack.imgur.com/dJsIi.jpg)


title: Toppling of a cylinder on a block

text: A uniform cylinder rests on a cart.The height and diameter is given.coefficient of static friction is given.How can i find the minimum acceleration of block such that the block topples? Morever what is the reason of toppling? ![cylinder on block](http://i.stack.imgur.com/eBbOx.png) Again, if i consider static friction to be the cause,then what is the point of rotation? If i consider the top to be te point of rotation,then why does the block topple with the lower end as point?I have conducted this experiment many times and if the block falls,it fell along the lower end.


title: How do I calculate the speed of a temperature change?

text: I know how to calculate the change of temperature in a copper wire depending on voltage and current, but not how fast the temperature would change.


title: Why are quark up and down an isospin doublet?

text: I have some difficulties in understanding the isospin of quarks. As far as I am concerned, the isospin formalism is used to express the physical property of electric charge. I mean: what I know about the isospin formalism is that it is used to describe the charge of a nucleon, proton and neutron are two different isospin state of the same particle (nucleon). Consider: $\tau ^2$ and $\tau_z$ the CSCO related to the isospin formalism. I will denote with $t$ and $t_z$ the quantum number that allows for determining the eigenvalue of the related operator. I know that $\tau_z|^1_1p>=+\frac{1}{2}|^1_1p>\Rightarrow t_z(^1_1p)=+\frac{1}{2}$ $t_z|^1_0n>=-\frac{1}{2}|^1_0n>\Rightarrow t_z(^1_0n)=-\frac{1}{2}$ and so I can write the charge operator: $q=e\left(\frac{1}{2}+\frac{1}{2}t_z\right)$ The problems comes with quarks. I know that the quark up, $q(u)=\frac{2}{3}$ and $q(d)=-\frac{1}{3}$. But I read that $u,d$ are an isospin doublet with $t_z=\frac{1}{2}$. Could someone explain me why? The charge operator does not return fractional charge, I think I made a mistake somewhere but I do not know where..


title: Uncertainty principle - momentum so precise that uncertainty of position is outside light-cone?

text: Thought experiment: what happens if we measure momentum of a particle so precisely, that the uncertainty of its position becomes absurd? For example, what if the uncertainty of the position exceeds 1 light year? We know for a fact that the particle wasn't a light year away from the measuring device, or else how could the momentum have been measured? What if the uncertainty extended beyond the bounds of the universe? Isn't there some point at which we know for certain the particle was closer than what the uncertainty allows for?


title: Quantum field theory defines its own bounds of applicability

text: I recall hearing in a lecture something along the following lines: "Due to some intrinsic feature of quantum field theory in general (or maybe it was the standard model?), we know where it is applicable, and thus we know that there will be no new physics at low energy (the part I care about is in day-to-day living), or at least not without huge (like throwing away relativity) changes in our models." What is this property at a formal level? Is there a concise way to explain it to non-specialists?


title: Is Tube light polychromatic?

text: Is tube light/fluorescent lamp monochromatic or polychromatic? Please justify.


title: Why isn't there a limit for a Euclidean rotation, as for a Minkowski rotation?

text: From invariance of the Minkowski scalar product, we get the Lorentz transformations. In addition, we get a constant $c$ preventing space-like and time-like intervals being rotated into one another. The Euclidean transformations are derived in the same way from invariance of the Euclidean scalar product, yet there isn't a constant that limits these rotations as above. Why?


title: Temperature coefficient of resistivity (resistance) in function of temperature

text: If I'm correct, the definition of the temperature coefficient of resistivity at a certain reference temperature is the derivative of the resistivity in function of the temperature, divided by the reference resistivity. Are there materials where the temperature coefficient is constant over different reference temperatures? I only seem to find its value at reference temperature 20 degree Celsius... If it would be constant you'd get: $\alpha=\frac{\partial R(T)}{R(T)\partial T} \iff \alpha R(T)=R'(T)$ Which is an easy to solve differential equation and gives an exponential function for $R(T)$. Could it be that simple or am I missing something?


title: HOMEWORK: Minimum angle to start the mop moving

text: Situation: I'm a janitor who wants to clean the floor with a mop. I need to find a minimum angle theta so that the mop starts moving. ![enter image description here](http://i.stack.imgur.com/VDl0Z.png) First, I constructed the Newton's equations: $$ma_x = F_\text{push} \sin(\theta) - F_f$$ $$ma_y = F_n - F_g - F_\text{push}\cos(\theta)$$ Then, I set both $ma_x$ and $ma_y$ equations equal to zero and simplified the equations: $$\mu_\text{static} F_n = F_p \sin(\theta)$$ $$F_n = mg + F_p \cos(\theta)$$ Then, I divided the equations: $$\mu_\text{static} = \frac{F_p \sin(\theta) } { mg + F_p \cos(\theta) }$$ Then, I looked at the answer and found that $\theta = \tan^{-1}(\mu_\text{static})$. So, I assumed that $F_p$ is infinitely large and used L'Hôpital's Rule and got the right answer. The question is: is it right to assume that $F_p \to \infty$?


