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<p>The EM tensor of the $bc$-CFT is
$$
T(z) = \colon \partial b c \colon - \lambda \partial \colon b c \colon
$$
After expanding in a mode expansion, we find
$$
T(z) = \sum_{m} \frac{1}{z^{m+2}} \sum_n \left( \lambda m - n \right) \colon b_n c_{m-n} \colon \implies L_m = \sum_n \left( \lambda m - n \right) \colon b_n c_{m-n} \colon ~~...(1)
$$
Now, one would like to rewrite this expression using creation-annihilation (CA) normal ordering ${}_\circ^\circ~~{}_\circ^\circ$ instead of conformal-normal order $\colon~~\colon$. A simple calculation reveals
$$
L_m = \sum_n \left( \lambda m - n \right) {}_\circ^\circ b_n c_{m-n}{}_\circ^\circ + \frac{\lambda \left( 1 - \lambda \right)}{2} \delta_{m,0}
$$
The above expression was derived by requiring that $L_m$ satisfy the Virasoro algebra. </p>
<p>I am now looking to provide an alternative derivation of the above expression. Here's my procedure. We start with the result of exercise 2.13(a) in Polchinski (which I have already verified)
$$
\colon b(z) c(w) \colon - {}_\circ^\circ b(z) c(w) {}_\circ^\circ = \frac{(z/w)^{1 - \lambda} - 1 }{z - w}
$$
Now, taking a limit of $w \to z$, we find
$$
\colon b(z) c(z) \colon = {}_\circ^\circ b(z) c(z) {}_\circ^\circ + \frac{1-\lambda}{z}
$$
from where we can derive the difference between the CA normal ordering and conformal ordering of the mode coefficients
$$
\colon b_m c_n \colon = {}_\circ^\circ b_m c_n {}_\circ^\circ + \left( 1 - \lambda \right) \delta_{m+n,0}
$$
Now, let us start from the expression $(1)$ and plug in the above. We find
\begin{equation}
\begin{split}
L_m &= \sum_n \left( \lambda m - n \right) \colon b_n c_{m-n} \colon \\
&= \sum_n \left( \lambda m - n \right) \left[ {}_\circ^\circ b_n c_{m-n} {}_\circ^\circ + \left( 1 - \lambda \right) \delta_{m,0} \right] \\
&= \sum_n \left( \lambda m - n \right) {}_\circ^\circ b_n c_{m-n} {}_\circ^\circ + \delta_{m,0} \sum_n n \left( \lambda - 1 \right) \\
\end{split}
\end{equation}
Comparing with the expression mentioned above, we are required to have</p>
<blockquote>
<p>$$
\sum_{n=-\infty}^\infty n \left( \lambda - 1 \right) = \frac{\lambda \left( 1 - \lambda \right)}{2}
$$
I am not able to show this. Anyone has any idea? </p>
</blockquote>
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<p>How can I find the amount of point force at the end of a cantilever plastic beam that produces e.g. 45deg slop at the end of the beam? Is this the right equation: $$F=\frac{2EI\theta ^2}{L^2 sin(\theta)}$$</p>
<p>I derived this equation by putting $x=\frac{L}{\theta}sin(\theta)$ and $\theta_x=\theta$ in the following eqation from the "Large and small deflections of a cantilever beam" paper: $$x=\sqrt{ \frac{2EI}{F} } \left(\sqrt{sin(\theta)}-\sqrt{sin(\theta)-sin(\theta_x)}\right)$$</p>
<p>where $\theta$ is the maximum slope at beam end and $\theta_x$ is the beam slop at x.</p>
<p>Thanks for your help in advance.</p>
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<p>I understand that faster-than-light communication is impossible when making single measurements, because the outcome of each measurement is random. However, shouldn't measurement on one side collapse the wave function on the other side, such that interference effects would disappear? Making measurements on "bunches" of entangled particles would thus allow FTL communication, by making observed interference effects appear or disappear. How does such an experiment not:</p>
<p>1) Clearly imply that faster-than-light communication is possible?</p>
<p>or</p>
<p>2) (if #1 is rejected) Imply that measurement of one half of an entangled pair does not cause the collapse of the other half's wave function.</p>
<p>Why doesn't this thought experiment clearly show that if we maintain that FTL communication is ruled out, we must also rule out "universal collapse" in the Copenhagen interpretation?</p>
<p><strong>EDIT: Here is an example of an explicit experiment (though I think experts could come up with something better):</strong></p>
<p>You can entangle a photon with an electron such that the angle of the photon is correlated with the electron's position at each slit of a double slit experiment. If the photon is detected (it's outgoing angle measured), then which-path information is known, and there is no interference. If the photon is not detected, the interference remains. </p>
<p>The experiment is designed such that the photon and electron go in roughly opposite directions, apart from the tiny deflection which gives which-path information. You set up a series of photon detectors 100 ly away on one side, and your double slit experiment 100 ly away in the opposite direction. Now you produce the entangled pairs in bunches, say of 100 entangled pairs, each coming every millisecond, with a muon coming between each bunch to serve as a separator. </p>
<p>Then the idea is that someone at the photon detector side can send information to someone watching the double-slit experiment, by selectively detecting all of the photons in some bunches, but not in others. If all of the photons are detected for one bunch, then the corresponding electron bunch 200 ly away should show no interference effects. If all of the photons are not detected for one bunch, then the corresponding electron bunch 200 ly away would show the usual double-slit interference effects (say on a phosphorus screen). (Note that this does not require combining information from the photon-detector-side with the electron-double-slit side in order to get the interference effects. The interference effects would visibly show up as the electron blips populate the phosphorus screen, as is usual in a double-slit experiment when which-path information is not measured.)</p>
<p>In such a way the person at the photon detectors can send '1's and '0's depending on whether they measure the photons in a given bunch. Suppose they send 'SOS' in Morse code. This requires 9 bunches, and so this will take 900 milliseconds, which is less than 200 years.
