title: Completing the trace in the Non-abelian Chern-Simons Term

text: I've been having a little trouble proving that [Page 138 of "Introduction to Topological Quantum Computation" by Jiannis K. Pachos]: $S_{CS} = \dfrac{k}{4 \pi} \int_{M} d^{3}x \, \epsilon^{\mu \nu \rho} tr \left( A_{\mu}\partial_{\nu}A_{\rho} + i\dfrac{2}{3} A_{\mu} A_{\nu} A_{\rho} \right)$ (with $A_{\mu} = A_{\mu}^{a} T_{a}$ and $T_{a}$ a generator of the gauge group's Lie alegebra) can be written as: $S_{CS} = \dfrac{k}{8 \pi} \int_{M} d^{3}x \, \epsilon^{\mu \nu \rho} \left( A_{\mu}^{a}\partial_{\nu}A_{\rho}^{a} - \dfrac{1}{3} f_{abc} A_{\mu}^{a} A_{\nu}^{b} A_{\rho}^{c} \right)$ where$f^{abc}$ is antisymmetric. * * * My attempt thus far: $tr \left( A_{\mu}\partial_{\nu}A_{\rho} + i\dfrac{2}{3} A_{\mu} A_{\nu} A_{\rho} \right)$ $= tr \left( A_{\mu}^{a} T_{a} \, \partial_{\nu} \, A_{\rho}^{b} T_{b} + i\dfrac{2}{3} A_{\mu}^{a} T_{a} \, A_{\nu}^{b} T_{b} \, A_{\rho}^{c} T_{c} \right)$ $= tr \left( A_{\mu}^{a} \partial_{\nu} A_{\rho}^{b} (T_{a} T_{b}) + i\dfrac{2}{3} A_{\mu}^{a} A_{\nu}^{b} A_{\rho}^{c} (T_{a}T_{b}T_{c}) \right)$ Using $tr(T_{a} T_{b}) = \dfrac{1}{2}\delta_{ab}$ and $(T_{a}T_{b}T_{c}) = \dfrac{1}{2}[T_{a}, T_{b}]T_{c} + \dfrac{1}{2}\\{ T_{a}, T_{b} \\}T_{c}$ we get: $= \left( \dfrac{1}{2}A_{\mu}^{a} \partial_{\nu} A_{\rho}^{a} + i\dfrac{1}{3} A_{\mu}^{a} A_{\nu}^{b} A_{\rho}^{c} \, tr \left( [T_{a},T_{b}]T_{c} + \\{ T_{a},T_{b} \\}T_{c} \right) \right)$ Using $$[T_{a}, T_{b}] = if_{abd}T_{d}$$ and $\\{T_{a}, T_{b}\\} = \dfrac{1}{N} \delta_{ab} + d_{abd}T_{d}$, where $d_{abc}$ is symmetric, we get: $= \left( \dfrac{1}{2}A_{\mu}^{a} \partial_{\nu} A_{\rho}^{a} + i\dfrac{1}{3} A_{\mu}^{a} A_{\nu}^{b} A_{\rho}^{c} \, tr \left( if_{abd}T_{d}T_{c} + \dfrac{1}{N} \delta_{ab} T_{c} + d_{abd}T_{d}T_{c} \right) \right)$ Using $tr(T_{A}) = 0$ and $tr(T_{a} T_{b}) = \dfrac{1}{2}\delta_{ab}$, we get: $$= \left( \dfrac{1}{2}A_{\mu}^{a} \partial_{\nu} A_{\rho}^{a} + i\dfrac{1}{3} A_{\mu}^{a} A_{\nu}^{b} A_{\rho}^{c} \, \dfrac{1}{2} \left( if_{abc} + d_{abc}\right) \right)$$ Now this would be precisely what we need if the $d_{abc}$ disappeared somehow but I'm not sure how to make that happen. If $A_{\mu}^{a} A_{\nu}^{b} A_{\rho}^{c}$ were anti-symmetric under permutation of the indices (abc) then if would cancel with the symmetric $d_{abc}$ but as far as I'm aware the $A_{\mu}^{a}$'s don't behave that way. Any help would be greatly appreciated. Cheers in advance.


title: What makes matter "solid"?

text: I'm a non-physicist with a basic high-school understanding of physics. I've always wondered what it is that makes things "solid". Why do molecules rebound from each other? There's just a bunch of tiny atoms there with (comparatively) large spaces between them. Why don't they just "slide" around or between each other?


title: Does opening or closing the window in a non-AC car in the summer affect how much heat is felt inside?

text: This will perhaps look like a very basic and trivial question. But I find it confusing. As an experience, when you are travelling in non-AC car in summer, have people felt if putting on or putting off the windowpane affect how much heat it feels inside to passengers. On one hand, I feel that heat can still come inside thru glass particles (This makes me ask if heat needs media to travel?) and so putting on windowpane should not reduce heat felt inside. But psychologically it feels like by closing the window, you are blocking heat. Also the windowpane will at least block the heated air from coming inside. But does this heated air really brings much heat apart from the heat coming through anyway or it actually in practical acts more like a ventillator thus reducing heat felt inside. I realize that this question may appear strange, but it pops up to my head everytime I travel in non-AC car in summer. I have tried to articulate it as well as I could. Do insulator windowpane glasses reduce heat felt inside? because they will transmit lesser heat inside. How do black shade glasses work?


title: Is a 5p to 5s transition possible in a multielectron atom?

text: Hey guys, I'm reading my physics book right now and I have a quick question about transitions in multielectron atom. I am posting a picture as well so you can see what I am talking about. How come the picture doesn't show a transition from the 5p state to the 5s state? I thought that as long as the different between the orbital quantum numbers was 1, then the emission could occur. In this case, 5s has l=0 and 5p has l=1, so the difference is one and I would think the transition would occur. Also in the picture, it doesn't show the transition of 4p to 4s even though the orbital quantum difference is also one. Is the picture wrong? ![enter image description here](http://i.stack.imgur.com/IgbO1.jpg)


I have a two-part question:   1. First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories". Here they give a general definition for the Chern-Simons action for a general $3$-manifold $M$. My question is if anyone knows of any follow-up to this, or notes about their paper?   2. To those who know the paper: They say that they have no problem defining the action modulo $1/n$ (for a bundle of order $n$) as $n\cdot S = \int_B Tr(F\wedge F)$ $(mod 1)$, but that this has an $n$-fold ambiguity consisting of the ability to add a multiple of $1/n$ to the action - What do they mean here? Also, later on they re-define the action as $S = 1/n\left(\int_B Tr(F\wedge F) - \langle \gamma^\ast(\omega),B\rangle\right)$ $(mod 1)$ - How does this get rid of the so-called ambiguity?  Basically my question is if anyone can further explain the info between equations 3.4 and 3.5 in their paper. Thanks.


title: Half-massive sphere in space

text: A sphere contains all the mass on one side while the other side is empty of matter. Both sides are separated by a strong material, you name it, that will keep the sphere together. Given that the mass side is more massive than the sun or at least very massive, how would it behave in respect to gravity? Would it spin?


title: Why does a rotating tire use the static, rather than the dynamic coefficient of friction?

text: The explanation I have heard of the difference between static and dynamic friction is that static friction is stronger because bonds form when one object is put on top of another object and these have to be overcome to get the movement started. For a rotating tire, although the point on the ground will be stationary for an instant, it would seem that bonds wouldn't have time to form. So, why isn't the dynamic coefficient of friction used?


title: How should I throttle my rocket to reach highest altitude?

text: _"Real world" problem._ > _Suppose we want to launch a rocket equipped with an engine which can be > throttled as we prefer. Suppose also that the amount of fuel burnt per time > is directly proportional to the upward force the engine exerts. How should > we throttle the engine to reach the highest altitude, taking into account > atmospheric drag?_ * * * _Physical formulation._ Let $m(t) = m_d + m_f(t)$ be the total mass of the rocket, where $m_d$ is its dry mass and $m_f(t)$ the mass of the remaining fuel at time $t$. Our engine provides a force directly proprtional to the amount of fuel burnt; so the force exerted at the moment $t$ will be $$ F = -\eta \dfrac{dm_f}{dt},$$ where $\eta$ is a positive constant which depends on the efficiency of the engine. For the drag force we use the fluid-dynamics formulation: $$ F_d = k(x) \dot x(t)^2, $$ where $x$ is our height and $k$ a coefficient which takes into account the change of density of the air with altitude. So by Newton's law we have $$\ddot x = \dfrac{F - F_d - mg}{m} = \dfrac{-\eta \dot{m_f} - k(x) \dot x ^2}{m_d + m_f}-g.$$ The problem has therefore become the following: Find a non-negative function $\theta(t)$ (the throttle, i.e. $\theta = - \dot{m_f}$) such that $$ \int_0^{\infty} \theta = m_f(0), \qquad \text{(we burn all the fuel, not more nor less)} $$ and such that the trajectory $x(t)$ solving the differential equation $$ \ddot x(t) = \dfrac{\eta \theta(t) - k(x(t)) \dot x(t)^2}{m_d + m_f(0) - \int_0^t \theta}-g$$ attains somewhere maximum value, that is for any other $\tilde \theta$ satsfying the previous requirements, its trajectory $\tilde x$ is such that $\sup_{t \in [0, \infty)} \tilde x(t) \leq \sup_{t \in [0, \infty)} x(t)$. * * * _Mathematical formulation and recap._ Given $g, \eta, C>c \in (0, \infty), k : [0, \infty) \rightarrow [0,\infty) $ a decreasing function, let \begin{align} S_{c,C} := \Bigg\lbrace f \in L^1([0, \infty)) \text{ s.t. } & \int_0^{\infty} f = c ,\\\ & f \geq 0 \text{ a.e.} ,\\\ & \exists x_f(t) \text{ s.t. } \ddot x_f(t) = \dfrac{\eta f(t) - k(x_f(t)) \dot x_f(t)^2}{C - \int_0^t f}-g \Bigg\rbrace. \end{align} Find $\theta \in S_{c,C}$ s.t. $$\forall \tilde \theta \in S_C, \sup_{t \in [0, \infty)} x_{\tilde \theta}(t) \leq \sup_{t \in [0, \infty)} x_{\theta}(t).$$


title: Example of two linearly independent, nowhere vanishing vector fields in $\mathbb{R}^{2}$

text: I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\mathbb{R}^{2}$. Is it necessary that these two vector fields commute? Would you give me an example for these two vector fields?


title: Tensions And Pulleys With Masses

text: The problem I am working on is: "A block of mass m1 = 1.80 kg and a block of mass m2 = 6.30 kg are connected by a massless string over a pulley in the shape of a solid disk having radius R = 0.250 m and mass M = 10.0 kg. The fixed, wedge-shaped ramp makes an angle of θ = 30.0° as shown in the figure. The coefficient of kinetic friction is 0.360 for both blocks." The provided diagram: ![enter image description here](http://i.stack.imgur.com/y0qm9.gif) Determine the acceleration of the two blocks. (Enter the magnitude of the acceleration.) Determine the tensions in the string on both sides of the pulley. * * * What I was wondering is why there are two different tension forces acting on the pulley? Could someone give me a descriptive answer? Also, does the mass of the pulley somehow affect the tension forces? Why exactly?


title: What does a de Broglie wave look like?