title: How do I determine a component of relative velocity?

text: > At a particular instant, a stationary observer on the ground sees a package > falling with speed v1 at an angle to the vertical. To a pilot flying > horizontally at constant speed relative to the ground, the package appears > to be falling vertically with a speed v2 at that instant. What is the speed > of the pilot relative to the ground? Ok, so the object falling relative to the observer has a different velocity then it does relative to the pilot. The y-component of the objects velocity will be identical for both reference frames, but the horizontal velocity is going to differ. Ground speed is equal to the vector sum of Airspeed and Windspeed. If the pilot observes the package to be falling vertically, he must be approaching it from behind. Assuming the package was dropped vertically, the x-component of its velocity should reflect the windspeed? So, the ground speed of the plane would be the sums of its constant velocity and the x-component of the objects velocity? I am asking for some guidance with this questions, but not the answer, if someone could give me some hints.


title: How do I calculate pressure drop across a reduction in pipe diameter?

text: If reducing from one pipe size to another through a reducing fitting, say 6.25cm to 5cm, how do I calculate the pressure drop? Also, how does this impact a change in volumetric flow rate (or does it affect the volumetric flow rate)?


title: The Time That 2 Masses Will Collide Due To Newtonian Gravity

text: My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r. Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as: $$F_g ~=~ G\frac{Mm}{r^2}$$ How do we find out at what time they will collide? What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$. I've tried taking a number of integrals, which haven't really yielded anything useful. Any guidance? No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :)


title: Were there any efforts made by early physicists to discover and explain how composite bodies fall?

text: At the dawn of the modern era, Galileo discovered and described how composite bodies fall through the air (or at least the discovery is usually attributed to him). I'm interested in whether this had been discovered earlier and how, particularly since it seems to me that there are good grounds for this result to hold true purely on the basis of continuity and symmetry. Imagine three balls of the same size and weight, and at equal distances from each other, dropped from a tower at the same time. By symmetry, all three must hit the ground at the same time. Now repeat the experiment, but move the left-hand ball next to the middle one. This makes no difference to the result—the three balls still hit the ground at the same time. Now repeat the experiment once more after slightly increasing the contact area of the two adjacent balls. Again, I would expect them to hit the ground at the same time. By repeating this, the left-hand and middle balls eventually merge into a single larger ball, which will fall at the same time as the right-hand one. Did anyone make this argument in the pre-modern literature? I'd be interested to know whether any of the ancient atomists came up with similar arguments when they considered how atoms moved under gravity. It ought to have then been a simple step via the above argument to see that composite bodies fall at the same rate.


title: Why define four-vectors to be quantities that transform only like the position vector transforms?

text: A four-vector is defined to be a four component quantity $A^\nu$ which transforms under a Lorentz transformation as $A^{\mu'} = L_\nu^{\mu'} A^\nu$, where $L_\nu^{\mu'}$ is the Lorentz transformation matrix, which including boosts, rotations and compositions. (In other words, as the components of a position vector $(x0,x1,x2,x3)$would transform). The useful property of four-vectors is claimed to be that if two four-vector expressions are equal in one frame, then they will be equal in all frames : $A^\mu = B^\mu \Leftrightarrow A^{\mu'} = B^{\mu'}$ and therefore we can express laws of physics in terms of four vectors, because they remain invariant in all frames. But this property will be true even for four component quantities that transform (across reference frames) as $A^{\mu'} = T_\nu^{\mu'} A^\nu$, where $T$ is _any_ transformation matrix (not necessarily a Lorentz one). As long as we can find a $T$ that will describe how the quantity's components transform, we can apply that T to both sides of an equality. So why require (i.e. define) four-vectors to only be quantities that transform under a Lorentz matrix?


title: Why does it spark when I push a plug in the electrical socket?

text: When I slowly push a plug into the electrical socket I can often see sparks. Can anybody explain why? Can this be possibly harmful for the devices I plug in?


title: Is there still mystery about spin crisis?

text: The gluon is a vector boson; like the photon, it has a spin of 1. If this is true and if we know the spin of protons and neutrons then why the news SPIN COMES FROM GLUONS?  is telling that, we aren't quite sure about overall nuclear spin?


title: How fast to travel to end up where i started from?

text: I've heard that if one travels far enough in one direction through the universe, one ends up at the starting point, due to the "geometry" of the universe. What rate is space expanding at? Is it in fact faster than light? How fast would one need to travel relative to expanding space/time to end up at the starting point?


title: Solid in Liquid Heat Transfer

text: If there is a solid immersed in a large (but finite) pool of water, where the solid has temperature $T_s$ and the water has temperature $T_w$, with $T_w>T_s$, how can I calculate $T_s(t)$ and $T_w(t)$? (Suppose that the masses and the material properties of the solid are known.)