The point is that such an experiment would only work if you assume that the measurement of the photon really does collapse the wave function nonlocally.</p>
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<p>If you line up the suns rays parallel to a Fresnel lens, the light is concentrated, and the focus directly underneath. However, what happens if the sun is off to the side, making the light hit at an angle (ei. 45 degrees)? Will the only difference be the focal point, of will the light be less concentrated? If so, by how much? What about a convex lens? Thanks</p>
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] |
<p>Is that true that the big bang caused the quantum entanglement of all the particles of the universe so every particle is entangled to each other particle of the universe?</p>
| 7,972 |
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<p>I have an exam in two days for first year university physics. Often for dynamics problems, I am required to solve algebraic systems of equations by hand, and this can be very daunting. </p>
<p>When I see the solutions, however, the steps that the solver took to seem very clean and almost obvious. Are there some rules of thumb that physicists use to solve small systems of equations, either by elimination or substitution?</p>
<p>Here is an example. Find $\frac{m_1}{m_2}$ in terms of only $\theta$ where the known quantities are $\theta$, $m_1$, $m_2$.</p>
<p>$$F_T \sin \theta = m_1 g$$
$$F_T \cos \theta = m_1 a$$
$$F_T \sin \theta + F_N \cos \theta = m_2 g$$
$$F_N \sin \theta - F_T \cos \theta = m_2 a$$</p>
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] |
<p>I have a slight difficulty understanding the solution to the following problem:</p>
<p>A light inextensible string with a mass $M$ at one end passes over a pulley at a distance $a$ from a vertically fixed rod. At the other end of the string is a ring of mass $m(M>m)$ which slides smoothly on the vertical rod as shown in the figure. The ring is released from rest at the same level as the point from which the pulley hangs. If $b$ is the maximum distance the ring will fall, determine $b$ using the principle of virtual work. </p>
<p>The solution is: </p>
<p>Let $l$ be the length of string. Therefore, </p>
<p>$$x+(a^2+b^2)^{1/2}=l\tag{1}$$
Imagine a vertical displacement $\delta b$ of the ring along the rod. </p>
<p>$$\delta x+b(a^2+b^2)^{-1/2}\delta b=0\tag{2}$$
Therefore,
$$\delta x=-b(a^2+b^2)^{-1/2}\delta b$$</p>
<p>The constraints over the pulley and rod do no work. By the principle of virtual work, $$Mg\delta x + mg \delta b=0$$
Substituting value of $\delta x$:
$$-Mgb(a^2+b^2)^{-1/2}\delta b = -mg\delta b$$
Since $\delta b$ is arbitrary,
$$b^2=\frac{m^2}{M^2}(a^2+b^2)$$
$$b=\frac{ma}{(M^2-m^2)^{1/2}}$$</p>
<p>My questions are: </p>
<ol>
<li>How is Equation 2 obtained from Equation 1? (especially with regard to the intermediate steps)</li>
<li>When I attempted the problem, I had the tension in the string doing work, because the displacement of the string is in line with the tension in the string. Why is this not being considered? Thanks in advance.</li>
</ol>
<p><img src="http://i.stack.imgur.com/AwaiU.jpg" alt="setup of problem"></p>
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<p>In string theory, we are told strings can split and merge if the string coupling is nonzero, even while the worldsheet action remains Nambu-Goto or Polyakov plus a topological term. However, a classical solution, in say the light cone gauge, shows that provided the worldsheet time increases with light cone cone time initially, adding topological terms will not change the solution, and the string will not split. So why and how do strings split?</p>
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<p>I read the Popular Science books, <em>The Brief history of time</em> (Hawking), <em>The fabric of the cosmos</em> (Greene), and <em>The Grand Design</em> (Hawking), etc. but did could not manage to understand what <a href="http://en.wikipedia.org/wiki/String_theory" rel="nofollow">string theory</a> is, even at a Popular level. I couldn't understand what exactly it is, even at a Popular Science level.
!</p>
<p>Can any one please briefly describe what are its' fundamental postulates? </p>
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<p>I have studied <a href="http://en.wikipedia.org/wiki/Chern%E2%80%93Simons_theory">Chern-Simons (CS) theory</a> somewhat and I am puzzled by the question of how diff. and gauge invariance in CS theory are related, e.g. in $SU(2)$ CS theory. In particular, I would like to know about the relation between large gauge transformations and large diffeos. If you know any good sources, I would be really grateful. Thank you!</p>
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<p>I am not from a physics background. I came to know this while studying probability? Is this because the planets are nearer than the stars so we get average behaviour rather than random. Is law of large number used here? Can anyone explain in detail. </p>
| 261 |
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<p>I am currently looking at changes in DOS when sampling recipocal space finely. More precisely, I am looking at the expressions</p>
<p>$$\rho_\text{1D}(E)\text{d}E = \frac{m}{\pi \hbar} \sum_i \text{H}(E-E_i)\text{d}E, \\
\rho_\text{2D}(E)\text{d}E = \frac{1}{\pi}\sqrt{\frac{2m}{\hbar^2}} \sum_i \frac{n_i\text{H}(E-E_i)}{\sqrt{E-E_i}}\text{d}E$$</p>
<p>where $E_i=\frac{\hbar^2k^2}{2m}$ is taken from the free-electron model. Here $H$ is the Heaviside step-function, and $n_i$ is a degeneracy factor (set to unity).</p>
<p>Let's now say that I can sample $k$-space arbitrarily fine. I want to see how the 1D and 2D cases converge. I can easily see that 2D converges to 3D (i.e. $\sqrt{E}$-behaviour) (see figure below), but what can one expect for the 1D system?</p>
<p>Usually, DOS explanations in books and online only focus on the actual expression and not what happens when you sum up all channels. At least it's not something I have seen.</p>
<p>Any ideas?</p>
<p><img src="http://i60.tinypic.com/352md8g.png" alt="2D DOS"></p>
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] |
<p>The following is a graph of the intensity of Bremsstrahlung generated by accelerating electrons to hit a target vs. its wavelength.</p>
<p><img src="http://i.stack.imgur.com/EbRI9.png" alt=""></p>
<p>I'm wondering what causes the additional peaks for high energies? I have read in my physics script and on Wikipedia that "these peaks are characteristic of the material used as target", but what exactly is going on at the level of atoms?</p>
<p>Does the electron maybe go into some sort of bound state and thus loses - in addition to its kinetic energy - also some potential energy? Why would that only happen for high enough energies?</p>