text: 1. What does a de Broglie wave look like?   2. Are de Broglie waves transverse or longitudinal?    3. Can they be polarized?    4. What about the de Broglie wave of a ground state neutral spin-zero Helium 4 atom?    5. What experimental evidence do we have that supports the detailed nature of a de Broglie wave?  I have always assumed that de Broglie waves were mathematically identical to electromagnetic waves, but I just realized there is no basis for this assumption, and in fact it must be false, unless there is an analogue to both the magnetic and the electric components of the electromagnetic wave. So what does a de Broglie wave "look like"?


title: What is difference between the different 'flavours' of neutrinos?

text: Moreover, how-come scientist know that muon-neutrino are different from electron-neutrino when they didn't even know what the difference was? Did they interact differently with other particles?


title: Electric potential energy and signs

text: I know that electric potential is negative near a negative charge and positive near a positive charge. But does this mean a small positive 'test' charge has a negative electric potential energy near a negative charge and a small negative charge would have a positive?


title: Reflection of light

text: > **Possible Duplicate:**   >  What is the difference between a white object and a mirror? When I look at a red object under white light, I see it as red because it absorbs the other colours and reflects the red light. When I wear black clothes in summer, I get hotter than if I wear white clothes because the black absorbs all colours of light and the energy associated with it. So my question is, what makes the difference between a white piece of paper and a mirror? The white paper doesn't absorb any colours, that's why it looks white. But what makes that different from a mirror, which also reflects the light that hits it? Or is my basic understanding of light and colour wrong? I'm not exactly experienced in Physics, I just like knowing how some things work.


title: Rolling ball which slips

text: A bowling ball of mass $M$ and radius $r_0$ is thrown along a level surface so that initially ($t = 0$) it slides with a linear speed $v_0$ but does not rotate. As it slides, it begins to spin, and eventually rolls without slipping. How long does it take to begin rolling without slipping? I am confused with the textbook solution. I understand that kinetic friction first acts on the ball. The linear velocity is given by $$ V_{CM} = v_0 - \mu g t$$ The angular acceleration is given by $$I \alpha = \Sigma \tau \implies \alpha = 5 \mu_k g/2r_0 $$ The book states this angular acceleration is constant (presumably from $t=0$). How can one arrive at this conclusion? I'm confused how this can this be true if the force causing the torque _changes_ from kinetic friction to static friction (when the ball starts to roll without slipping)? See the full book solution here. Source: Giancoli's _Physics for Scientists and Engineers_.


title: Intuition behind Work

text: I have a doubt in understanding the intuition behind the concept of work. First of all, I think this isn't duplicate, I've searched on the site, and the closest thing I've found was this post which is different than what I'm struggling here to understand. Well, let me show an example of what I mean: momentum. For me, defining the "quantity of motion" to be $p = mv$ is pretty obvious, the amount of motion should be proportional to how fast the thing is moving and how much thing that's making the movement. Position, velocity, acceleration, those all have obvious interpretations too. But what about work ? I've already heard that "work is the amount of force that was used to make the movement", however, if that's the case, shouldn't we define work simply as the component of the force in the direction of movement ? Why do we multiply by distance ? In other words, how do we get a "feeling" of what's work intuitively so that the equation defining it simply states mathematically our intuition ? Thanks very much in advance.


title: Why is the ground state of the ferromagnetic tetrahedron threefold degenerate?

text: I'm preparing a presentation on Spin-Ice, but something's been bugging me for a while. On the Wikipedia page for Geometrical Frustration, it says the following about easy spins on a tetrahedron with ferromagnetic interactions: > There are three different equivalent arrangements with two spins out and two > in, so the ground state is three-fold degenerate. ![](http://i.stack.imgur.com/7CRYL.png) I just don't understand why.   * We're considering easy spins with ferromagnetic interactions, so any ground state needs 2 spins pointing towards the center (in) and 2 spins pointing away from the center (out).    * Given a tetrahedron, I can think of 6 different ways to distribute 2 spins in and 2 spins out on the vertexes. $(iioo,ioio,oiio,iooi,oioi,ooii)$    * You can rotate any of the above states to reproduce another, which might mean they are not different.  **But then, why does wiki state there are three _different_ arrengements?** **Note** Each of the above configurations has a total magnetic moment in a different direction. We can place our $(x,y,z)$ axis so that each one of the first three cases has a total moment in the positive direction of one of the axis, and each of the last three has a total moment in the negative direction of one of the axis.


title: Difficulties with understanding total entropy change and unavailabillty

text: Of course, I know the fact that the entropy of an isolated system never decreases. Neverthless what makes me confused about the entropy(or change of entropy) of an isolated system is the explanation used to tell the unavailable heat/work in a irreversible process. With a closed system and its surrounding imagine that the system expanding isothermally(at T) then for 1) reversible process Surely the total change of entropy is zero. Here I say the process is from a state A to a state B and the heat system absorbs is q. 2) irreversible process (from state A to B) With very large surrounding compared to the system, the temperature of the surrounding can be assumed to be constant and same as that of system. Because the entropy is a state function the entropy change of the system is still q/T. It is obvious that q and q'(heat absorbed by system in irreversible process) are different. The problem is that a lot of books and lecture notes explain that the entropy change of the surrounding is -q'/T. By the definition of entropy I cannot handle this statement because -q' is the heat extracted from the surrounding "irreversibly". If q' can be treated as a heat transferred reversibly why not use this quantity for the system? In the view of 2nd law of thermodynamics the difference between the magnitudes of entropy change of the system and the surrounding should exist in irreversible process but if we can say that -q'/T is the change of entropy of the surrounding we can also say that the change of entropy of the system is q'/T. As a result the total entropy change will be zero, which doesn't obey the law of thermodynamics. Books/lecture notes examples for this problem: Introduction to the thermodynamics of materials, D. R. Gaskell, 5th ed., p.44. & A note for lost work In addition if the entropy of the system is a state function can't we regard the entropy of the surrounding as a state function? The entropy changes of the system are same for both cases, reversible and irreversible processes because the first and final states are unchanged. In this situation I think the surrounding also have the same first and final states for both reversible and irreversible processes. In this way the total entropy change will be zero since q/T - q/T = 0. If reversible this result is not strange, but state function doesn't depend on paths but states... In short, my troubles are   1. Many books/notes argue that q-q' is the heat unavailable or consumption of free energy, and I think this statement is plausible but the reason that we can take q' is not permissible thing. q' is not the reversibly transferred heat. It has irreversible property and thus we can't use q' for calculating entropy.   2. Entropy is a state function and this is true for both entropies of systems and surroudings. If we take an irreversible path between states(A to B) the total entropy change would be zero for we can always draw a reversible path between the states(also A to B). However this violates the law of thermodynamics.


title: What are the conserved charges related to the Virasoro generators?

text: I have just learned from reconsidering my demystified book, that when conformally maping the worldsheet of a closed string to the complex plain by using the transformation $z = e^{\tau + i\sigma}$ and $\bar{z} = e^{\tau - i\sigma}$ the Virasoro generators can be calculated as $$ L_m = \frac{1}{2\pi i} \oint \frac{dz}{z}z^{m+2}T_{zz}(z) $$ and $$ \bar{L}_m = \frac{1}{2\pi i} \oint \frac{d\bar{z}}{z}\bar{z}^{m+2}T_{\bar{z}\bar{z}}(\bar{z}) $$ with $T_{zz}(z)$ and $T_{\bar{z}\bar{z}}(\bar{z})$ being the holomorphic and antiholomorphic component of the energy-momentum tensor. By the deformation of path theorem, the contour of integration can be shrinked and expanded in the complex plain, which means that the Virasoro generators are invariant under a time translation because the mapping to the complex plain has transformed the time coordinate into the radial coordinate. So these Virasoro operators should commute with the Hamiltonian and they have conserved charges related to them. My question now is: What are these conserved charges and what is their physical meaning?


title: The direction of electric field in a diode

text: ![PN-junction diode](http://i.stack.imgur.com/pqtaI.png) Why electric field pointing to left? (The middle area is depletion region.) (This is without any external electrical voltage applied.)


title: What are the consequences of relativistic angular velocities?

text: If I take a rod of some radius $r$ and length $L$, and I spin this rod with angular velocity $\omega$. How would the geometry of the rod appear to an observer as one converges to $c$? What are the consequences of this to, say, electrons in a solenoid?


title: How a spacecraft travelling near light speed avoid asteroids?

text: How would a spacecraft traveling near light speed avoid a (relatively dense group of) asteroids? Or suppose such spacecraft is designed, how would the physics work for steering it inside such a "cloud" of asteroids? If it simply by designing the spacecraft to have huge acceleration, wouldn't that mean lots of constraints on the material the spacecraft can be made of?


title: Vacuum to vacuum transition amplitude

text: I have two questions about Vacuum to vacuum transition amplitude.   1. Can any particle stay in $|0\rangle$?   2. I was studying this topic from Srednicki's QFT book. He writes in eq.$(6.22)$ $$\langle0|0 \rangle_{f,h} = \int \mathcal{D}p \mathcal{D}q \, \exp \Big[i\int_{-\infty}^\infty dt (p \dot{q} - H_0(p,q) - H_1(p,q) + fq + hp \Big]$$ $$ =\exp \Big[-i \int_{-\infty}^\infty dt H_1 \Big(\frac{1}{i} \frac{\delta}{\delta h(t)}, \frac{1}{i} \frac{\delta}{\delta f(t)} \Big) \Big] \times \int \mathcal{D}p \mathcal{D}q \,\exp \Big[i\int_{-\infty}^\infty dt (p \dot{q} - H_0(p,q) + fq + hp \Big] $$ Here in the second line the arguments of $H_1$ suddenly changes from $p$,$q$ to $\frac{1}{i} \frac{\delta}{\delta h(t)}, \frac{1}{i} \frac{\delta}{\delta f(t)}$. How can I derive this line from the first line?


title: Relation between symmetries and Killing vectors by Weinberg

text: In his book, "Gravity and Cosmology", Weinberg talks about relations between homogeneous metric spaces and Klling vectors. First he says about infinitesimal isometrics $$ x^{\alpha}{'} = x^{\alpha} + \varepsilon k_{\lambda}{}^\alpha. \qquad (1) $$ Then he gets Killing equations and a fact that each of Killing vectors $k_{\lambda}$ can be represented near point X as Taylor series $$ k_{\lambda}{}^\mu (X, x) = A_{\lambda}{}^\nu (X, x) k_{\nu}{}^\mu + B_{\lambda}{}^{\rho \delta}D_{\rho}(X, x) k_{\delta}{}^\mu, \quad A, B \neq f(k_{\lambda}). $$ Then he gives the definition of homogeneous spaces: they are the spaces which refer to the existence of isometries $(1)$ which make parallel shifts from point X to any point of its surroundings. By the other words, he says, it means that this property of space requiers that the **metric of corresponding space must have Killing vectors which can take all possible values at any given point X**. So it leads to the statement that **in N-dimensional space we can choose N Killing vectors which have the property** $$ k_{\lambda}{}^\mu(X, X) = \delta_{\lambda}^{\mu}. $$ I don't understand words in italics. Can you explain them to me?