title: Existence and Uniqueness of spurious states in Bosonic String Theory

text: Green, Schwarz and Witten introduce so called spurious states $\phi$, that fulfill $$ (L_0 -a )\phi = 0\quad \text{and} \quad (\phi,\psi)=0 \text{ for all physical states }\psi $$ where the $L_n$ are the Virasoro generators, that can be expressed through the exitation creation/annihilation operators of the String: $$ L_m=\frac{1}{2} \sum_{-\infty}^\infty \alpha_{m-n} \cdot \alpha_n $$ Now for $m=0$ this has an odering ambiguity which is resolved by defining $$ L_0=\frac{1}{2} \alpha_0^2 + \sum_{n=1}^\infty \alpha_{-n} \cdot \alpha_n $$ and adding this parameter $a$, which is just a c-number, in every formula containing an $L_0$. The suprious states however can be written as $$ \phi = \sum L^\dagger_n ~\chi_n = \sum L_{-n} ~\chi_n $$ with $$ (L_0 -a +n) \chi_n = 0 $$ Does anybody know whether the $\chi_n$ are unique and how do I know that such states exist in the first place?


title: Solving the soliton equation without energy

text: In this passage from Srednicki's _Quantum Field Theory_ (page 576) > The solution of interest is time independent, so we can set $\dot\varphi = > 0$. We can also rewrite the remaining terms in $E$ as \begin{align} E &= > \int_{-\infty}^{+\infty}\mathrm{d}x\Bigl[\tfrac{1}{2}\Bigl(\varphi' - > \sqrt{2V(\varphi)}\Bigr)^2 + \sqrt{2V(\varphi)}\varphi'\Bigr] \\\ &= > \int_{-\infty}^{+\infty}\mathrm{d}x \tfrac{1}{2}\Bigl(\varphi' - > \sqrt{2V(\varphi)}\Bigr)^2 + > \int_{-v}^{+v}\sqrt{2V(\varphi)}\mathrm{d}\varphi \\\ &= > \int_{-\infty}^{+\infty}\mathrm{d}x \tfrac{1}{2}\Bigl(\varphi' - > \sqrt{2V(\varphi)}\Bigr)^2 + \tfrac{2}{3}(m^2/\lambda)m.\tag{92.5} > \end{align} Since the first term in eq. (92.5) is positive, the minimum > possible energy is $M \equiv \tfrac{2}{3}(m^2/\lambda)m$; this is much > larger than the particle mass $m$ if the theory is weakly coupled ($\lambda > \ll m^2$). Requiring the first term in eq. (92.5) to vanish yields $\varphi' > = \sqrt{2V(\varphi)}$, which is easily integrated to get $$\varphi(x) = > v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{92.6}$$ How do they get the solution (92.6) where the energy E is missing? I want to get the equation (92.6) from equation(92.5), can you do some process?


title: For how long must a molecule remain stable to be considered "stable"?

text: In the _Star Trek: Voyager_ episode _The Omega Directive_, Seven of Nine says that the Borg synthesized a molecule which was "kept [] stable for one trillionth of a nanosecond before it destabilized". That's $10^{-21}$ seconds, which is pretty brief. Of course the point of that part of the script is to illustrate how unstable the molecule in question is. But this got me thinking: in our world, for **how long must a molecule remain stable to be considered "stable"?** Is there a generally applicable time limit or range (even order of magnitude)? I did find Lifetimes of stable particles, but both the question and accepted answer talks about fundamental particles, whereas I am considering molecules.


title: Upthrust vs Drag

text: > _24: A ball is falling at terminal speed in still air. The forces acting on > it areupthrust, viscous drag and weight. What is the order of increasing > magnitude of these 3 forces?_               Answer :upthrust  < viscous drag  < weight      Could some one please explain why in still air upthrust of a ball is smaller than the viscous drag ?


title: How do substances with like charges repel each other?

text: I have read in a number of places how substances with opposite charges attract each other: The excess electrons in the one substance repels the electrons in the other substance so that they move away from the surface, leaving the protons closer to the surface which are then attracted to the excess electrons in the one surface. But I have not found any parallel explanation of what happens when substances with like charges are brought near each other. I'd also be grateful for an explanation of what happens when two electrically neutral substances are brought near each other.


title: Graphene +1 extra carbon bond

text: I'm not a physicist just a curious mind, so please go easy! I was just watching a BBC Horizon Documentary that featured a piece on the recently discovered material Graphene. One of the facts mentioned on the documentary was that a sheet of graphene, despite being a lattice of carbon atoms just one atom thick, could hold the weight of a household cat without breaking. They showed an illustration of graphene's hexagonal structure and how a crude form of graphene could be created using Sellotape and piece of graphite. OK so here's my question.. From what I can remember from Chemistry lessons in school, carbon has a valency of 4, so it can bind to 4 more carbon atoms. Surely (in theory at least) a sheet of carbon could exist that adopts a square grid lattice, as oppose to graphene's hexagonal structure. If so wouldn't such a material be stronger than graphene? Also would such a lattice naturally form under any circumstance or does carbon always assume a hexagonal structure when reduced to layers?


title: How to find the Higgs coupling with a mixing matrix?