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<p>I'm currently writing a simulation in python with scipy and matplotlib to reproduce an one dimensional driven diffusive system described in this <a href="http://prl.aps.org/abstract/PRL/v74/i2/p208_1" rel="nofollow">paper</a> from M.R. Evans et al.</p>
<p>The system consists of positive, negative and hole particles. On the left side of the system the positive (negative) particles are produced on the left (right) side with a possibility a and destroyed at the right (left) side with the possibility b. </p>
<p>In some cases the system should show a flip between positive and negative high density states and I'm trying to reproduce this behaviour with my <a href="http://pastecode.org/index.php/view/81369818" rel="nofollow">simulation</a>. But all I can see is an increasing current within my simulation and I can't find any problems in my code explaining such behaviour.</p>
<p>Does anyone here have some experience simulation such or similar systems?</p>
<p>Cheers,
Florian</p>
| 7,980 |
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<p>Wikipedia has this page on <a href="http://en.wikipedia.org/wiki/Gravity_assist">gravity assists</a> using planets. In some cases this effect was used to accelerate the spacecraft to a higher velocity. This diagram shows this in a very <em>oversimplified</em> manner. <img src="http://i.stack.imgur.com/6O45U.png" alt="enter image description here"></p>
<blockquote>
<p>That got me thinking that if light is affected by gravity, and if it slingshots around the black hole/massive object, can't it gain a higher speed than $c$? </p>
<p>What limitations are stopping it from doing this?</p>
</blockquote>
<p>Forgive me if the answer to this question is pretty straight-forward or staring-you-in-the-face kind. I haven't fully understood the mechanics of the gravitational slingshot fully yet, but I couldn't wait to ask this.</p>
<p><strong>Note:</strong> If it is possible (I highly doubt that!), could you provide an explanation using Newtonian mechanics, I'm not very familiar with general relativity because I'm in high school.</p>
| 619 |
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<p>The equations of motion of a double pendulum are well-known. Usually you'd have the them expressed in the rotations $\theta_1(t)$ and $\theta_2(t)$. There are two degrees of freedom. Now consider the case in which the motion at the bottom is constrained in the $x$-direction:</p>
<p><img src="http://i.stack.imgur.com/cQRKL.png" alt=""></p>
<p>There is now only one degree of freedom $\Delta(t)$. Instead of deriving the equation of motion for this system is there a way to use the EOMs of the original (unrestrained) system with a sort of constraint relation to derive the new EOM?</p>
<p>EDIT: Lagrangian</p>
<p><img src="http://i.stack.imgur.com/nVcU6.png" alt=""></p>
<p>resulting in the equation of motions</p>
<p><img src="http://i.stack.imgur.com/J5blx.png" alt=""></p>
<p><img src="http://i.stack.imgur.com/1jOup.png" alt=""></p>
<p>taken from <a href="http://psi.nbi.dk/@psi/wiki/The%20Double%20Pendulum/files/projekt_2013-14_RON_EH_BTN.pdf" rel="nofollow">here</a> (note that $\theta_2$ is drawn with the incorrect sign in the figure above). </p>
<p>And then the constrained equation is (I think)</p>
<p>$$h_1 \sin(\theta_1) = - h_2 \sin(\theta_2) = \Delta(t)$$</p>
| 7,981 |
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] |
<p>Can someone explain to me the concept of symmetric, antisymmetric, and mixed symmetry when talking about the states of identical particles?</p>
| 7,982 |
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<p>Time symmetry is often explained by the example of orbiting objects... What I can't find an explanation for is the moment when an object enters into orbit around another object. That clearly breaks time symmetry, since once object is in orbit (and you reverse time), it will never leave of it's own. Does this mean laws of gravitational motion are not time symmetric? Or is there some other explanation (e.g. entropy of the system)?</p>
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<blockquote>
<p><strong>Possible Duplicate:</strong><br>
<a href="http://physics.stackexchange.com/questions/25378/how-vacuous-is-intergalactic-space">How vacuous is intergalactic space?</a> </p>
</blockquote>
<p>The <em>emptiness</em> of space is explained in many articles... But, space does contain some matter due to these possible reasons:</p>
<ul>
<li><p>During the explosion of supernovae, some elements are created. Not all supernovae nucleosynthesis are successful. So, there should be some scattering of matter into outer space.</p></li>
<li><p>Cosmic rays are coming from interstellar space (and even from sun) in all directions. Would all of the particles have enough efficiency to reach their destination (I mean, take earth or any other interstellar object)?. Some protons or neutrons could be scattered into space.</p></li>
<li><p>Even the gaseous molecules in atmosphere of celestial bodies have possibility to reach escape velocity and go into space.</p></li>
</ul>
<p>Are my assumptions correct? If so, then space must contain elements up to some extent. Isn't it? Or is it due to the infiniteness of space, that these scattered particles are ignored?</p>
| 262 |
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<p>This is part of an admission exam problem, found at
<a href="http://www.sissa.it/mp/admission/tests//2008_common.pdf" rel="nofollow">http://www.sissa.it/mp/admission/tests//2008_common.pdf</a></p>
<p>Consider the Hamiltonian of Kepler problem
$$H(\boldsymbol{r},\boldsymbol{p})= \frac{|\boldsymbol{p}^2|}{2\mu} +\frac{\alpha}{|\boldsymbol{r}|}, \qquad \mu>0>\alpha,$$</p>
<p>where $\boldsymbol{r}\in M=\mathbb{R}^3\setminus\{ 0 \}, \ (\boldsymbol{r},\boldsymbol{p})\in T^*M$
and
$|\boldsymbol{r}|=\sqrt{r_1^2+r_2^2+r_3^2}$.</p>
<p>The quantities
$$\boldsymbol{m}=\boldsymbol{r}\times\boldsymbol{p}, \qquad \boldsymbol{W}=\boldsymbol{p}\times\boldsymbol{m}+ \mu\alpha\frac{\boldsymbol{r}}{|\boldsymbol{r}|}$$
are constants of motion, as is well known.</p>
<p>It is stated then that the flows generated by the functions $m_i$ and $W_i,\ i=1,2,3$ are canonical transformations. </p>
<p><strong>I don't understand</strong> is what is meant by this statement: I mean, I know what a canonical transformation is, but I would appreciate some explanation or reference about this precise statement.</p>
<p>Thanks in advance for the help, and I hope this is formulated in compliance with the rules of this community.</p>
| 7,984 |
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<p>The field must have properties such as <code>planar wave</code> and <code>moving in glass</code> so</p>