title: Work done by rolling vs skidding friction force

text: Two identical bicycles having equal weight riders are traveling along a level road adjacent to each other with the same non-zero velocity. Bike A, (the "skidder"), applies the rear brake strongly enough to lock up the rear wheel which begins to skid. Bike B, (the "roller"), also applies the rear brake, but not so strongly and in such a manner that the same frictional braking force is achieved at the rear wheel. Bike A and Bike B both slow at the same rate and come to a stop at the same time. The time spent and distance traveled as they slow to a stop is the same. The only difference is that Bike A skids to a stop with a locked up rear wheel and Bike B rolls to a stop without skidding at all. For both of the following two questions, assume that there is zero friction in the bicycle drivetrains and zero air resistance and zero rolling resistance. Also, ignore the infinitesimal energy transferred from the bikes to the Earth.   1. Bike A (the "Skidder"): From the ground reference frame, does the (kinetic) skidding friction force of the road acting on the tire do any work on the bike as a whole?   2. Bike B (the "Roller"): From the ground reference frame, does the (static) rolling friction force of the road acting on the tire do any work on the bike as a whole? My understanding is that the rolling friction force is constrained and can do no work on the _wheel_ of the bike, but does it do work on the bike as a whole? And if so, how?


title: Level of calculus required for physics

text: First time for me here so kindly let me know if I violate the rules - especially if this is a duplicate. After reading the page how to become a good theoretical phycist, I started a serious revision of calculus. For exercises, Math.SE is a good place. I was attempting exercises from this user (question s/he asked and answered) and realized I couldn't solve the majority of them. So the question is what is the level of mastery of calculus required for physics? What is(are) the best book(s) for that? I'm interested in limits, integration and infinite series for now. Background: standard mathematics with limits, integration, infinite series, differential equations (ODEs and PDEs) and numerical analysis and optimization. Everything was fast paced because professors wanted to finish the program as fast as possible ( during 3 semesters ) so I really never got good at any of the listed above.   Note: self-study kind of books will be most helpful.


title: evidence on the equation of state for dark energy?

text: If dark energy contributes mass-energy density $\rho$ and pressure $p$ to the stress-energy tensor, then you can define $w=p/\rho$, where $w=-1$ gives a cosmological constant, $w<-1$ gives a big rip, and $w<-1/3$ if we want to use dark energy to explain cosmological acceleration. The WP "Big Rip" article cites a paper that dates back to 2003 http://arxiv.org/abs/astro-ph/0302506 , which states that the empirical evidence was at that time only good enough to give $-2 \lesssim w \lesssim -.5$. Have observations constrained $w$ any more tightly since 2003? I've heard many people express the opinion that $w<-1$ is silly or poorly motivated, or that "no one believes it." What are the reasons for this? Other than mathematical simplicity, I don't know of any reason to prefer $w=-1$ over $w\ne -1$. Considering that attempts to calculate $\Lambda$ using QFT are off by 120 orders of magnitude, is it reasonable to depend on theory to give any input into what values of $w$ are believable?


title: Physical Interpretation of Poynting Vector

text: I'm looking for a physical interpretation of the Poynting Vector. I understand that it should be thought of as an energy flow due to the electromagnetic field, but would I be correct in saying that in some sense it's equivalent to a photon flow/packet? After all, for an electromagnetic wave, the poyntig vector yields the direction of propogation, or the direction in which the photon is moving. Does this analogy apply in general?


title: Virasoro TT OPE in Polchinski's book

text: I'm trying to understand eq. 2.2.11 in Polchinski's first book. He's computing $$:\partial X^\mu(z)\partial X_\mu(z): :\partial' X^\nu(z')\partial' X_\nu(z'):$$ Now, I understand why this expression can be written as $$\text{expression above}~=~:\partial X^\mu(z)\partial X_\mu(z)\partial' X^\nu(z')\partial' X_\nu(z'):\quad - 4\alpha'/2 (\partial\partial' \ln|z-z'|^2):\partial X^\mu(z)\partial'X_\mu(z'): + 2\eta_\mu^\mu(-\alpha'/2 \partial\partial'\ln|z-z'|^2)^2.$$ However, he then states to do a Taylor expansion inside the normal ordering to get the OPE in standard form, i.e. $$\sim~ \frac{D\alpha'^2}{2(z-z')^4}-\frac{2\alpha'}{(z-z')^2}:\partial'X^\mu(z')\partial'X_\mu(z'): - \frac{2\alpha'}{z-z'}:\partial'^2X^\mu(z')\partial' X_\mu(z'): + \text{non- singular terms.}$$ I don't understand the last step. How exactly does he insert the Taylor expansion? Could someone please illuminate? For instance, I don't see where the first term goes? Does that disappear when he Taylor-expands?


title: Bound state in a potential well?

text: Reading from http://quantummechanics.ucsd.edu/ph130a/130_notes/node151.html It says: > This means that the solutions separate into even parity and odd parity > states. We could have guessed this from the potential. Why A or B is zero, then we will have even and odd parity? What does this sentence mean? And then, from the graph of the link above, it says: > Where the curves intersect (not including the asymptote), is an allowed > energy. There is always one even solution for the 1D potential well. In the > graph shown, there are 2 even and one odd solution. The wider and deeper the > well, the more solutions. The solution of the graph shows the energy. How do we know a wider and deeper well will have more solutions?


title: How much solar energy hits a rooftop in Israel?

text: I was asked to today about the available plans (in Israel) to install photo- voltaic receptors on the roof and sell the energy. This is a legitimate plan and there are several companies that do this. The claim is that the initial investment is repaid after roughly 5 years which sounds too optimistic to me. I want to carry the calculation myself and check the results. Where can I find some data about the light flux that hit the earth in a typical year? _note_ I'm not sure, since this is not my specific field of expertise, about the commonly used term.


title: Real world usage; pump flow vs pressure vs pump head, what is more useful in this situation?

text: I am currently making a water cooling system for my computer and decided to see if I could get some better help here than what people think they know on the computer forums. So for my cooling system I have three, what are called, water blocks. They have a inlet and an outlet and are the biggest source of restriction. I specifically have three, and they each have 1/4in input and output barbs, I then connect them all with two manifolds. The manifold is made of 3 PVC T fittings and an end cap. So all the T's are setup such to look like this TTT . One of the ends of the tops on the T's is capped and the other is the input/output, while the bottom of the T's goes to and from the water blocks. (I hope this is very easy to visualize, if you cannot ask me and I will try to explain more) The input for each manifold is 1/2in tubing and the T's run to x3 1/4in tubing lines. So the water enters in a 1/2in tube, is split to x3 1/4in tubes, then combines back into a 1/2in output tube. Each waterblock is responsible for, at max, 190watts of heat to dissipate. So the entire loop must remove ~600watts of heat + the pumps heat = ~625 watts. This is where the radiator comes in, I found a nice chart that tells me the size I need to remove X watts from a system so I know I have the right size radiator. The radiator has 1/2in tubing inputs and outputs (but it is restrictive cause of the inside geometry. So my question is, what should I consider in a pump when trying to find one for this system? People talk of flow rate, vs system pressure, vs pump head. I know the head is how hard the pump can push with no restriction in its flow. So I figure pressure is combined in the pump head figure. What no one talks about though is how to find out how much flow and pressure is needed for the water cooling loop to work as it should. Even if I have the right size radiator to remove the amount of heat I have it is worthless if the water is not moving in the right manner. So can I possibly buy a pump that has too much flow; how can I figure the minimum amount of flow needed for my loop to work? EDIT: Ok I have found out a little bit more.. So I figure part of what I need to know is the bernoulli equation, which will describe the flow and pressure of the water at a certain point in the system. What I guess I need to do is equate the amount of heat water can remove at a specific velocity. So each water block gives off 190watts, and water has a thermal conductivity of .6Watts/m*K at 0C. How do I equate these two figures to figure out how much flow and head my pump should have?


title: Does a ski racer with a larger mass have an advantage?

text: Does a ski racer with a greater mass have an advantage over a racer with a lesser mass? If mass of one racer is 54 kg and the mass of a more slender racer is 44 kg I know the speed at which they descend should be equal if they were to fall in a vacuum. How does friction force, air resistance and momentum play a part in determining the advantage or disadvantage of larger mass in ski racing?


title: Skiing downhill

text: The other day on skiing holiday we've been arguing about whether an adult has weight advantage over a child when skiing downhill. I was claiming that gravity is a constant regardless of object's weight. I was claiming that weight doesn't affect skier's speed. I can still remember primary school experiment where we dropped same sized objects of different matter in vacuum. Styrofoam and iron ball of the same size and they both fell equally. It is true that skier's maximum speed is determined by his air drag (and snow/ski resistance) and heavier objects have higher persistence making them move through resistive medium easier. But compared to a child an adult would have more drag due to larger _surface_ so I suppose this somehow evens out. I was basically claiming that weight doesn't matter at all but that other factors are much more important when it comes to skier's speed like: ski stiffness (weight force much better distributed over the whole ski surface than with softer skis) and ski sliding surface smoothness and especially skier's position (to reduce drag as much as possible). I also got a bit confused when I started to think about force vectors where greater weight results in longer vectors resulting in stronger movement pull. ## Question If we approximate is it safe to say that a child can ski just as fast as an adult? Or that a thinner adult would ski faster than his same high companion that is overweight due to larger drag surface? Or does his weight benefits his speed? **What is the correct answer here?**


title: How do I find angular and linear velocity after normal force and "infinite" friction force?

text: Look at this picture: http://i.imgur.com/mY8ShkV.png Here we have a ball resting on top of a platform (the contact point is marked red). I know the normal vector (marked green). I also know the mass, radius, and current angular and linear velocity of the ball. Let's say I set the new angular velocity after friction which is always big enough ("infinite") has been applied to: $V_a = V_l \times rN$ where $V_l$ is the linear velocity, $r$ is the radius of the ball and $N$ is the normal vector. Does this math check out, is this correct? Because this model doesn't work if I leave the ball leaning on the side of the platform where it should obviously roll of the platform and fall. But it doesn't. What is the correct mathematical model for simulating "infinite" friction? I suppose you will also have this problem when you want to simulate finite friction and the friction doesn't give it's max force.


title: About free quarks and confinement

text: I simply know that a single free quark does not exist. What is the reason that we can not get a free quark? If we can't get a free quark then what is single-top-quark?


title: Sliding ruler on table top

text: I've got a standard foot-long flat plastic ruler (about an inch wide) and a desk with a smooth formica-like surface. When idly passing time reading on the computer, I will pick up one side of the ruler (keeping one of the four edges still leaning on the table) and drop it to see how far it will slide. It can travel about a foot before stopping. Why is it sliding? It doesn't slide nearly as far if I give it a little push. Has it trapped some air temporarily between the ruler and the table? And does it stop because it is hitting some irregularities on the table? Or because it has rolled off all of the trapped air molecules? Thanks.


title: Conservation of probability in Jaynes-Cummings model

text: I was reading about Jaynes-Cummings model. I've studied Bachelor level QM.   In the interaction picture the solution with _rotating wave approximation_ and _initial conditions_ $C_g(0)=0$ and $C_e(0)=1$ is of the form $$|\Psi(t) \rangle = \sum_{n} \\{ -iC_{n-1} \sin(\lambda \sqrt{n} t)|g\rangle + C_n \cos(\lambda \sqrt{n+1} t)|e\rangle \\} |n \rangle$$ where $\sum_{n}C_n |n\rangle$ is the state of the field at $t=0$. From this, to me it seems like the probability is not conserved --- I mean, should it not be so that $$ |iC_{n-1} \sin(\lambda \sqrt{n}t)|^2 + |C_n \cos(\lambda \sqrt{n+1} t)|^2 = 1?$$ Clearly it isn't. Why? What does it mean?