text: It is known that the couplings to the Higgs are proportional to the mass for fermions; $$g_{hff}=\frac{M_f}{v}$$ where $v$ is the VEV of the Higgs field. I'm trying to figure out why this is true _without explicitly constructing all the interaction terms_. First I can say that, the lagrangian, being massless requires the Higgs field $\vec{\phi}=\left( \begin{array}{c} \phi^1\\\ \phi^2\\\ \end{array} \right)$ (weak doublet) which adds a symmetry breaking part and a coupling to the fermions. The coupling is proportional to $$g(\bar{\psi}_L\vec{\phi}\psi_R+\bar{\psi}_R\phi^\dagger\psi_L)$$ where $\psi_L$ is a weak doublet $\psi_R$ a weak singlet. So, after the symmetry breaking, we can write the Higgs field as $\vec{\phi}=\left( \begin{array}{c} 0\\\ v+h(x)\\\ \end{array} \right)$ because of the local $SU(2)_W$ symmetry. Therefore the coupling becomes $$gv(\bar{\psi}^{(2)}_L\psi_R+\bar{\psi}_R\psi^{(2)}_L)+gh(\bar{\psi}^{(2)}_L\psi_R+\bar{\psi}_R\psi^{(2)}_L)$$ where the $(2)$ exponent indicates the component in the doublet. Then, we can rewrite the fields as Dirac fields: $\psi=\psi^{(2)}_L+\psi_R$ which gives $$gv\bar{\psi}\psi+gh\bar{\psi}\psi$$ From which we identify the mass of the fermionic field as $M_f=gv$ which implies that the coupling to the higgs is $$g_{hff}=\frac{M_f}{v}$$ Of course I only considered one family of fermions, so my question is how do I generalize the argument when the coupling includes family mixing? $$g(\bar{L}\vec{\phi}\Lambda R+\bar{R}\Lambda^\dagger\vec{\phi}^\dagger L)$$ where $\Lambda$ is a family mixing matrix. Is it straightforward from the diagonalization of the mixing matrix?


title: How exactly is the propagator a Green's function for the Schrodinger equation

text: Sakurai mentions that the propagator is a Green's function for the Schrodinger equation because it solves $$\left(H-i\hbar\frac{\partial}{\partial t}\right)K(x,t,x_0,t_0) = -i\hbar\delta^3(x-x_0)\delta(t-t_0)$$ I don't see that. First of all, I don't understand where the $-i\hbar$ comes from. And if I recall correctly, a Green's function is used to solve inhomogeneous linear equations, yet Schrodinger's equation is homogeneous $$\left(H-i\hbar\frac{\partial}{\partial t}\right)\psi(x,t) = 0$$ i.e. there is no forcing term. I do understand that the propagator can be used to solve the wave function from initial conditions (and boundary values). Doesn't that make it a kernel? And what does Sakurai's identity mean?


title: Quantum uncertainty can explain the Riemann Hypothesis?

text: In the recent paper "Riemann Hypothesis as an Uncertainty Relation" (http://arxiv.org/abs/1304.2435) the author claims that the presence of zeros out of the critical line may lead to the violation of a Heisenberg-type uncertainty relation. Is that work a proof of Riemann's hypothesis?


title: Amateur Stargazing (30 degrees N)

text: When I look at the sky I easily recognize Orion, The Big Dipper, and the North Star (in relation to the Big Dipper). Besides these (and the moon^^), what **_easily identifiable_** (unaided eye) astronomical objects should I look for? (Latitude = 30 degrees North)


title: How did micrometeorite flux change with the age of the solar system?

text: Do we also have functions for the micrometeorite flux on the Moon, Mars, and even any random body in space as a function of solar distance too?


title: Why should mass be attractive in nature?

text: Why does a mass attract all the masses around it? Why should't it repel or just stay calm? Why should it be like that?


title: What real experimental systems are well-described by Glauber-Ising spins?

text: I'm hoping for references to actual physical systems in which all or at least most of the following can be simultaneously characterized: the spin flip rate, the temperature, and a relaxation or similar timescale. I'm also specifically looking for Glauber vs Kawasaki or some other dynamics. Isolated spins are especially interesting to me; spin glasses are not.


title: Minkowski to Euclidean

text: When dealing with solutions to Einstein's equations given by a 4d metric with signature $(-,+,+,+)$, we're able to move to Euclidean space using some transformation so that our signature is now $(+,+,+,+)$ and our metric is changed slightly different. What I'm worried about is what this does to the properties of my metric. When I go to Euclidean space, I am able to deduce some nice relations, however, $\textbf{do these relations hold for my original metric?}$ e.g. let's say I'm interested in the metric $$ds_1^2=-V_1dt^2+\frac{dr^2}{V_1}+W_1d\Omega^2,$$ where $d\Omega^2$ is metric on $S^2$. Now my metric in Euclidean space is $$ds_2^2=+V_2dt^2+\frac{dr^2}{V_2}+W_2d\Omega^2.$$ where $V_1(a,b,c)=V_2(i a,b,c)$ and for $V_2$ I have some cool relation (e.g. $V_2(a=b)=0$, for $i$ the imaginary number. How can I relate this to the properties of $V_1$ if $a$, $b$ and $c$ are real parameters? The reverse transformation gives $V_1(-i a=b)=0$, but I don't know how to interrupt this $a$ and $b$ are real. Perhaps I haven't throughout deeply enough about it yet and it's a naive question, but I don't think I yet comprehensively understand what this transformation means for the properties of our space.