<p>$$\bar E=E \left( \hat i t + \hat j \sin(kz-wt) + \hat k \cos(kz-wt+\phi) \right)$$</p>
<p>that is also moving in the x-axis and right-hand-circular-polarized so the phase difference $\phi=\pm \pi/2$ (more on p.557 in Understanding Physics -book and <a href="http://en.wikipedia.org/wiki/Phase_%28waves%29#Phase_difference" rel="nofollow">here</a>). I need some natural constants to express the permittivity with the electric field $\bar E$. The wave must move in glass. It has refractive indices so that $n_1 \sin(\theta_1)=n_2 \sin(\theta_2)$, more <a href="http://en.wikipedia.org/wiki/Refractive_index" rel="nofollow">here</a>.</p>
<p><strong>I. How can I express electric field in Glass?</strong></p>
<blockquote>
<p>$$ \bar D= \epsilon_0 \bar E + \bar P = \epsilon_0 (1+\chi) \bar E =
\epsilon_r \epsilon_0 E$$</p>
<p>where the permittivity applies to the electric field. Now you have the
electric field in terms of permittivity and $\bar D$ is the density of
the wave in the material. $\bar E$ is the outside -field while the
$\bar P$ is the inner polarization -thing.</p>
</blockquote>
<p><strong>II. How can I get planar wave?</strong></p>
<blockquote>
<p>According to this <a href="http://en.wikipedia.org/wiki/Plane_wave" rel="nofollow">here</a>, the
planar wave can be expressed as </p>
<p>$$U(\bar r, t) = A_0 e^{i \left( \bar k \cdot \bar r - wt + \phi \right)}$$</p>
<p>but now I am confused how I can get the phase difference there.</p>
</blockquote>
<p><strong>Perhaps related</strong></p>
<blockquote>
<ol>
<li><a href="http://physics.stackexchange.com/questions/10101/help-me-to-visualize-this-wave-equation-in-time-to-which-direction-it-moves">Help me to visualize this wave equation in time, to which direction it moves?</a></li>
</ol>
</blockquote>
| 7,985 |
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] |
<p>Most of the 5-dimensional Higgs models can be seen, if I understand correctly, as models where the Higgs is a composite. </p>
<p>Now, is this true for <a href="http://en.wikipedia.org/wiki/Noncommutative_standard_model" rel="nofollow">Connes models</a>? It is a model of extra dimensions too, in some sense. And when you look at the papers, at some moment a product of two symbols is substituted by a single $\phi$, so it seems that some composition is at work.</p>
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] |
<p>Knowing that astronomical twilight (i.e. astronomical dawn) is when the sun is 18 degrees below the horizon, I am calculating the astronomical twilight time this way:</p>
<pre><code>Sunrise of day1 - [ (Sunrise of day1 - Sunset of day0) / 180° ] * 18°
</code></pre>
<p>Is that correct?</p>
| 7,987 |
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<p>I guess the Hilbert space of the theory is precisely the space of all gauge invariant operators (mod equations of motion..as pointed out in the answers) </p>
<ul>
<li><p>Is it possible that in a gauge theory the Wilson loops are the only observables? </p>
<p>(...I would vaguely think that if a set of Wilson loops one for every cohomology class of the space-time is the complete set of observables then this is what would be "a" way of defining a Topological Field Theory but may be this is also possible for pure gauge theories in some peculiar limit or on some special space-time geometries..)</p></li>
<li><p>When the above is not true then what are all the pure gauge theory observables?..I guess its only the local observables that is missed by the Wilson loops..</p></li>
<li><p>In general is it always true that all gauge invariant observables are precisely all the polynomials in the fields which are invariant under the action of the gauge group? (..and this is a well studied question in algebraic geometry under the name of Geometric Invariant Theory?..) </p></li>
<li><p>If one has matter in the theory then I guess the baryons and the mesons are the only matter observables? I guess there is no gauge group dependence on their existence? </p>
<p>(..though baryons can always be defined for any anti-symmetric combination of the flavour indices I guess mesons can be defined only if equal amount of matter exists in the conjugate representation of the gauge group also..right?..) </p></li>
<li><p>Why are gauge traces of arbitrary products of matter fields neither baryons nor mesons? (...in arbitrary gauge theories is it legitimate to identify these states as ``chiral primaries" in any sense?..) </p></li>
</ul>
| 7,988 |
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<p>I'm having problem understanding how to compute a functional derivative when it's involved more than one integral, such as the coulomb potential energy functional:</p>
<p>$$ J[\rho] = \frac 12\int \frac{\rho(r)\rho(r')}{|r - r'|} drdr' $$</p>
<p>According to the functional derivative formula I should do something on these lines:</p>
<p>$$ \frac {\delta J}{\delta \rho (r)} = \frac{\partial}{\partial \rho(r)} [ \frac 12 \int \frac{\rho(r) \rho(r')}{|r - r'|}dr' ]$$</p>
<p>In my wrong reasoning I would simply take $\rho(r) $ out of the integral and apply the derivative:</p>
<p>$$\frac {\partial \rho(r)}{\partial \rho(r)} \frac 12 \int \frac{\rho(r')}{|r - r'|} dr' = \frac 12 \int \frac{\rho(r')}{|r - r'|} dr'$$</p>
<p>Which is wrong because the correct result should be:</p>
<p>$$\int \frac{\rho(r')}{|r - r'|} dr'$$</p>
<p>1) I'm quite confused by the notation and how to treat a partial derivative by $\rho(x)$</p>
<p>2) What's the correct way to handle and compute functional derivatives in these cases? I'm actually in a similar situation with a much complex derivative, such as the same thing in the density matrix formalism:
$$ J[\gamma_1] = \frac 12 \int \frac{\gamma_1 (x_1', x_1) \gamma_1 (x_2', x_2) \delta(x_1' - x_1) \delta(x_2' - x_2) dx_1 dx_1' dx_2 dx_2'}{|x_1 - x_2|}$$
$$\frac {\delta J[\gamma_1]}{ \delta \gamma_1 (x_1'. x_1)}$$</p>
| 7,989 |
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] |
<p>Here is the thermal resistance data for three speaker coils disengaged from the speaker cone. Any ideas? I would think it would be a horizontal line.</p>
<p><img src="http://i.stack.imgur.com/kV0Sd.png" alt="http://i.stack.imgur.com/pjKyE.png"></p>
<p><a href="http://en.wikipedia.org/wiki/Thermal_resistance" rel="nofollow">http://en.wikipedia.org/wiki/Thermal_resistance</a>
Absolute thermal resistance of the part, K/W</p>
<p>edit: I also want to point out that the same trend occurred with the loudspeaker functioning as a normal speaker (before I tore out the voice coils to test separately). The reason for this test was that I thought the drop in thermal resistance was due to unwanted cooling effects (actually desirable, but not at the moment) from diaphragm movement, but the drop in thermal resistance vs temperature is still there.</p>