title: How "to take" this integral?

text: When I learned anharmonic model of crystal, I read that considering anharmonic oscillations and Boltzmann distribution for the "atoms" of crystal we can get the dependence of distance between the "atoms" from a temperature as $$ \langle r \rangle = r_{0} + \alpha T. $$ As I understood the words below, it's like $$ \langle r \rangle = \frac{\int \limits_{0}^{\infty}re^{-\frac{U}{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{U}{kT}}dr} \approx \frac{\int \limits_{0}^{\infty}re^{-\frac{U_{0} + a(r - r_{0})^{2} + b(r - r_{0})^{3} }{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{U_{0} + a(r - r_{0})^{2} + b(r - r_{0})^{3} }{kT}}dr} = |x = r - r_{0}| = \frac{\int \limits_{0}^{\infty}(x + r_{0})e^{-\frac{ax^{2} + bx^{3}}{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{ax^{2} + bx^{3}}{kT}}dr}, $$ and then - $$ \langle r \rangle \approx r_{0} + \frac{\int \limits_{0}^{\infty}xe^{-\frac{ax^{2} + bx^{3}}{kT}}dr}{\int \limits_{0}^{\infty}e^{-\frac{ax^{2} + bx^{3}}{kT}}dr}. $$ What can I do on the next step?


title: Derivation of the quadratic form of the Dirac equation

text: I am asked to derive the quadratic form of the Dirac equation in an electromagnetic field, $\left[\left(i\hbar \partial - \frac{e}{c}A\right)^2 - \frac{\hbar e}{2c} \sigma^{\mu\nu} F_{\mu\nu} - m^2c^2\right] \psi = 0,$ where $F_{\mu\nu} =\partial_{\nu} A_{\mu} - \partial_{\mu} A_{\nu}$ is the usual e-m field tensor and $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^{\mu}, \gamma^{\nu}].$ A hint in my textbook suggests that I left-multiply the Dirac equation, $\left[-\gamma^{\mu} \left( i \hbar \partial_{\mu} - \frac{e}{c} A_{\mu} \right) + mc\right] \psi = 0,$ by the expression $\gamma^{\nu} \left( i \hbar \partial_{\nu} - \frac{e}{c} A_{\nu} \right) +mc$ and use the commutation relations for the gamma matrices. I have been unsuccessful to this end. I get 0 = $\left[-\gamma^{\nu} \gamma^{\mu} \left( i \hbar \partial_{\nu} - \frac{e}{c} A_{\nu} \right) \left( i \hbar \partial_{\mu} - \frac{e}{c} A_{\mu} \right) + m^2 c^2 \right] \psi = - \left[( \gamma^{\mu} \gamma^{\nu} + 2 i \sigma^{\mu\nu}) \left( i \hbar \partial_{\nu} - \frac{e}{c} A_{\nu} \right) \left( i \hbar \partial_{\mu} - \frac{e}{c} A_{\mu} \right) - m^2 c^2 \right] \psi = -\left[ -\left(i\hbar \partial - \frac{e}{c}A\right)^2 + 2 i \sigma^{\mu\nu} \left( i \hbar \partial_{\nu} - \frac{e}{c} A_{\nu} \right) \left( i \hbar \partial_{\mu} - \frac{e}{c} A_{\mu} \right) - m^2 c^2 \right] \psi, $ where in the last step I have used $(\gamma^{\mu})^2=-\mathbb{1}$. From here I have tried expanding the expression $\left( i \hbar \partial_{\nu} - \frac{e}{c} A_{\nu} \right) \left( i \hbar \partial_{\mu} - \frac{e}{c} A_{\mu} \right)$ and rewriting the appropriate cross-term in terms of $F_{\mu\nu}$ but I don't know what to do with the other terms. I can't seem to extract the term $\frac{\hbar e}{2c} \sigma^{\mu\nu} F_{\mu\nu}$. Also, the sign of the $\left(i\hbar \partial - \frac{e}{c}A\right)^2$ term appears to be backwards. Can anyone help me with this?


title: Is there a conserved quantity that enforces planar orbits in central force motion?

text: From what I remember, one of the first steps in finding the equations of motion for an orbiting body is to argue that the body's motion has to be restricted to a plane, because the central force has no component perpendicular to the plane defined by the initial position and velocity vectors. (Or equivalently, because the direction of the total angular momentum has to be constant, since there is no torque.) This means that you can choose coordinates such that $\dot\phi = 0$, which makes the third term of this general Lagrangian vanish: $$L = \frac{1}{2}m\dot r^2 + \frac{1}{2}mr^2\dot\phi^2 + \frac{1}{2}mr^2\dot\theta^2\sin^2\phi - U(r)$$ For example, this is done in this derivation around equation (11). (Note: in this post I use the Zwillinger convention for spherical coordinates listed on MathWorld.) Now, I could be wrong, but I think there should be a total of three conserved quantities in this problem. If I use the argument in the previous paragraph to impose a coordinate system in which $\dot\phi = 0$, then I can easily identify the energy as the Noether invariant corresponding to time translation invariance, and the angular momentum as the Noether invariant corresponding to rotational invariance around the $z$ axis ($\theta\to\theta + \epsilon$). But I can't think of a third one, and I suspect that the restriction $\dot\phi = 0$ (by the reasoning of the first paragraph) eliminates that third conserved quantity. So what if I _don't_ use the physical argument from the first paragraph, so that I'm _not_ limited to $\dot\phi = 0$? From the general Lagrangian I've written above, if my intuition is right, I should be able to obtain three conserved quantities from Noether's theorem _alone_. Of course, the energy is one, and that's still easy to compute; another one should be angular momentum, although that no longer corresponds to $\theta\to\theta + \epsilon$ but to some more complicated transformation. So I guess my question has two parts,   1. **Prerequisite:** What is the symmetry transformation that generates angular momentum when $\dot\phi \neq 0$?   2. **Main question:** Is there another spatial symmetry that generates another conserved quantity? If so, what is it? P.S. I do know there's basically no practical value to this question, since you can always choose coordinates such that $\dot\phi = 0$, but I'm just curious.


title: Bohr Model of the Hydrogen Atom - Energy Levels of the Hydrogen Atom

text: Why the allowed (stationary) orbits correspond to those for which the orbital angular momentum of the electron is an integer multiple of $\hbar=\frac {h}{2\pi}$? $$L=n\hbar$$ **Bohr Quantization rule of the angular momentum** that leads to energy levels of the Hydrogen atom: $$E_n=-\frac {Z^2}{1+\frac {m_e}{M_p}} \frac {\Re}{n^2}$$ the question is **Why quantum numbers are natural numbers**? $$E=nh\nu$$ $$n=0,1,2,3....$$


title: Reference for the ${\cal N}=3$ Chern-Simons Lagrangian at general $N_c$, $N_f$

text: I was wondering if someone could give me a reference where someone has explicitly written the Lagrangian for ${\cal N}=3$ $SU(N_c)$ Chern-Simons theory coupled to $N_f$ fundamental hypermultiplets. Any degree of explicitness would be great given that I haven't been able to trace it anywhere. If its written out in component fields with the matter and multiplet content made clear then thats really great! Reference to any other Lagrangian which is "close" to this would also be fine.   * Appendix A on page 31 of this paper seems to do what I want but the notation is highly unusual as in that $s_{ab}$ and the symplectic form and the $US_p(2N_f)$, the $Q,\tilde{Q}$ (and the $A1$, $A2$) are very unfamiliar coming from a Wess-and-Bagger and Weinberg-vol-3 backgorund. May be it will help if someone can reference instead a pedagogic background literature which will help parse the statement in this appendix.    * Lastly is this theory somehow automatically superconformal?


title: Representation of indistinguishability in quantum mechanics

text: I was wondering that if particles are indistinguishable in quantum mechanics, then why do we still express their states $\left| \uparrow \downarrow \right\rangle$, as meaning particle 1 (in the first position) in state "up" and particle 2 (in the second position) in state "down", and so this is a different state than $\left| \downarrow \uparrow \right\rangle$. We then go onto constructing a linear combination which is symmetric and so forth. But we are saying it is symmetric under exchange of particles 1 and 2. Why don't we just talk about two particles, why do we have to give them labels, even if they don't affect measurement? Why aren't $\left| \uparrow \downarrow \right\rangle$ and $\left| \downarrow \uparrow \right\rangle$ the same state? Surely, if they are truly indistinguishable, labels are just a redundancy. Is there some mathematical reason behind this, or is there a deeper physical reason?


title: Decay of Cobalt-60 isotope

text: How does the Gamma decay of Cobalt-60 occur? Motivation: A research team led by D. Habs made contributions to our understanding of the gamma decay of Ca-40 and Zr-90: http://prl.aps.org/abstract/PRL/v53/i20/p1897_1. You don't need to own this article to answer, but does anyone have any idea how to describe a similiar process for Cobalt-60? For those of you who don't have the article, I am asking about the perturbation background during the gamma decay of Cobalt-60.


title: How much Bicarbonate of Soda and Vinegar would I need to reach space?

text: So here is my problem - as part of my job I present some science demonstrations to children and one of the tricks I regularly use is the bicarb/acetic acid rocket. I thought the other day that a really fun fact to give them to go along with it would be the amount of bicarb and vinegar you would need to, say, escape the Earth's atmosphere. I'm a bit rusty but I think it's less trivial than it sounds. I think you need the rocket equation, which means you need the exhaust velocity, which is in turn quite complex to calculate. Does anyone have any simpler approaches which might work? A few nuggets which might help: 1 mole of bicarb produces 291 cc of CO2. Escape velocity is ~11000 m/s Any answers will be very appreciated!


title: Why isn't average speed equal to displacement over time?

text: I'm in an introductory Physics course and I need help! > During a one-hour trip, a small boat travels 80.0km north and then travels > 60.0km east. What is the boat's average speed during the one-hour trip? I found that the displacement is 100km. But I did not arrive at the right answer by dividing 100km/1hour. Why doesn't this give me the average speed?


title: Dirac Parenthesis and redefinition of constraints

text: As I know in the analyzing a constrained system always we have freedom to change a constraint by adding another constraint to it or by multiplying it by anything except zero. Because this modification does not change the constraint surface. Now assume that I change one the primary constraints. Let me explain by an example. Suppose that $\phi_i$ where i=1,...,n are all of the constraints. The complete Hamiltonian is \begin{equation} H=H_0+\lambda_n\chi_n \end{equation} where $\chi_n's(n<i)$ are primary constraints. Now I do a redefinition for example \begin{equation} \chi_1\rightarrow\alpha\chi_1+\phi_3. \end{equation} What do happen to the complete Hamiltonian and Dirac Parenthesis? As I understand if I substitute the new constraint into the complete Hamiltonian the equations of motion may change!


title: Who are "we", and what counts as a "question" in consistent histories?