title: Recommendation on mathematical physics book of Symplectic geometry

text: I want to learn the applications of symplectic geometry in physics. Which mathematical physics textbook will have a detailed and heuristic explanation of this aspect?


title: Why isn't angle of launch used in this kinetic energy equation?

text: Question: A stick is thrown from a cliff 27 m high with an initial velocity of 18 m/s at an angle of 37 degrees above the horizontal. a) use the law of conservation of energy to determine the speed of the stick just before it hits the ground. solution: ![a busy cat](http://i60.tinypic.com/2qmdz74.png) Just wondering why the vertical initial speed vector isn't used? Should 18sin37 be used? I think I am confusing myself with projectile motion.


title: Why is the $\theta$ term of QCD violating charge and parity (CP) symmetries?

text: From the non-trivial nature of the QCD vacuum, the Lagrangian is augmented with a term like \begin{equation} \theta \frac{g^2}{32 \pi^2} G_{\mu \nu}^a \tilde{G}^{a, \mu \nu} \end{equation} where $ \tilde{G}^{a,\mu \nu} = \frac{1}{2} \epsilon^{\mu \nu \rho \sigma} G^a_{ \rho \sigma} $ is the dual field strength tensor. This term is said to violate CP, giving rise to the strong CP problem. I understand the CP violation comes from the epsilon tensor in the dual field strength but I am looking for a simple straightforward demonstration of the CP violating nature of a term like $G \tilde{G}$.


title: Calculate orbit height from tangential velocity

text: Is it possible to calculate the altitude of a satellite orbiting a planet, just from the planet's mass and the satellite's velocity, assuming that the satellite is in a perfect circular orbit? If so, what would the equation be? I've derived the following equation: $G\frac{M}{v^2} = r$ , however I don't think it's correct.


title: EM wave properties

text: Here are two small questions about EM wave properties: **Question 1:** Why do E and B propogate simultaneously? See picture ![enter image description here](http://i.stack.imgur.com/RcVwS.jpg) I mean why do not works a paradigm “changing E makes B and changing B makes E?” For such scenario we would have phase shift beatween E and B by $\pi/2$. I am asking because of f.e. maximum of B = dE/dt it is when E=0. So why not E = E_max*sin(x) and B=B_max*cos(x)? **Question 2:** But for standing wave we have phase shift beatween E and B by $\pi/2$ (by theory). http://www.youtube.com/watch?v=tbR93plA1WU ![enter image description here](http://i.stack.imgur.com/FlZVj.png) So do you know some experiments to confirm that maximums of B are really shifted by $\pi/2$ for standing wave?


title: Correlation between color of light visible to us and the intensity of transmitted light by the material

text: I wanted to know if there is any relation between the color of light observed by the eye and the intensity of light transmitted by a material. Let us assume that I have a table of intensities of light transmitted at different wavelengths by a material for 2 materials. Is it possible to determine which material will appear more colored and which is that color? I know that if a material transmits with more intensity at the wavelength of green light, it would be more 'greener' but what will be the effect of a range of wavelengths?


title: Hamiltonian Operator for Harmonic Oscillator

text: I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both of them I ham having a small confusion while defining the Hamiltonian operator Tannoudji used this expression $\frac{H}{\hbar w}$ for defining the Hamiltonian operator where $H$ is classical Hamiltonian while Griffiths have not used this expression at all. The confusion is how this expression from Tannoudji came across and why Griffiths did not use this expression at all.


title: Why did the ancients fail to discover that the Earth orbits the Sun?

text: The ancients observed that the Sun and the 'fixed' stars rotated about the Earth. They were also aware that the Earth was spherical. They performed many astronomical measurements on the planets - which are far less 'obvious' than the Sun and the stars. Presumably they knew that the speed of rotation of the Sun and the 'fixed' stars differed and that difference happened to be one day a year. It would seem (in retrospect) obvious that this was no coincidence and no other heavenly motion was so clearly linked to any other. Could they not have deduced that the difference was due to the annual path of the Earth around the Sun?