<p>edit: I am going to run the same experiment using a 1/4W resistor today and report my findings.</p>
<p>For x_C = 40, 60, 80:</p>
<ol>
<li>Heat oven to x_C</li>
<li>Measure electrical resistance</li>
<li>put .2W of power into resistor</li>
<li>Let the resistance measurement settle (thermal time constants)</li>
<li>Measure increased electrical resistance</li>
<li>Calculate resulting change in temperature of resistor</li>
<li>Calculate resulting thermal resistance (dT/dR)</li>
</ol>
| 7,990 |
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<p>I did a lab today in Physics in which we launched ball from a spring loaded cannon directly into a pendulum that captured the ball, held it, and swung upwards with it (representing a totally inelastic collision). One question in particular has confused me:</p>
<blockquote>
<p>If the collision between the projectile and pendulum had lasted 1 millisecond, what would the average force have been which the projectile exerted on the pendulum for the long-range case? </p>
</blockquote>
<p>My attempt at a solution is as follows: From all the searching I've been doing online, I've found the equation $F = {{p_f}-{p_i}\over {t}}$. I know $p_f$, $p_i$, and I'm given t. Is my understanding right? Can I go right ahead and crunch these numbers, or do I have an incorrect equation?</p>
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] |
<p>You can dip your hands into a bowl of non-Newtonian fluid but if you are to punch it, it goes hard all of a sudden and is more like a solid than anything else. </p>
<p>What is it about a non-Newtonian fluid that makes it go hard when having a force suddenly exerted on it?
How does it go from being more like a liquid to a solid in such a short amount of time?
Does it change its state as soon as the force has made contact with it?</p>
| 496 |
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<p>I was wondering if anyone here has calculated before the partition function for the Jaynes-Cummings Hamiltonian:
$H_{JC}=\omega_0 (a^\dagger a + \sigma^+ \sigma^-)+g (a \sigma^+ +a^\dagger \sigma^-)$
(Here I have assumed resonance and $a,a^\dagger$ are bosonic operators and $\sigma^-,\sigma^+$ are ladder operators for a spin one half particle)
For the Hamiltonian above the energy of the ground state is zero and corresponds to 0 excitations in the harmonic oscillator and the spin being down.
The excited eigenenergies come in pairs and are given by:
$\omega_{n\pm}=n \omega_0 \pm \sqrt{n}g$.
I am interested in knowing the partition function:
$\mathcal{Z}=\text{tr}( \exp(-\beta H_{JC} ))=1+2\sum_{n=1}^\infty\exp(-\beta \omega_0 n )\cosh(\beta g \sqrt{n}) $
I tried Mathematica to get an analytic expression for the above sum and it did not work.
Any thoughts on whether the summation can be expressed in terms of some special function or how to calculate it numerically in an efficient and reliable way?</p>
| 7,992 |
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] |
<p><img src="http://i.stack.imgur.com/o8Kjj.jpg" alt="enter image description here"></p>
<p>Hey, im having a bit of trouble with the problem in the added photo.
So, there is the cylinder which is attached by a massless rope to a massless pulley, to a box (assume it is a pointed object).
Now, Im supposed to find the acceleration (linear) of the cylinder assuming its going down.
So, I know I can write torques and Newton's laws equations but i could not figure out why my professor said that the acceleration of the box (upwards) is twice the size of the linear acceleration of the cylinder.
Can anybody help figure this issue please.</p>
| 7,993 |
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<p>I have trying to show that the continuum limit of N quantum harmonic oscillators gives rise the the klein-gordon field. However, instead of a usual finite string, I want to do it on a ring. Hence, my Lagrangian is </p>
<p>$$L=\frac{m}{2}(\dot{q_1}^2+\dot{q_2}^2+.... \dot{q_n}^2)-\frac{m \omega^2}{2}[(q_1-q_2)^2+(q_2^2-q_3^2)+....(q_n-q_1)^2]$$</p>
<p>So that the matrix for V is </p>
<p>$V=\begin{pmatrix} 2 & -1 &0 &. &. &. &-1\\-1&2&-1\\0 &-1 &2 &-1 \\.\\.\\.\\-1 &&.&.&.&-1 & 2
\end{pmatrix}$</p>
<p>So that $L=\frac{m}{2}[\dot{x}^2-\omega^2 x^{T}V{x}]$. All the quantities here are matrices.</p>
<p><strong>How do I find the eigenvalues of this matrix?</strong></p>
<p>I tried to find a recursion relation between the characteristic polynomial of $N$ and $N-1$ dimensional matrix, but I failed. Is this the correct method? What other method is there?</p>
<p>After finding the eigenvalues, the lagrangian can be written separated into its normal modes, and the propagator or kernel can be found out easily using that of the free particle. The limit of this as $N\to \infty$ should be the klein gordon field. </p>
<p>But I am stuck on this. Any help will be appreciated. </p>
| 7,994 |
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<p>For our project we have to study an infrared filter. This filter is composed of glass and several layers (nanolayers of titanium oxide, silver and cupper deposited on one side of the glass).<br>
Now we have to determine the reflection coefficient of the coating (i.e. the nanolayers alone) given that of the glass and that of the filter as a whole. </p>
<p>Now our supervisor claims that this is calculated by the formula: $R_{filter} = R_{coating}R_{glass}$. This formula seems a bit odd to us though, f.i. because it doesn't hold in general (one can imagine a material composed of two materials, where one has 100% reflectance and the other 50%, so this material should have 100% reflectance, while according to the formula it has 50%). I wonder if somebody maybe knows if this formula holds, and maybe can explain which assumptions one could make under which one can derive the formula.</p>
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<p><em>If you think that this question is likely to get closed then please do not answer and only say that in the comments since this system doesn't let me delete the question once it has answers.</em> </p>
<p>Everytime I am faced with an analysis of the spectral function it looks like a "new" unintuitive set of jugglery with expectation values and I am unable to see a general picture of what this construction means. I am not sure that I can frame a coherent single question and hence I shall try to put down a set of questions that I have about the idea of spectral functions in QFT. </p>
<ul>
<li>I guess in a Dirac field theory one defines the spectral function as follows, </li>
</ul>
<p>$\rho_{ab}(x-y) = \frac{1}{2} <0|\{\psi_a(x),\bar{\psi}_b(y)\}|0>$ </p>
<p>Also I see this other definition in the momentum space as, </p>
<p>$S_{Fab}(p) = \int _0 ^\infty d\mu^2 \frac{\rho_{ab}(\mu^2)}{p^2 - \mu ^2 + i\epsilon}$</p>
<p>Are these two the same things conceptually?