text: If the preferred basis in quantum mechanics and/or choice of consistent histories in consistent histories is arbitrary, and can only be determined by the "questions we ask", just who exactly is this nebulous "we", and what counts or doesn't count as a "question"? This is a _dead_ serious question. PS: Suppose we are closed (i.e. no external interactions) quantum simulations in a mega quantum computer. We are simulated without being measured or observed or decohered in any way at the "substrate level" of our programmer/sysadmin gods. Then, these gods decide to unsimulate and uncompute the entire simulation. Suppose we asked some questions before being "uncomputed". Are these questions wiped out after uncomputation? These "gods" don't know what we asked or observed precisely because they avoid observing/measuring us in any way.


title: Can the charge of particles spontaneously flip from positive to negative or vice versa?

text: I'm thinking of matter antimatter annihilation, are there reactions where normal matter converts to antimatter?


title: Does the stress-energy tensor contain the equations of motion?

text: Derivatives $\nabla_i T^{ik}=0$ of a stress-energy tensor of physical system express conservation laws. Whether contains a stress-energy tensor also the information on the equations of motion of system?


title: Is the speed of light the ultimate speed limit?

text: As we all know nothing can go faster than the speed of light as mentioned by most of our pioneer's in physics. But as I was listening to one of the statements of Sir. Stephen Hawkins he stated that at the time of big bang the universe expanded to about a thousand light years in just few trillionth of a second, how it's possible. Are our laws of physics wrong as if they are correct they should be applicable everywhere in every conditions. Please answer thihs question as answer in above link was not about it.


title: How to calculate the Darcy-Weissbach friction factor for shear thinning laminar flow in a pipe?

text: The Darcy-Weissbach friction factor for laminar flow would be $\frac{64}{Re}$ Now, having a shear thinning (non-newtonian) fluid where the viscosity is not constant how do I arrive at $Re$?   To know an apparaent viscosity, I'd need to know the shear rate, but that is not constant over the diameter of the pipes. Obviously I need to make allowances anyway (like assuming that my fluid obeys a power law over the relevant shear rates), so the method doesn't neet to be uber-exact. Bu I will want to know where I'm off. Googling this, I only ound numerical/CFD solutions to far more complex problems an I couldn'T draw my answer from there.


title: Null lines and degenerate plane

text: Can anyone explain me what null lines are and degenerate plane? I don't know anything about it, I don't have physics background and I am a mathematics student and please tell me if there is any good reference for it so that a beginner can understand it.


title: Frasca's mapping of classical Yang-Mills to $\phi^4$ theory

text: I recently came across an article on the arXiv 0709.2042 written by Marco Frasca, where he provides a mapping between classical Yang-Mills theory to $\phi^4$ theory. Has his idea been fruitful in understanding the infrared regime of quantum Yang-Mills theory, as he claims?


title: What is decay associated spectra?

text: What is decay associated spectra? Suppose we measure the fluorescence intensity over different wavelengths and over time, we get: $$I(\lambda,t) = \sum_i^n \alpha_i(\lambda) \exp(\frac{-t}{\tau_i}).$$ The assumption is that there are n component,species, in the $I(\lambda,t)$. If we fit the right hand to the experimentally obtained $I(\lambda,t)$, and get $\alpha_i$ and $\tau_i$, then people call $\alpha_i$ the decay associated spectra. Now, if we integrate over time we get the steady state emission spectra. The thing that I cannot understand is the decay associated spectra. What does it mean? If it is the steady state spectra of species i, then why does it become negative sometimes? People, say that when it becomes negative, it indicates energy transfer between species. Could someone please elaborate this concept more?


title: Why are $\mu_0$ and $\epsilon_0$, which appear in electrostatics and magnetostatics, related to the speed of light which appears in electrodynamics?

text: $\epsilon_0$ and $\mu_0$ appear in electrostatics and magnetostatics. When we include time varying fields we have electrodynamics and the appearance of c which turns out to be related to $\epsilon_0$ and $\mu_0$. My question is: is there an intuitive way to understand why although $\epsilon_0$ and $\mu_0$ are associated with non-time-varying phenomenas, yet they are related to c which appear when we have time varying fields.


title: How do electrons interact if one of them had just exited the two slits of the double-slit experiment?

text: Consider the following experiment: a double-slit set-up for firing electrons one at a time. Let's now add a second electron (orange), which is fired parallel to the first one, but in the opposite direction, and so as to pass closer to one slit than the other, and, importantly, above the plane in which interference occurs: ![alt text](http://i.stack.imgur.com/RCGXh.png) The firing of the two electrons is timed so that the orange electron is "closest" to the purple electron somewhere around point P, i.e. after the latter has passed through the two slits. Presumably the electrons will repel due to having charge. Presumably an interference pattern will still be produced by the purple electron if the experiment is repeated thousands of times. Now my question is, how exactly does the orange electron get deflected? **If the experiment is repeated, would it always hit the same point on the opposite wall, or would it also start producing a sort of an interference pattern?** Thanks to user:Timwi for the diagram!


title: 'Warp'- type travel; is it possible?

text: I just re-read an article about space travel, and the same article claimed that Star Trek-type 'warp' nuclear engines ARE theoretically possible, but that it will take around 500 years for humans to develop and adapt them for use in the engines of space craft. How true is this statement?


title: How to check units?

text: I've got: $Q=\frac{Er^2}{k}$ how to check the units? I start with $\left[\frac{\text V}{\text m} \, \text m^2\right]$, tried replacing $[ \text V ]$ with $\left[ \frac{\text J}{\text C} \right]$, but it's not leading me to $[\text C]$.


title: Relativistic Mass including exponential decay

text: So from what I gather, relativistic mass = $m_0\gamma$ where $\gamma$ is the lorentz factor. So if i have a mass that is .5 at rest then it is safe to say that the relativistic mass will be 1 if it goes at $\frac{\sqrt{3}}{2}c$ My question is what happens if that .5 is actually a radioactive isotope and is decaying while speeding up? Then at what speed will it approximately 'equal' 1? Thanks


title: Why do rockets accelerate fastest horizontally?

text: I've heard that rockets accelerate fastest when travelling horizontally to the ground, not downwards or upwards. Is that true, and why?


title: Wavelength and resolution

text: I'm reading some texts that seam to assume knowledge of light that I'm not too familiar with. How does wavelength of light relate to the minimum distance span that can be observed (i.e. you cannot make a lens big enough to resolve individual atoms), and is this a light phenomena or an intrinsic wave phenomena?


title: What is the physical interpretation of the density matrix in a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?

text: (a) Any textbook gives the interpretation of the density matrix in a single continuous basis $|\alpha\rangle$:   * The diagonal elements $\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha \rangle$ give the populations.   * The off-diagonal elements $\rho(\alpha, \alpha') = \langle \alpha |\hat{\rho}| \alpha' \rangle$ give the coherences. (b) But what is the physical interpretation (if any) of the density matrix $\rho(\alpha, \beta) = \langle \alpha |\hat{\rho}| \beta \rangle$ for a double continuous basis $|\alpha\rangle$, $|\beta\rangle$? I know that when the double basis are position and momentum then $\rho(p, x)$ is interpreted as a pseudo-probability. I may confess that I have never completely understood the concept of pseudo-probability **[*]** , but I would like to know if this physical interpretation as pseudo-probability can be extended to arbitrary continuous basis $|\alpha\rangle$, $|\beta\rangle$ for non-commuting operators $\hat{\alpha}$, $\hat{\beta}$ and as probability for commuting ones. **[*]** Specially because $\rho(p, x)$ is bounded and cannot be 'spike'. * * * **EDIT:** To avoid further misunderstandings I am adding some background. Quantum averages can be obtained in a continuous basis $| \alpha \rangle$ as $$\langle A \rangle = \int \mathrm{d} \alpha \; \langle \alpha | \hat{\rho} \hat{A} | \alpha \rangle$$ **(a)** Introducing closure _in the same basis_ $| \alpha \rangle$ $$\langle A \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \alpha' \; \langle \alpha | \hat{\rho} | \alpha' \rangle \langle \alpha' | \hat{A} | \alpha \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \alpha' \; \rho(\alpha,\alpha') A(\alpha',\alpha)$$ with the usual physical interpretation for the density matrix $\rho(\alpha,\alpha')$ as discussed above. **(b)** Introducing closure _in a second basis_ $| \beta \rangle$, we obtain the alternative representation $$\langle A \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \beta \; \langle \alpha | \hat{\rho} | \beta \rangle \langle \beta | \hat{A} | \alpha \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \beta \; \rho(\alpha,\beta) A(\beta,\alpha)$$ When the **two** basis are momentum $| p \rangle$ and position $| x \rangle$ the density $\rho(p,x)$ is the well-known Wigner function whose physical interpretation is that of a pseudo-probability. My question is about the physical interpretation of $\rho(\alpha,\beta)$ in **two** arbitrary basis $| \alpha \rangle$, $ | \beta \rangle$.


We know that hydrogen and helium are lighter that air. Therefore it is certain that they would create a lifting force when enclosed in a balloon. How do we describe the relationship of balloon size and related lift?


title: Calculation of the G-force

text: I have a formula which is $\text{G-force} = \frac{v\omega}{9.8}$, where $v$ is speed and $\omega$ is the angular velocity. I've seen on the internet that G-force is actually $\text{acceleration}/9.8$. I'm confused as to which formula is correct. For simulating the motion of particle taking a turn, would omega simply be velocity divided by radius of turn? Assuming Cartesian coordinates.   Another funny thing I noticed is that while simulating particle motion, a 7G turn showed up as an almost straight line (while using a constant turn motion model) with a velocity of 900m/s and time interval of 1second. Am I simulating wrong or is my use of the first equation wrong?


title: Electron Decay, Why are there P and higher orbitals?

text: Related: Decay from excited state to ground state My confusion arose initially from the definition of binding energy being the lowest energy state (n=1) in the hydrogen atom. This, I assume, is simply because hydrogen only has one atom, and electrons don't exist in higher energy states stably. Then my next question was, why not? If these higher energy states exist, why can't electrons maintain those orbits? The above question seems to answer that question, but then I don't understand why bigger atoms CAN hold these higher energy eigenstates. Is is just because the lower electrons "prevent" the higher ones from decaying? I can see where the exclusion principle comes from, but I can't see how it would prevent decay, only how it would prevent more than two electrons from inhabiting the same orbit.


title: Which PPN parameters are least well determined?

text: There are by now (late 2010) many tests of GTR. GTR is holding up quite well! But, surely, we do not yet have it cornered in every possible way. In terms of the PPN parameters used to characterize low-order deviations of any theory of gravity from Newtionian ideals, where are the loosest screws? Which PPN parameter is least constrained by experiment? What kinds of experiments would help constrain it?


title: White Holes and Time-Reversed Oppenheimer-Snyder collapse

text: So, the canned explanation that I always hear about why the white hole solution of the extended Schwarzschild solution is non-physical is that "The matter distribution cuts off the white hole solution." If I look at Oppenheimer-Snyder collapse, however, the cosmological solution that I paste to in the dust's interior is just the closed pressureless model of the Robertson-Walker metric, which is perfectly time-symmetric about the point of maximal expansion. If I look at this naïvely, then, the fluid in the Oppenheimer-Snyder solution originated in a singularity, which then disintegrates, to form an expanded fluid, which reaches some maximal extent, and then recollapses to form a singularity in the future, after some finite time. That past singularity sure seems like a white hole to me. What am I misinterpreting here? Or is it just a pathology of the Oppenheimer- Snyder construction that it is over-simple, and more physically realistic matter models WOULD cut off the past singularity?