title: Power divergences from loops

text: I do not know what I should think about power divergences from loops. Most QFT textbooks tell us how to deal with logarithmic divergences from loops $\sim\ln(\Lambda^2/\Delta)$: we can set a counterterm $\sim\ln(\Lambda^2/\mu^2)$ to cancel the divergence at some scale $\mu$, then at any other scale this stuff in terms of renormalized quantities are finite $\sim\ln(\mu^2/\Delta)$. However, sometimes we encounter power divergences like $\sim\Lambda^2s$ in theories where we have momentum dependent interactions (such as a non- renormalizable theory with interaction $\phi^2\partial_\mu\phi\partial^\mu\phi$), where $s$ is one of the Mandelstam variables which depend on the external momenta. If we try to apply the same method to add counterterms, we may have a counterterm $\sim\Lambda^2\mu^2$ such that the power divergence is cancelled at energy scale $\mu$. But if we go to another scale with $s\neq \mu^2$, we will have $\Lambda^2(\mu^2-s)$, which is divergent and invalidates perturbation theory. This is very bad and it means we cannot cancel the infinity if we only have this type of counterterms. Fortunately, there is another possibility: we may be able to add counterterms that gives $s\Lambda^2$ (for example, by field strength renormalization in the theory with interaction $\phi^2\partial_\mu\phi\partial^\mu\phi$), which is able to cancel the power divergences at all scales. This is nice, and it is even nicer than the case with logarithmic divergences, because we can cancel power divergences identically at all scales but we can cancel logarithmic divergences only at a particular scale that we choose to be our renormalization scale. However, my concern is that because we can only add terms that is compatible with the symmetries of the system, we may not be able to add some counterterms that are necessary for cancelling some power divergences but inconsistent with the symmetries. My questions are: 1) Is this (the second type of counterterms) the only way that power divergences will be cancelled? 2) If yes, are we always able to set counterterms with respect to the symmetries of the system in this way, so that power divergences can always be cancelled at all scales (then we do not have to worry about power divergences at all!)? 3) If no, how should we cancel power divergences? Or if it cannot really be cancelled, what is its effect? By the way, one may say if we use dimensional regularization, power divergences appear in the same way as logarithmic divergences (as $1/\epsilon$ where $\epsilon=4-d$), so we can apply the method of dealing with logarithmic divergences to them. But I think this is too tricky and I want to face this problem with other regularization schemes which have a cutoff.


title: Spectral energy distribution fitting tools or routines

text: I have observed magnitudes and fluxes for an object in different wavelengths from optical to mm. Now I need a tool, routine or something like that to fit a spectral energy distribution (SED) and derive some physical parameters from the fit. I'm looking for a ready tool which can be run remotely or installed locally. Mind you, I'm aware of this: http://caravan.astro.wisc.edu/protostars/


title: Different definition of SL(2,R) algebra?

text: I'm looking into $SL(2,\mathbb{R})$ group and it's algebra. I found on line that the $sl(2,\mathbb{R})$ algebra is given by the two by two real matrices of trace zero. This Lie algebra has dimension three; a standard basis is given as $$X=\begin{pmatrix}1 & 0\\\ 0 & -1\end{pmatrix}, Y=\begin{pmatrix}0 & 1\\\ 0 & 0\end{pmatrix}, Z=\begin{pmatrix}0 & 0\\\ 1 & 0\end{pmatrix}$$ with commutation relations $[X,Y]=2Y$, $[X,Z]=-2Z$, $[Y,Z]=X$, and Jacobi identity is satisfied (I calculated via Mathematica). Now, in an article about Kerr/CFT correspondence, the Near-Horizon Extreme Kerr (NHEK) geometry, has an enhanced $SL(2,\mathbb{R})\times U(1)$ isometry group, with Killing vectors that generate $SL(2,\mathbb{R})$ group \begin{equation} \tilde{J}_0=2\partial_\tau \end{equation} \begin{equation} \tilde{J}_1=\frac{2r\sin\tau}{\sqrt{1+r^2}}\partial_\tau-2\sqrt{1+r^2}\cos\tau\partial_r+\frac{2\sin\tau}{\sqrt{1+r^2}}\partial_\varphi \end{equation} \begin{equation} \tilde{J}_2=-\frac{2r\cos\tau}{\sqrt{1+r^2}}\partial_\tau-2\sqrt{1+r^2}\sin\tau\partial_r-\frac{2\cos\tau}{\sqrt{1+r^2}}\partial_\varphi \end{equation} with algebra that satisfies \begin{equation} [\tilde{J}_0,\tilde{J}_1]=-2\tilde{J}_2,\quad [\tilde{J}_0,\tilde{J}_2]=2\tilde{J}_1,\quad [\tilde{J}_1,\tilde{J}_2]=2\tilde{J}_0 \end{equation} and Jacobi identity is also satisfied. Now, my question is, since this is $SL(2,\mathbb{R})$ group, shouldn't the algebra be the same? That is, shouldn't Lie brackets be identical in the first and second case? Why is there a difference? In one case I have $[X,Y]=2Y$, and in other $[X,Y]=-2Z$ basically. Is this because of how we defined the generators? I'm kinda confused because how do I know the latter are indeed generators of $SL(2,\mathbb{R})$. I mean, all I need for Lie algebra is to have the basic axioms fullfilled, and that's it, right?