I tried but couldn't prove an equivalence. </p>
<p>(I define the Feynman propagator as $S_{Fab} = <0|T\big(\psi_a(x)\bar{\psi}_b(y)\big)|0>$)</p>
<ul>
<li><p>Much of the algebraic complication I see is in being able to handle the quirky "_" sign in the time-ordering of the fermionic fields which is not there in the definition of the Feynman propagator of the Klein-Gordon field (..which apparently is seen by all theories!..) and to see how the expectation values that one gets like $<0|\psi_a(0)|n><n|\bar{\psi}_b(0)|0>$ and $<0|\bar{\psi}_b(0)|n><n|\psi_a(0)|0>$ and how these are in anyway related to the Dirac operator $(i\gamma^\mu p_\mu +m)_{ab}$ that will come-up in the far more easily doable calculation of the spectral function for the free Dirac theory. </p></li>
<li><p>Is it true that for any QFT given its Feynman propagator $S_F(p)$ there will have to exist a positive definite function $\rho(p^2)$ such that a relation is satisfied like,</p></li>
</ul>
<p>$S_F(p) = \int _0^\infty d\mu ^2 \frac{\rho(\mu^2)} {p^2 - \mu^2 +i\epsilon}$ </p>
<p>So no matter how complicatedly interacting a theory for whatever spin it is, its Feynman propagator will always "see" the Feynman propagator for the Klein-Gordon field at some level? (..all the interaction and spin intricacy being seen by the spectral function weighting it?..) </p>
<ul>
<li>One seems to say that it is always possible to split the above integral into two parts heuristically as, </li>
</ul>
<p>$$\begin{eqnarray}S_F(p) &=& \sum (\text{free propagators for the bound states})\\
&&+ \int_\text{states} \big( (\text{Feynman propagator of the Klein-Gordon field of a certain mass})\\
&& ~~~~~~~~~~~~~~\times(\text{a spectral function at that mass})\big)\end{eqnarray}$$ </p>
<p>Is this splitting guaranteed irrespective of whether one makes the usual assumption of "adiabatic continuity" as in the LSZ formalism or in scattering theory that there is a bijection between the asymptotic states and the states of the interacting theory - which naively would have seemed to ruled out all bound states? </p>
<p>To put it another way - does the spectral function see the bound states irrespective of or despite the assumption of adiabatic continuity? </p>
| 7,996 |
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<p>please look at the fig first
<img src="http://i.stack.imgur.com/KXMi6.jpg" alt="enter image description here"></p>
<p><br></p>
<p>1) How can you claim that the triangle ABC is same as the triangle PQR?
2) How can you claim that the angle between V1 and V2 is same as the angle between AC and AB?</p>
<p>I think i have forgotten some basic concept from geometry.</p>
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<p>I read in a book the following about <a href="http://en.wikipedia.org/wiki/Pendulum#Compound_pendulum" rel="nofollow">compound pendulum</a> and small displacements: </p>
<ol>
<li><p>There are two points only for which the time period is minimum. </p></li>
<li><p>there are maximum 4 points for which the time period is same. </p></li>
</ol>
<p>Why is this? Can someone please explain? I am familiar with maximum time period being when $k=l$.<br>
In general, time period is $$T=2\pi \sqrt\frac{k^2+l^2}{lg}$$ for small angle approximation.</p>
<p>$k$=Radius of gyration about the centre of gravity, $l$=distance of point of suspension from Centre of Gravity, $g$=gravity</p>
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] |
<p><strong>Background:</strong> Classical Mechanics is based on the Poincare-Cartan two-form </p>
<p>$$\omega_2=dx\wedge dp$$</p>
<p>where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, the so-called Born-reciprocal relativity is based on the "phase-space"-like metric</p>
<p>$$ds^2=dx^2-c^2dt^2+Adp^2-BdE^2$$</p>
<p>and its full space-time+phase-space extension:</p>
<p>$$ds^2=dX^2+dP^2=dx^\mu dx_\mu+\dfrac{1}{\lambda^2}dp^\nu dp_\nu$$</p>
<p>where $$P=\dot{X}$$</p>
<p>Note: particle-wave duality is something like $ x^\mu=\dfrac{h}{p_\mu}$.</p>
<p>In Born's reciprocal relativity you have the invariance group which is the <em>intersection</em> of SO (4 +4) and the ordinary symplectic group Sp (4), related to the invariance under the symplectic transformations leaving the Poincaré-Cartan two-form invariant. The intersection of SO(8) and Sp(4) gives you, essentially, the unitary group U (4), or some "cousin" closely related to the metaplectic group. </p>
<p>We can try to guess an extension of Born's reciprocal relativity based on higher accelerations as an interesting academical exercise (at least it is for me). In order to do it, you have to find a symmetry which leaves spacetime+phasespace invariant, the force-momentum-space-time extended Born space-time+phase-space interval </p>
<p>$ds^2=dx^2+dp^2+df^2$ </p>
<p>with $p=\dot{x}$, $ f=\dot{p}$ in this set up. Note that is is the most simple extension, but I am also interested in the problem to enlarge it to extra derivatives, like Tug, Yank,...and n-order derivatives of position. Let me continue. This last metric looks invariant under an orthogonal group SO (4+4+4) = SO (12) group (you can forget about signatures at this moment). </p>
<p>One also needs to have an invariant triple wedge product three-form </p>
<p>$$\omega_3=d X\wedge dP \wedge d F$$</p>
<p>something tha seems to be connected with a Nambu structure and where $P=\dot{X}$ and $F=\dot{P}$ and with invariance under the (ternary) 3-ary "symplectic" transformations leaving the above 3-form invariant. </p>