title: Wave with mass transport?

text: I don't think this question has been asked on this forum before (at least I didn't find it). In the case of a tsunami, an earthquake generates a wave which will travel with the sea/ocean as the medium. However, what I remember from high school is that mechanical waves transfer energy but energy only. On a sectional view we represented molecules of water going up and down as the wave travels but the molecules never moved horizontally. Then why does a tsunami can flood half of a country? in that case the water does move horizontally. what am I missing? I have read that for regular waves, the problem is different and their momentum comes from wind streams, but I don't think this is the case for tsunami. I would appreciate some pieces of answer.


title: What is a non linear $\sigma$ model?

text: What exactly is a non linear $\sigma$ model? In many books one can view many different types of non linear $\sigma$ models but I don't understand what is the link between all of them and why it is called $\sigma$.


title: Why isn't the force modelled which confines excess charge to remain inside a conductor?

text: Excess charge placed on a conductor distributes itself on the surface of a conductor. But why isn't the force which confines the charge to remain there, modelled in some way?


title: Is the free energy bounded

text: > Can we say that the free energy of a mixture of a finite number of fluids > situated in a bounded container $\Omega$ is bounded? I was thinking of this as an optional hypothesis for one mathematical theorem I want to prove, and I wondered if the energy of some object can be considered to have an upper bound (maybe very large, but still finite). This seems reasonable if the object is bounded. Still, I was thinking of the black holes as mass concentrated in a point, and their energy is huge (I guess).


title: Why do sea levels not rise uniformly around the world?

text: While reading the article _Hawaii’s Beaches Are in Retreat, and Its Way of Life May Follow_ on The New York Times's website, I was surprised to find out that sea levels do not rise around the world at a uniform rate. Assuming that the global warming is true (I think there's enough credible evidence for it - but I don't want a debate on the veracity of global warming - therefore, just assume it's true), what may be the reason for the non-uniformity?


title: Kinematics Problem

text: The question asks me to find the angular velocity. ![figure](http://i.stack.imgur.com/18FaV.jpg) Now I do not want you to solve my homework, I want explanation please. It states that the acceleration of point P is $\vec{a}= -3.02 \vec{i} -1.624 \vec{j}$ when $\theta=60$ It also states that the diameter of the flywheel is 600mm. My question is : I know that the acceleration is split into 2 components, tangential and normal. I know that $a_n=r \alpha$ and $a_t=-r(\omega)^2$.   * Is everything I mentioned until now correct ?   * How can I know which value does an and at take from a given above ?    * How do I decide the i and j terms respectively ? Again, please do not answer the question and find the angular velocity, but please explain the correct approach and whether my deductions are correct.


title: Is there any situation in Physics where the Right Hand Rule is not arbitrary?

text: We use Right Hand Rule in calculating Torque not because that's the direction torque is pointing in the real, physical world, but because it's a convenient way to indicate the "sign" of the rotation and its axis, and as long as we pick one coordinate system and stick to it, everything works out. I think (I might be incorrect) we use the Right Hand Rule in Electromagnetism arbitrarily, but it happens to be very convenient only because of the right combination of Benjamin Franklin's arbitrary choice for designating current flow and the way our magnetic field was aligned at the time of the convention choice for North. But essentially, it is arbitrary -- if Benjamin Franklin had chosen current flow to match the flow of electrons, we'd either use another choice of axes or say that the magnetic field is negative. What I'm saying is that I don't think that nature itself is inherently "right handed" in this situation. Is there any case/situation in physics where the Right Hand Rule and choice of coordinate axes is not arbitrary, but rather inherent in nature itself? How so?


title: Problem on speed of light

text: As I was reading special relativity, my book says the speed of light is $c$ with respect to any other thing. Does that mean the speed of an individual photon is $c$ even with respect to another photon? I mean, shouldn't the relative velocity be zero?


title: What is the physical difference between states and unital completely positive maps?

text: Mathematically, completely positive maps on C*-algebras generalize positive linear functionals in that every positive linear functional on a C*-algebra $A$ is a completely positive map of $A$ into $\mathbb{C}$. Furthermore, we have the Stinespring construction as a powerful generalization of the GNS construction. Certainly, the relationship between completely positive maps and positive linear functionals can only go so far. I am curious about what physics has to say about this analogy/generalization. It seems that completely positive maps should serve as generalized states of a quantum system, but I've mostly seen cp maps arise in the discussion of quantum channels and quantum operations. I'd like to know precisely in what sense a completely positive map can be viewed as a generalized physical state. > **Question:** What is a completely positive map, physically? Particularly, > in what precise sense can a completely positive map be regarded as a > generalized (physical) state? If there are nice survey papers discussing the above relationship, such a reference may serve as an answer to my question.


title: Some questions about Wilson loops

text: Let $G$ be the gauge group whose Yang-Mill's theory one is looking at and $A$ be its connection and $C$ be a loop in the space-time and $R$ be a finite- dimensional representation of the gauge group $G$. Then the classical Wilson loop is defined as, $W_R(C)(A) = Tr_R[Hol(A,C)]$, the trace in the representation $R$ of the gauge field $A$ around the curve $C$.   * I want to know why the above can be written as, $W_R(C) = e^{i\int _D F}$ where $F$ is the curvature of the connection when $G$ is Abelian and $C$ I guess is the boundary of a contractible disk $D$.  {..the above claim reminds me of heuristic calculations (far from a proof!) that I know of where one shows that in the limit if infinitesimal loops, the eventual deviation of a vector on being parallel transported along it by the Riemann-Christoffel connection is proportional to the product of the corresponding Riemann curvature tensor and the area of the loop..}   * In the above proof kindly indicate as to what is the subtlety with $G$ being non-Abelian? Isn't there a natural notion of "ordering" in some sense along the loop given by a parametrization or the trajectory of a particle?    * In relation to discussions on confinement, what is the motivation for also looking at the cases where $R$ is a representation not of $G$ but of its simply connected cover? I mean - how does the definition for $W_R(C)$ even make sense if $R$ is not a representation of $G$?  I don't understand what is meant by statements like (from a lecture by Witten), _"..if $R$ is a representation of $G$, then there are physical processes contributing to $ <W_R(C)>$ in which large portions of the Wilson line have zero-charge i.e carry trivial representations of $G$, because some particles in the theory have annihilated the charges on the Wilson line.."_ I would have thought that its only a representation of $G$ that can be fixed and I don't see this possible imagery of seeing a representation attached to every point on the loop.   * What is the subtlety about Wilson loops for those representations of $G$ which can come from a representation of its universal cover? If someone can precisely write down the criteria for when this will happen and then what happens...


title: Unknown quantum state with promise of classical data

text: I am trying to solve a problem in the measurement and identification of quantum states with a promise as to what states it could be. Here is the problem. Imagine a system that produces qubits in one of four states $S = \\{a,b,c,d \\}$, evenly distributed. In one shot, I can receive $K$ copies of a state in $S$. However, a random unitary, evenly distributed in $SU(2)$ has been applied, so, at the detectors I receive $S^{\prime} = \\{Ha,Hb,Hc,Hd \\}$. I have a detector system that includes $2M$ detectors and they represent the projection operators onto any pair of basis vectors in any basis (thus we can choose which $M$ bases we want to use the detectors in). All I want to do is decide if the state is $a,b,c,$ or $d$. How many copies of the state do I need (ie what is a minimum for $K$)? What is the smallest $M$? Another, more general question is this: what should I use to represent the state before I do a measurement? Should it not be a density matrix which is an integral over all states? Since there is a random unitary applied, that means that I can receive, in a fixed basis, any state with equal probability. What would be the update rule for this density matrix after the $j^{th}$ measurement result? I realize there are pieces missing in my question. I will revise it this post this evening, but hopefully it will give the general idea.


title: What factors cause the velocity saturation to occur at different electric fields for different materials?

text: In semiconductors the velocity of carriers gets saturated after a certain value of electric field. In _silicon_ it occurs at around $10^4 kV/cm$ and in _GaAs_ at some other value.   * What factors are involved here?    * Is it related to the value of band-gap alone or other parameters are at play? Furthermore, suppose the value of saturated velocity of an electron in _Si_ at room temperature is $10^7 cm/s$, for an electron in GaAs it is even higher.   * Why is that so?


title: Assumptions when calculating $\vec{B}$ using Ampère's (circuital) law

text: When considering the same setup as in this question, i.e. a straight, infinitely long wire carrying the current $I$, Ampère's circuital law $$\oint_C \vec{B} \cdot \mathrm{d}\vec{r} = \mu_0 I_\text{enc}$$ is often used to calculate $\vec{B}$, as this is a cylindrically symmetric problem. Due to the symmetry, $\vec{B} = B(\rho) \vec{e}_\varphi$ (cylinder coordinates $\rho, \varphi, z$) is assumed and, by integrating over a circle with radius $\rho$, one finds $$B(\rho) \cdot 2 \pi \rho = \mu_0 I \Leftrightarrow B(\rho) = \frac{\mu_0 I}{2 \pi \rho}$$ My question is whether the assumption $\vec{B} = B(\rho) \vec{e}_\varphi$ is actually valid. I can accept that $B$ has to be independent of $\varphi$ and $z$ due to the symmetry, but I'm not so sure about the direction of $\vec{B}$. In particular, $\vec{B}$ does depend upon $\varphi$, as $\vec{e}_\varphi = \begin{pmatrix} -\sin(\varphi)\\\ \cos(\varphi) \\\ 0\end{pmatrix}$, so the assumption that $\vec{B}$ is independent of $\varphi$ (as I've heard or seen in books) really only applies to the absolute $|\vec{B}|$, doesn't it? In fact, with this rationale, could $\vec{B}$ not also have components in other directions which are not "seen" by the line integral? Assuming $$\vec{B} = B_\rho(\rho) \vec{e}_\rho + B_\varphi(\rho) \vec{e}_\varphi + B_z(\rho) \vec{e}_z,$$ the above would only calculate $B_\varphi(\rho)$. Does one have to use additional reasoning to be able to assume $\vec{B} = B(\rho) \vec{e}_\varphi$?


title: How does quantization solve the ultraviolet catastrophe?