title: How to get a $\mathcal{N}=2$ SuperYang-Mills Lagrangian from a quiver

text: How can one write down the $\mathcal{N}=2$ SuperYang-Mills Lagrangian given a quiver graph? For concreteness consider the quiver $$(2)-(4)-[6]$$ where the node $(2)$ corresponds to a $U(2)$ factor of the gauge group, the $(4)$ node is a $U(4)$ factor and the $[6]$ node is a $SU(6)$ global symmetry (flavour). The lines correspond to two $\mathcal{N}=2$ hypermultiplets charged in the fundamental representations of one node and the antifundamental representation of the other node into which the line ends. In particular, how can one read the superpotential from this graph? How can one read the Kahler potential? Is this latter one always assumed to be canonical? In case you would like to choose an easier quiver to make an easier example it is perfectly fine.


title: Change of operator in the Hamiltonian

text: We are told that the particle has mass m and charge e and is moving in 2 dimensions. The position operator $\mathbf{X}=(X_{1},X_{2})$ and momentum operator $\mathbf{P}=(P_{1},P_{2})$ We are given that the Hamiltonian is: $H=\frac{1}{2m}((P_{1}+\frac{1}{2}eBX_{2})^{2}+(P_{2}-\frac{1}{2}eBX_{1})^{2})$ Where $B$ is a constant and $eB\neq 0$. We are required to show that the energy levels are of the form $(n+\frac{1}{2})\frac{\hbar\left | eB \right |}{m}$. My thoughts so far: the energy levels look fairly similar to that of a harmonic oscillator, so I'm thinking I may need to define new operators to try and get to a Hamiltonian that looks more like that. I'm just not sure how I go about doing that? Thanks for your help in advance!


title: How does Newtonian mechanics explain why orbiting objects do not fall to the object they are orbiting?

text: The force of gravity is constantly being applied to an orbiting object. And therefore the object is constantly accelerating. Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas? I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it?


title: Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

text: I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and possible non-commutative rules are mentioned too. I do not get it: given a path $q(t)$ and its variation $\delta q(t)$, the equivalence between the variation of the derivative $\delta \dot{q}$ and derivative of the variation $\dot{\delta q}$ seems to me a fact descending straight from calculus, not an arbitrary choice.


title: Is it possible to extract the index of refraction from reflection/transmission measurements like this?

text: I'm trying to manipulate some data to see if my analysis method is reliable: I want to use transmission and reflection measurements within a certain wavelength range to get the index of refraction (real _and_ imaginary parts) of a material of very well established index of refraction, like regular silica glass. The information I get out of the apparatus is the normalized transmission $T(\lambda)$ (the intensity transmitted through the material divided by the intensity transmitted through only air) and the normalized reflection $R(\lambda)$ (the intensity reflected off the material divided by the intensity reflected off a nearly perfect reflector). I also have the width of the sample, and the light is at normal incidence so I don't need to worry about any angle stuff or polarization. So from what we learned in class the standard procedure is to write out the fields in each medium and make the boundary conditions ($E$ continuous and $dE/dx$ continuous) match up at the boundaries. My notation is visible in this diagram: ![slide](http://i.stack.imgur.com/Ijk5D.png) Where $k_1$ and $k_2$ are determined from $n_1$ and $n_2$ through $k = \frac{\omega}{c}n$. Applying the boundary conditions gives us 4 equations for the 5 coefficients and allows us to get $F/A$ and $B/A$ (which are the electric field amplitudes of the transmitted and reflected waves, respectively) in terms of only $k_1$,$k_2$, and $L$. Now for a given wavelength, $T = \frac{|F|^2}{|A|^2}$ and $R = \frac{|B|^2}{|A|^2}$. So, that gives me two equations for two unknowns (the real and imaginary parts of $n_2$). So, is there any reason this shouldn't work for solving (numerically, definitely not analytically) for both parts of $n_2$ if I have $k_1$, $L$, $T(\lambda)$ and $R(\lambda)$?


title: Could this mean that Schrodinger's Cat thought experiment is false?

text: Since it is uncertain on whether or not the cat could be dead or alive, it is concluded that it depends on the observer to make it either dead or alive. But lets look at a clock. It doesn't take observance of the gears to make it function. The clock ticks despite there being an observer. Could this mean that the famous thought experiment is false?


title: Hamilton-Jacobi Equation

text: In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and hence I do not understand how we can partial differentiate $S$ with respect to time? A simple example would also be helpful.


title: What exactly causes an increase in resistivity?

text: > Explain in terms of charge carriers why the resistance of copper is less > than that of CuSO4 solution, which is less than that of hydrogen gas This is the question directly asked for homework, and I'm not quite sure if I have got the right end of the stick, nor how to answer it. I understand that 'charge carriers' in metals are delocalised electrons, positive and negative ions in salt solutions and ions & electrons in gases when they are made into plasmas by heating. Is the increase in resistivity down to size of charge carriers, and/or how much they are vibrating due to temperature (i.e. for gases to conduct they must be very hot)? I know this question is a bit stupid but neither my teacher nor the textbook are very helpful, and I would really appreciate any help. Many thanks!


title: Does a scale model with 1/2 the linear length have 1/8 the mass?

text: If a scale model has 0.5x the original length in all directions, should its mass be 1/8th of the original mass?


title: Does a celestial system exhibit a collective magnetic field?