<p><strong>My Question(s):</strong> I am trying to discover some (likely nontrivial) Born-reciprocal like generalized transformations for the case of "higher-order" Born-reciprocal like relativities (I am interested in that topic for more than one reason I can not tell you here). I do know what the phase-space Born-reciprocal invariance group transformations ARE (you can see them,e.g., in this nice thesis <a href="http://eprints.utas.edu.au/10689/2/Whole.pdf" rel="nofollow">BornRelthesis</a>) in the case of reciprocal relativity (as I told you above). So, my question, which comes from the original author of the extended Born-phase space relativity, <strong>Carlos Castro Perelman in</strong> <a href="http://vixra.org/abs/1302.0103" rel="nofollow">this paper</a>, and references therein, is a natural question in the context of higher-order Finsler-like extensions of Special Relativity, and it eventually would include the important issue of curved (generalized) relativistic phase-space-time. After the above preliminary stuff, the issue is: </p>
<blockquote>
<p>What is the intersection of the group SO (12) with the <em>ternary</em> group which leaves invariant the triple-wedge product </p>
<p>$$\omega_3=d X\wedge dP \wedge d F$$</p>
</blockquote>
<p>More generally, I am in fact interested in the next problem. So the extra or bonus question is: what is the (n-ary?) group structure leaving invariant the (n+1)-form</p>
<p>$$ \omega_{n+1}=dx\wedge dp\wedge d\dot{p}\wedge\cdots \wedge dp^{(n-1)}$$</p>
<p>where there we include up to (n-1) derivatives of momentum in the exterior product or equivalently</p>
<p>$$ \omega_{n+1}=dx\wedge d\dot{x}\wedge d\ddot{x}\wedge\cdots \wedge dx^{(n)}$$</p>
<p>contains up to the n-th derivative of the position. In this case the higher-order metric would be:</p>
<p>$$ds^2=dX^2+dP^2+dF^2+\ldots+dP^{(n-1)}=dX^2+d\dot{X}^2+d\ddot{X}^2+\ldots+dX^{(n)2}$$</p>
<p>This metric is invariant under SO(4(n+1)) symmetry (if we work in 4D spacetime), but what is the symmetry group or invariance of the above (n+1)-form and whose intersection with the SO(4(n+1)) group gives us the higher-order generalization of the U(4)/metaplectic invariance group of Born's reciprocal relativity in phase-space?</p>
<p>This knowledge should allow me (us) to find the analogue of the (nontrivial) Lorentz
transformations which mix the </p>
<p>$X,\dot{X}=P,\ddot{X}=\dot{P}=F,\ldots$</p>
<p>coordinates in this enlarged Born relativity theory. </p>
<p><strong>Remark:</strong> In the case we include no derivatives in the "generalized phase space" of position (or we don't include any momentum coordinate in the metric) we get the usual SR/GR metric. When n=1, we get phase space relativity. When n=2, we would obtain the first of a higher-order space-time-momentum-force generalized Born relativity. I am interested in that because one of my main research topics are generalized/enlarged/enhacend/extended theories of relativity. I firmly believe we have not exhausted the power of the relativity principle in every possible direction. </p>
<p>I do know what the transformation are in the case where one only has X and P. I need help to find and work out myself the nontrivial transformations mixing X,P and higher order derivatives...The higher-order extension of Lorentz-Born symmetry/transformation group of special/reciprocal relativity. </p>
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<p>We are able to look directly at the sun near sunset and sunrise, which clearly demonstrates the fact that our atmosphere attenuates visible light. Let's imagine it follows the typical attenuation profile.</p>
<p>$$ I = I_0 \, e^{-(\mu/\rho)\rho \ell} $$</p>
<p>Where $(\mu/\rho)$ is the mass-attenuation coefficient in units of $(m^2/kg)$, and $\rho l$ is the mass-thickness (or the area-density, I think it has a few names) in units of $(kg/m^2)$.</p>
<p>The effect of the "soft" sunlight is then predicted as a result of the fact that the mass-thickness of the atmosphere between our eyes and the sun diverges fairly fast as the angle to the sun falls to zero degrees above the horizon.</p>
<p>Let's say that a person is standing on (perfectly spherical) Earth, with their eyes at a known elevation, looking at the sun which lies at a known angle above the horizon. What is the expression for the mass-thickness of air in that line of sight?</p>
<p>The reason I find this non-trivial is that I can't figure out if the density profile of the atmosphere should matter or not. You could reduce it to simple geometry and get an answer, but is there a coherent argument for that being correct? With a clear expression, I'm actually kind of curious if you could measure the mass-attenuation coefficient with just a digital picture. The sun's intensity starts out constant over the circle, and you know the angle between the top and bottom of the sun exactly. So if you could extract intensity data over the vertical diameter maybe you could then do a least-squares function fit to extract out that attenuation coefficient, and even do it for each of the 3 colors. I don't plan on doing that, but it would be a cool science project.</p>
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<p>Acceleration is directed towards the center of the circle in a uniform circular motion. Is it same for the non-uniform circular motion?</p>
| 8,001 |
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<p>In <em>Peskin & Schroeder</em>, chapter 9 introduces the functional methods.</p>
<p>The idea, to recall, is simply to sum over all the possible paths:</p>
<blockquote>
<p>$U(x_a,x_b;T) = \sum_{\text{all paths}} e^{i . \text{phase}} = \int Dx(t) e^{i . \text{phase}} $</p>
</blockquote>
<p>Then, it happens that the phase could be <strong>identified</strong> with the action. But I have not understood how this could be done ... Could anyone clarify that point?</p>
<p>Thanks in advance.</p>
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] |
<p>Consider a pair of (possibly rotating) charged black holes with masses m1, m2 and like charges q1, q2. It seems that under certain conditions gravitational attraction should exactly cancel electrostatic repulsion and a stationary spacetime will result. </p>