text: I understand how classical physics leads to the UV catastrophe. But I cannot understand how quantization solves it.   * How can quantization prevent the body from radiating a lot of energy? I know this question is similar to many other questions but I need a layman's explanation (without too much mathematical details).   * Finally why does the peak reach zero intensity after a certain frequency? ![enter image description here](http://i.stack.imgur.com/2vACu.png) Image source:- Wikipedia


title: Question on the logical structure of the EPR argument and Bell's inequalities

text: Recently I have read a lot online about the EPR argument and Bell's inequalities and its implications. When comparing what people write there online with the actual research articles of Einstein and Bell, it seems to me - put drastically - that people online talk about something but not about Bell and Einstein. Let me explain, what I mean. Online most of the times the logical structure of the argument is roughly given in the following way (see, e.g., here in the first few paragraphs): Assuming locality _and_ pre-existing properties, Bell's inequalities follow. Quantum mechanics' predictions (and experiments) are in contradiction to the inequality. Hence, one can abandon _either_ locality _or_ pre-existing properties. Reading the EPR paper (available here) however, the structure seems to be quite different. They consider the following _criterion_ of an element of the physical reality (and a criterion is not an assumption!): If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity. This, I think, is very reasonable. One can even very well argue that this is an analytical statement because of the phrase 'without in any way disturbing a system'. So one does not have to _assume_ the correctness of the criterion as it is true anyhow! It is employed in the following situation (Bohm's version of the EPR set-up): Think of the singlet state of two spin-1/2 particles with total spin zero. If one measures the spin in the x direction of particle A, one can predict with certainty the spin in x direction of particle B, no matter how far the particles are separated (think of a space-like separation). For the sake of the argument it is not needed that we consider different directions for the two particles. Now assuming locality, they must conclude by their criterion that there really exist pre-existing properties. That is Einstein. So, the correct reasoning should be (also in Bell's understanding as he pointed out repeatedly) that locality _implies_ the pre-existing properties. The consequence of Bell's inequalities then is that one cannot abandon either locality or pre-existing properties but _must abandon locality_. So, who's right? Why are there two different conclusions? Are Einstein and Bell missing an essential point? I often read that counterfactual definiteness is tacitly assumed. But that isn't an assumption right? It is simply a property of the singlet state if one measures the same spin direction for both particles which is sufficient here. It would be very much appreciated if it could be pointed out in Einstein's and Bell's original papers where the mistakes or debatable passages are (if there are any), since I think they are a very good basis for this discussion. Thanks.


title: Are there 2 kinds of photons, one that mediate the electromagnetic interaction and the other the quanta of light?

text: It is usually said that photons are the force carriers or the mediators of the electromagnetic forces between electric charges. At the same time we know also that electromagnetic waves on the quantum level are described in terms of photons. It seems to me that we have 2 types of photons:   * the photons which mediate the electromagnetic interactions which are offshell hence their masses do not necessarily have to be zero, I suppose    * and the photons which are the quanta of electromagnetic waves which are massless. Am I correct, or am I confused and missing something?


title: Semi-conductors

text: Suppose there is a semiconductor with Fermi energy $E_f$ and that there are $N$ bound electron states. I'd like to know why the mean number of excited electrons takes the form $$\bar n={N\over \exp\beta(\mu-E_f)+1}$$ where $\mu$ is the chemical potential. * * * I can see that the Fermi Dirac statistics say that for one fermion the mean occupation number is $$\bar n = {1\over \exp\beta(E-\mu)+1}$$ I am not sure however as to how for the semi-conductor it should assume the above form. The $N$ is clearly due to the $N$ bound $e^-$ states. I am not quite so confident as to why $$\mu\to E_f$$ $$E\to \mu$$ Could someone please explain? * * * Anybody? :(


title: Force when acceleration is zero

text: i tried to find it in books but i was unable to do so and i am not able to understand this and I couldn't think of a better example. i repeat. its just an example. if i punch someone's face with my arm having constant velocity, it will mean that the acceleration of my arm is zero. F=ma, therefore force exerted by my arm is zero. Now if the force applied is zero the why does the person whom i punched experiences pain. please help. thanks.


title: Prove an action expansion's even-indexed terms have to be integrated, where the odd-indexed terms are only derivatives of the potential (WKB)

text: After assuming a wavefunction of a form: $$ \psi \approx A \exp{\left(i \frac{S(x)}{\hbar}\right)}$$ and letting $$S = \hbar^0 S_0 + \hbar^1 S_1 + \hbar^2 S_2 +...$$ The odd-indexed terms of the action for a one-dimensional potential in the time-independent Schrodinger equation do not require integration if $p$ is known and differentiable. However the even-terms require non-trivial integration(which is extensively more computationally taxing). p is defined by: $$ p = \sqrt{2 m ( E - V(x))} $$ The terms: $$ S_0^\prime = p$$ $$ \boxed{S_0 = \int dx\, p =\pm \int dx \sqrt{2m(E-V(x))} } $$ $$ S_1^\prime = \frac{i}{2}\frac{1}{p}\frac{d p}{d x} $$ $$ \boxed{S_1 = \frac{i}{2} ln(p) } $$ $$ S_2^\prime = \frac{1}{8 p^3} \left(\frac{d p}{d x} \right)^2 - \frac{1}{4} \left(\frac{1}{p^2} \frac{d^2 p}{d x^2} - \frac{1}{p^3} \left(\frac{d p}{d x}\right)^2\right)$$ $$ \boxed{S_2 = \int dx \left(\frac{1}{4 p^2} \frac{d^2 p}{d x^2} + \frac{3}{8 p^3} \left( \frac{d p}{d x}\right)^2 \right)}\mathrm{\,requires\,\,integration} $$ Now for $S_3$: $$ S_3^\prime = -\frac{i}{8 p^3} \frac{d^3 p}{d x^3} + \frac{3}{4} \left( \frac{i}{p^4} \frac{d p}{d x} \frac{d^2 p}{d x^2} - \frac{1}{p^5} \left( \frac{d p}{d x}\right)^3 \right)$$ After making an educated guess that: $$ \frac{d}{dx} \left( -\frac{i}{8 p^3} \frac{d^2 p}{d x^2} + \frac{3 i}{16 p^4} \left(\frac{d p}{d x}\right)^2\right) = -\frac{i}{8 p^3} \frac{d^3 p}{d x^3} + \frac{3}{4} \left( \frac{i}{p^4} \frac{d p}{d x} \frac{d^2 p}{d x^2} - \frac{1}{p^5} \left( \frac{d p}{d x}\right)^3 \right)$$ Then $$\int dx S_3^\prime = \int dx \frac{d}{dx} \left( -\frac{i}{8 p^3} \frac{d^2 p}{d x^2} + \frac{3 i}{16 p^4} \left(\frac{d p}{d x}\right)^2\right)$$ Clearly because of my guess $$\boxed{S_3 = -\frac{i}{8 p^3} \frac{d^2 p}{d x^2} + \frac{3 i}{16 p^4} \left(\frac{d p}{d x}\right)^2} $$ How do we know that $S_2$ cannot be retrieved the same way? Surely I cannot try an ansatz for every possible function that could be a candidate for $S_2$. This problem has come up in general for different types of problems outside the scope of quantum mechanics. Is it only a consequence of this equation(1-D time-independent Schrodinger equation) that we have odd-indexed and even-indexed terms this way? Would it be different for 3-D?


title: Conformal Invariance and the OPE

text: I am reading Polchinski's String Theory now and am having trouble with a couple of things mentioned on Page 45 under "Conformal Invariance and the OPE". Here is the paragraph I am reading. The sentences in italics are my questions. He states Conformal invariance puts strong constraints on the form of the OPE, and in particular on the OPEs of the energy-momentum tensor. Consider the OPE of $T$ with the general operator ${\cal A}$. Because $T(z)$ and ${\tilde T}({\bar z})$ are (anti)holomorphic except at insertions _(What does this mean?)_ , the corresponding coefficient functions must also have this property _(What if the operator is not holomorphic? Are the coefficient functions still holomorphic?)_


title: Can the Montevideo interpretation of quantum mechanics do what it claims?

text: Partly inspired by the great responses to a my previous physics.SE question about "reversing gravitational decoherence, today I was rereading the intriguing papers by Gambini, Pullin, Porto, et al., about what they call the "Montevideo interpretation" of quantum mechanics. They've written lots of papers on this subject with partly-overlapping content; see here for a list. The overall goal here is to try to identify, within the (more-or-less) known laws of physics, a source of decoherence that would be _irreversible for fundamental physics reasons_ , rather than just staggeringly _hard_ to reverse technologically. One can argue philosophically about whether anyone should care about that, whether such a decoherence source is either necessary or sufficient for "solving the measurement problem", etc. Here, though, I'm exclusively interested in the narrower issue of whether or not such a decoherence source _exists_. Gambini et al.'s basic idea is easy to explain: quantum-gravity considerations (e.g., the Bekenstein bound) very plausibly put fundamental limits on the accuracy of clocks. So when performing a quantum interference experiment, we can't know _exactly_ when to make the measurement---and of course, the energy eigenstates are constantly rotating around! So, for that reason alone (if no other!), we can think about any pure state we measure as "smeared out a little bit" into a mixed state, the off-diagonal entries in the density matrix a little bit less than maximal. More precisely, Gambini et al. claim the following rough upper bound for the magnitudes of the off-diagonal elements. Here, T is the elapsed time between the beginning of the experiment and the measurement, Tplanck is the Planck time, and EA-EB is the difference in energy between the two things being kept in superposition (so that $\frac{E_A - E_B}{\hbar}$ is the Bohr frequency). (1) $\left| \rho_{offdiagonal} \right| \lt \exp \left( -\frac{2}{3} T_{planck}^{4/3}T^{2/3} \left( \frac{E_A - E_B}{\hbar} \right)^{2} \right).$ If EA-EB were equal to (say) the mass-energy of a few million protons, then (1) could certainly lead to observable effects over reasonable timescales (like a second). Now, it might be that there's an error in Gambini et al.'s analysis or in my understanding of it, or that the analysis relies on such speculative assumptions that one can't really say one way or the other. If so, please let me know! If none of the above holds, though, then my question is the following: **Can the bound (1) really do anything like the work that Gambini et al. claim for it---that is, of preventing "macroscopic interference" from ever being observed? More concretely, is it really true that anything we'd intuitively regard as a "macroscopic superposition" must have a large E A-EB value, and therefore a relative phase between the two components that rotates at unbelievable speed? In principle, why couldn't we prepare (say) a Schrodinger cat in an energy eigenstate, with the alive and dead components having the same energy (so that EA-EB=0)? Would such a state not constitute a counterexample to what Gambini et al. are trying to do?**