text: Sol exhibits a magnetic field, most of the planets in orbit around Sol exhibit a magnetic field - strong and weak both. Does the solar system as a whole exhibit a magnetic field? Does the paradigm scale upwards to larger celestial systems such as a cluster or galaxy?


title: Do an action and its Euler-Lagrange equations have the same symmetries?

text: Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations. Can it happen that the equations of motion derived by this procedure have different kinds and/or numbers of symmetries than the action one has started with? And if yes, are there underlying principles that state why which kind of symmetries the action does not have can emerge in corresponding EOMs or which kind of symmetries of the action can potentially disappear in the EOMs derived from the Euler-Lagrange equations?


title: Can path integrals be used to understand entanglement?

text: I like path integrals. I prefer to try to understand quantum phenomena in terms of path integrals rather than Hamiltonian mechanics. However, most of the standard texts on quantum mechanics start from Hamiltonian mechanics and mention the path integrals as a side note. This makes it difficult for me to gain insight into common phenomena (with a few exceptions like certain areas of optics) from the perspective of path integrals formulations. I am aware of some exceptions to this rule (books on path integrals in particular), but I have only seen entanglement explained from the perspective of eigenstates of a Hamiltonian. I was wondering how entanglement would be interpreted if you started from a path integral formulation. I would be more than happy to hear from a quantum field theoretic perspective as well, but the usual quantum mechanical path integral would certainly satisfy my curiosity.


title: Turboshaft Turbine Mathematical Model

text: Are there any simplified mathematical models for how two gas coupled turbines (also called a free power turbine) should interact with one another as the speed of the driving turbine changes. (i.e.) How is the speed of a turboshaft engine's power turbine affected by the speed of it's gas turbine?


title: How to evaluate commutator with angular momentum?

text: I need to evaluate the commutator $[\hat{x},\hat{L}_z]$. I believe the $L_z$ is referring to the angular momentum operator which is: $L_z = xp_y - yp_x$ using this relationship i end up with: $[x,L_z] = x(xp_y - yp_x)-(xp_y - yp_x)x$ my next step is substituting in for the p operator but i still dont get anywhere. any suggestions???


title: Einstein Field Equations and Electromagnetic Stress-Energy Tensor

text: My question is: if we write Einstein field equations in this form: $$R_{\mu\nu} - \dfrac{1}{2}g_{\mu\nu}R=8\pi \dfrac{G}{c^4}T_{\mu\nu}$$ Then the left hand side is one statement about the geometry of space-time and the right hand side is one statment about the distribution of momentun and energy right? My point is: what if we use the electromagnetic stress-energy tensor as the energy-momentum tensor? My thought was: if I understood correctly, does this says that electromagnetic fields can also change the geometry of space-time making it bend as does the presence of matter? Sorry if it makes no sense, or if it's completely nonsense, it's just a thought that came out, I'm just starting to study those things.


title: My conundrum with Gauss’ law in electrostatics

text: If I use Gauss’ law to calculate the electric field outside of a charged (conducting or insulating) sphere or a point charge, the fields are the same. However, as a test approaches a point charge, the field becomes infinite whereas the test charge approaching the surface of the sphere reaches a finite value of $Q/4πϵ_0R^2$, where $R$ and $Q$ are the radius and charge of the sphere. Why wouldn’t the field also go to infinity as I get infinitesimally close to the surface of sphere? Those are just “point charges” on the surface. What ultimately led me to this question was trying to physical understand why Gauss’ law predicts the same field for a charge point and charged sphere (obviously $r > R$). I don’t understand it.


title: Calculating Pressure in CGS units

text: > calculate pressure in CGS units using following data : $$ \mbox{Specific > gravity of mercury},\gamma_{Hg}=13.6\\\ \mbox{Density of water}, > \rho=10^3{\rm kg/m^3}\\\ \mbox{Gravity}, g=9.8{\rm m/s^2}\\\ \mbox{height}, > h=75{\rm cm}$$ I know, $P=h\rho g$ i have also converted all the data into CGS units $\gamma_{hg}=13.6,\rho=10^6{\rm g/m^3},\ g=980{\rm m/s^2},\\\ h=75{\rm cm}$ After that I thought it is easy, i just have to substitute the value in the equation. But then i saw specific gravity of mercury (frankly i saw this for first time) i thought what is this. Now i don't know what to do. I think it it has some relation with $g$ as there is word specific gravity. I have no idea what to do.


What are some common errors when it comes to measuring hall voltage of a semiconductor? I've thought of two errors:   1. Adjusting the potentiometer so that the width of the conductor would start with 0 voltage was very difficult.   2. The magnetic field was strongest in the center but was weak on the sides of the wafer. This caused some uncertainty. Can anyone else think of some good errors for this lab? Lab: https://dl.dropboxusercontent.com/u/764231/Lab.pdf Prelab (for the diagram): https://dl.dropboxusercontent.com/u/764231/Pre- lab.pdf


title: Is carrier charge density and carrier mobility constant in a given material?

text: If we assume the semi-conductor is doped by a variable amount, is there some way I can look up carrier charge density for the material in a reference somewhere? What about carrier mobility?