title: Pressure change due to fan removing air from a non-airtight room

text: The following problem occurred to me today: > Suppose a $100\mathrm{cfm}$ fan is pushing air out of a large room which is > airtight except for a $10 \mathrm{cm}^2$ hole. The air pressure outside the > room is $101.3\mathrm{kPa}$, and the atmosphere consists of $80\%$ nitrogen > and $20\%$ oxygen. All temperatures are $300\mathrm{K}$. What is the > equilibrium pressure inside the room? I believe I have given enough information to determine the resulting pressure. My inclination is to take a naïve statistical mechanics approach to the problem. This leads me down the following train of thought:   * Particles are distributed evenly inside and outside the room, with a cubic meter containing $N_i=\frac{P_i}{k_BT}$ and $N_o=\frac{P_o}{k_BT}$ molecules respectively, where $P_i$ is the pressure inside and $P_o$ the pressure outside.   * Velocities are distributed symmetrically and speeds satisfy a Boltzmann distribution, so for a fixed type of molecule $M$ the distribution of $v_z$, the $z$-coordinate of velocity squared, is $$D_M(v_z)=\int_{S^2}\mathrm{exp}\left(\frac{-m_Mv_z^2}{2k_BT\cos\theta^2}\right)\mathrm{d}\theta\mathrm{d}\phi$$ where $m_M$ is the mass of the molecule $M$.   * The number of molecules of $M$ that strike an area $A$ over time $t$ from inside is $$\int_0^\infty Ax_MN_i\int_{d/t}^{\infty} D_M(v_z)\mathrm{d}v_z\mathrm{d}d$$ where $d$ denotes the distance along the $z$-axis of the molecule from the area and $x_M$ denotes the fraction of all molecules which are $M$. Similarly, the number of molecules of $M$ that strike an area $A$ over time $t$ from outside is $$\int_0^\infty Ax_MN_o\int_{d/t}^{\infty} D_M(v_z)\mathrm{d}v_z\mathrm{d}d.$$   * The difference between these two, summed for oxygen and nitrogen, must be equal to the number of molecules of $M$ removed by the fan, which is $.0472\cdot N_it$.  Thus we have $$.0472\cdot N_it=A(N_i-N_o)\int_0^\infty \int_{d/t}^{\infty} \left(x_{N^2}D_{N^2}+x_{O^2} D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d$$ so solving for $N_i$ we get $$\begin{align} N_i &=\frac{AN_o\int_0^\infty \int_{d/t}^{\infty} \left(x_{N^2}D_{N^2}+x_{O^2} D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d}{A\int_0^\infty \int_{d/t}^{\infty} \left(x_{N^2}D_{N^2}+x_{O^2} D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d-.0472t}\\\ &=\frac{.1\mathrm{m}^2\cdot\frac{101.3\mathrm{kPa}\cdot \mathrm{m}^3}{1.38e-23 \mathrm{J}/\mathrm{K}\cdot 300\mathrm{K}}\int_0^\infty \int_{d/1\mathrm{s}}^{\infty} \left(.8D_{N^2}+.2 D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d}{.1\mathrm{m}^2\int_0^\infty \int_{d/1\mathrm{s}}^{\infty} \left(.8D_{N^2}+.2 D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d-.0472\mathrm{s}}\\\ &=\frac{2.45e24 \mathrm{m}^{2}\int_0^\infty \int_{d/1\mathrm{s}}^{\infty} \left(.8D_{N^2}+.2 D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d}{.1\mathrm{m}^2\int_0^\infty \int_{d/1\mathrm{s}}^{\infty} \left(.8D_{N^2}+.2 D_{O^2}(v_z)\right)\mathrm{d}v_z\mathrm{d}d-.0472\mathrm{s}}\\\ \end{align}$$ which I could probably evaluate using Mathematica if my desktop were working, but sadly it is not. From there it would be trivial to calculate $P_i$. My question is whether my reasoning up to this point is correct, and whether there are any factors I have left out. Additionally, is there an easier way to calculate $P_i$?


title: How can a non-conservative field be a scalar multiple of a conservative field?

text: Okay so I was reading this from University Physics by Freeman and Young and on the topic of inductors as circuit element, they wrote that $\mathbf{E_c} + \mathbf{E_n} = 0$ which makes no sense to me Here is an excerpt of the text and underlined the confusion with green ![enter image description here](http://i.stack.imgur.com/Ae1lO.jpg)


title: Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter

text: The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations. Say I consider an Euclidean space and a time parameter. How does the Euclidean metric (which in canoncial coordinates is $\delta_{ij}$, with $i,j\in \\{1,2,3\\}$), transform under a Galilean transformation, more specifically a velocity boost $t\mapsto t,x\mapsto x-vt$? In a way, from the three dimensional pov, which doesn't see $t$, this is just a translation. So here is what I'd like a derivation for: > $$(t\mapsto t'=t,\ x\mapsto x'=x-vt)\ \Longrightarrow\ > h_{ij}\equiv\delta_{ij}\mapsto\ ?$$


title: On Einstein's equivalence principles

text: There are two foundative Equivalence Principles in General relativity: **Weak Equivalence Principle (WEP):** the dynamics of a test particle in a gravitational field is independent of its mass and internal composition. (WEP is equivalent to say that the ratio between the _gravitational mass_ $m_g$ and the _inertial mass_ $m_i$ has a universal value $k$. This value can be considered as the scalar $1$). **Einstein equivalence principle (EEP)** : A frame linearly accelerated relative to an inertial frame in special relativity is LOCALLY identical to a frame at rest in a gravitational field. Most textbooks say that "obviously" (EEP) implies (WEP) but the converse is not true. I don't understand why the implication (EEP) $\Rightarrow$ (WEP) is true, and moreover I'd like a counterexample showing that (WEP) $\not\Rightarrow$ (EEP).


title: Does cosmic censorship rule out stable toroidal black holes? How?

text: I'm having a hard time understanding what the arguments against stable toroidal black holes are saying. For many of these, I can't figure out if they're talking about:   1. A non-rotating toroidal event horizon   2. A toroidal event horizon, for which it's matter is moving in the tangential direction (of its center ring) such that the gravitational force is balanced by its acceleration I am unimpressed and uninterested in #1, since this is predicted by basic Newtonian physics, and the matter starts out violently accelerating toward the center of mass. A number of academic arguments about the viability of toroidal black holes in the book Black Hole Physics, page 164-165. I _think_ that the term "non- stationary" might be referring to my #2 above, but this isn't explicitly established in that page or two. The most detailed mechanistic logic I found was this quote: > Jacobson and Venkataramani (1995) pointed out that a black hole with > toroidal surface topology provides a potential mechanism for violating > topological censorship. Specifically, a light ray sent from past null > infinity to future null infinity and passing through the hole in the torus > would not be deformable to a ray which did not come close to the black hole. > Thus, topological censorship implies that a toroidal horizon (if it exists) > must close up quickly, before a light ray can pass through. The 1995 Jacobson and Venkataramani paper does seem to make very strong statements along this thread. But nothing I read about the Cosmic censorship theorem itself sounds even remotely convincing. It's more-or-less saying you can't have naked singularities or causality violations. Neither of those are obvious to me from the above quote's argument. In layman's terms, it sounds like the arguments say that a stable toroidal black hole would allow you to time travel by going through the donut hole. The notation is probably just beyond my grasp. Does the argument really apply to #2 as I've defined it? And, if so, how can I convince myself of the proposition?


title: Weight of a tensor density

text: Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with the weight of **$\frac{-1}{2}$** : $$\overset{\sim}{g_{ab}} = \sqrt{({\det g})} ^{\frac{-1}{2}}~~g_{ab}$$ In the paper _The geometry of free fall and light propagation_ by Ehlers and his colleagues ( _Gen. Relativ. Gravit._ **44** no. 6, pp. 1587–1609 (2012)), on page 1599 there is an example. You can also look at the relation **$2.4$** of this article. I don't understand why this special weight is chosen for the tensor density constructed from metric.


title: Is it possible to remove the Higgs Field from an object to make it have no mass?

text: Can you accomplish this in theory? We found the Higgs boson, but can we remove it from particles?


title: Surface Area Relative to Element Distribution

text: We tend to find heavier elements like iron and others toward the center of celestial bodies. Lighter elements like carbon, hydrogen, and oxygen are almost always towards the surface. That makes sense because heavier elements have a greater gravitational effect and would naturally want to move toward the center. Now think about the chaos that ensued during initial formation of the celestial body. Heavier elements could get jumbled in with lighter elements throughout the depths. Even to this day, we have lighter elements being pushed to the surface through volcanic gases and ocean fissures. **Question** : If elements were not always arranged from heaviest to lightest during creation, would this in anyway introduce volatility in the overall surface area of the celestial body? Imagine the worst case scenario that would be utterly unimaginable: gold and iron wrapped around the surface with oxygen and hydrogen trapped deep within the depths. Would the overall surface area be any different over time as heavier elements pushed their way in and lighter elements rise to the surface? Intuitively, I would say it would change over time, but to what degree and at what rate of change depends on how elements were initially distributed.


title: Increase the signal-to-noise ratio of photomultiplier

text: It is a GRE physics problem. A technique would most likely increase the signal- to-noise ratio of a photomultiplier tube is to   1. operate the tube at a lower temperature   2. operate the tube at higher voltage per dynode than usually specified    3. use a radioactive source to saturate the noise level   4. use a retarding potential on the first dynode   5. use a photocathode of lower work function I tried searching for information regarding photomultiplier, but none seems to fit any of the above options.


title: Could arc reactors eventually be real?

text: Could arc reactors eventually be real? I have wondered about this, there is not much about it online. If they could how would they work?


title: Dyson Schwinger equation

text: given the Dyson equations $ \frac{\delta S}{\delta \phi(x)}\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Z[J]=0 $ is true that they are a solution or differential representation of the Generating functional ?? $ Z[J]=\int d[\phi(x)]exp(iS[\phi(x)])-\int dxZ[J]\phi(x)$ ?? if i can solve these set of equation can i 'renormalize' a physical theory ?? How are the Schwinger dyson equation solved ?? , thanks. are these set of equation similar to 'Schwinger's quantum action principle ' ?? $ \delta <A|B>_{J}=i<A|\delta S |B>_{J}$ the variation is made respect an external source $ J(x) $


title: Polyakov action: difference induced metric and dynamical metric

text: The Polyakov action is given by: $$ S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\gamma_{\alpha\beta}, $$ where $\gamma_{\alpha\beta}$ is called the induced metric and $g_{\alpha\beta}$ the dynamical metric on the world sheet. I have difficulties understanding the differences between these two metrics. I know that the latter is introduced in order to be able to remove the square root in the Nambu-Goto action, but I don't know what it means. The space in which the string propagates has just the Minkowski metric $\eta_{\mu\nu}$, if I am not mistaken. Furthermore, I think that the induced metric is derived by demanding $ds^2$(whole space) = $\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ = $ds^2$(world sheet) = $\gamma_{\alpha\beta}d\sigma^{\alpha}d\sigma^{\beta}$ Is this correct? I am really confused by all these different metrics.


title: quantum curvature

text: If a state can be a superposition of energy states, and mass equals energy (special relativity), and mass curves space-time (general relativity), then could we say that space-time around a quantum system that is in a superposition of states is also in a "superposition of curvatures"?


title: Physics of Carving on a Skateboard

text: How does carving on a skateboard work, why is it easier to carve on a longboard as opposed to a shortboard? I feel like I'm more prone to hurt myself trying to carve or throw my weight out on a shortboard, why is this? Is it just because you have more control over your weight distribution on a longboard? Can anyone give me an indepth analysis of this?


title: Is there an equation to calculate the average speed of liquid molecules?

text: I seem to remember from first year physics that we can calculate the RMS speed of a stationary, ideal gas with $v=\sqrt{\frac{3RT}{M}}$. Does a similar equation exist for liquids?


title: What is highest water pressure at which electrolysis can be performed?

text: What is highest water pressure at which electrolysis can be performed to derive hydrogen and oxygen? Does the dielectric constant of water, which which lowers as pressure increases, have an effect on this?


title: Why doesn't this model plane fly?

text: I have been designing a model plane for Design Technology for the past month or so, and today I laser cut my final design and assembled, it then tested it. Upon testing the plane does not get any lift, whereas the previous testing model which was virtually the same did. The plane is built using Balsa Wood, and Assembled with hot glue (I used as little glue as possible to reduce weight :) ) Any Ideas? Images: http://imgur.com/a/fdKxZ


title: Scale set by cosmological constant

text: Following on Jim Graber's answer to: Can "big rip" rip apart an atomic nucleus? If the cosmological constant is large enough, even the ground state of a hydrogen atom can be affected. So why is the energy scale for quantum gravity set by the planck mass and not by the cosmological constant? Is it because the cosmological constant can be associated with other theories (inflatons, or vacuum energies of the matter fields, etc.) and thus is just considered an ingredient and not gravity itself? If this does come down to a semantics issue, I'd still be interested to hear if the scale set by the cosmological constant suggests we may be able to see interesting quantum effects at that scale depending on what the cosmological constant 'is'.