<blockquote>
<p>What are these conditions?</p>
</blockquote>
<p>The point charges analogy suggests the equation</p>
<p>k q1 q2 = G m1 m2</p>
<p>However, it is by no means obvious this equation is the correct condition except in the large distance limit. Also:</p>
<blockquote>
<p>Is it possible to write down this solution of Einstein-Maxwell theory in closed form?</p>
</blockquote>
| 8,003 |
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<p>In the next attachements are:
1. Exercise 0.2.5 which I want help with.</p>
<ol>
<li>Proposition 0.2.1 and its proof.</li>
</ol>
<p>Now, basically a few things are changed in the theorem, I don't think I can use here the definition of s(t) in the proof of prop0.2.1 cause its s(t)=0, I don't think I can use this trick here.</p>
<p>Other thoughts that I had, obviously if I plug m=0 into prop0.2.1 I get that I should have:
$$\frac{d\gamma^1}{du}=\pm \frac{d\gamma^2}{du}$$, and $$\frac{d\gamma^2}{du}=a$$.</p>
<p>My question is how do I satisfy condition b in the theorem, I guess this x should be $$\pm Id +constant$$</p>
<p><img src="http://i.stack.imgur.com/uNJZS.png" alt="1"></p>
<p><img src="http://i.stack.imgur.com/Gftvm.png" alt="2"></p>
<p><img src="http://i.stack.imgur.com/Eo0PX.png" alt="3"></p>
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<p>I am trying to understand the difference between Bessel functions and modified Bessel functions (simply googling is yielding complicated, non-intuitive answers). I was under the impression that one allowed for a complex parameter while the other did not - is this true? </p>
<p>My question stems from trying to understand the radial part of the Hydrogen eigenproblem (with $u = rR(r)$):</p>
<p>$$ \frac{d^2u}{dr^2} = \left[ \frac{l(l+1)}{r^2} - k\right] u(r) $$</p>
<p>which is solved by a linear combination of Spherical Bessel functions and Neumann functions:</p>
<p>$$ u(r) = Ar j_l(kr) + Brn_l(kr) $$</p>
<p>Is this solution valid for both real and imaginary $k$? </p>
<p>For reference, this linear combination is from Griffiths' <em>Introduction to Quantum Mechanics</em>, Equation 4.45.</p>
| 8,005 |
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<p>Is it possible to have a membrane that will not let a liquid through it at normal pressures due to gravity, but pass that liquid when substantially pressurised?</p>
<p>For instance, a few inches of water (say 0.1psi) would be blocked, but 100psi would pass through. </p>
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] |
<blockquote>
<p>Quantum entanglement occurs when particles such as photons, electrons,
molecules as large as buckyballs, and even small diamonds
interact physically and then become separated; the type of interaction
is such that each resulting member of a pair is properly described by
the same quantum mechanical description (state), which is indefinite
in terms of important factors such as position,momentum, spin,
polarization, etc. <a href="http://en.wikipedia.org/wiki/Quantum_entanglement" rel="nofollow">Source: Wikipedia</a></p>
</blockquote>
<p>All objects can get quantumly entangled, but to get a more sustained quantum entanglement with a long <a href="http://en.wikipedia.org/wiki/Quantum_decoherence" rel="nofollow">dechoherence time</a>, the objects has to have equal properties. Do all properties have to be equal or could some properties be different?</p>
<p>For instance: Form seem to be important for quantum entanglement, but is size important? Could a <a href="http://www.nature.com/news/entangled-diamonds-vibrate-together-1.9532" rel="nofollow">small diamond be quantum entangled</a> with a larger diamond with the same shape?</p>
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<p>I am currently reading "Magnetic Monopoles" of Ya. Shnir. My problem is I can not retrieve a result the author provides in the first chapter of the first part. In this chapter, he studies the non-relativistic scattering of an electric charge on a magnetic one. </p>
<p>The author writes [p.5, near eq. (1.13)]:</p>
<blockquote>
<p>... the appearance of an additional term in the definition of the angular momentum $(1.11)$ originates from a non-trivial field contribution. Indeed, since a static monopole is placed at the origin, its magnetic field is given by $(1.1)$. Then the classical angular momentum of the electric field of a point-like electric charge, whose position is defined by its radius vector $\mathbf{r}$, and the magnetic field of a monopole is a volume integral involving the Poynting vector</p>
<p>\begin{align}
\tilde{\mathbf{L}}_{eg} &= \dfrac{1}{4\pi}\int \mathbf{r'} \times \left [ \mathbf{E} \times \mathbf{B}\right] d^3r'\tag{L.1}\\& = - \dfrac{g}{4\pi} \int d^3r' \left( \mathbf{\nabla}'\cdot \mathbf{E}\right) \hat{\bf r}' \tag{L.2}\\ &= -eg\hat{\bf r} \tag{L.3}
\end{align}</p>
<p>where we perform the integration by parts, take into account that the fields
vanish asymptotically and invoke the Maxwell equation</p>
<p>\begin{equation}\left(\mathbf{\nabla}' . \mathbf{E} \right) = 4 \pi e \delta^{(3)}\left( \mathbf{r} - \mathbf{r}'\right)\end{equation}
...</p>
</blockquote>
<p>The magnetic field is </p>
<p>$\mathbf{B} = \dfrac{g}{r^3} \mathbf{r} \tag{1.1}$</p>
<p>The generalised angular momentum is </p>
<p>$\mathbf{L} = \mathbf{r} \times m\mathbf{v} - eg \hat{\bf r} \tag{1.11}$</p>
<p>The author gives how he got $(L.2)$ from $(L.1)$ but I do not know how to do? Have you any idea?</p>
| 8,008 |
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<p>My understanding of pseudovectors vs vectors is pretty basic. Both transform in the same way under a rotation, but differently upon reflection. I might even be able to summarize that using an equation, but that's about it.</p>
<p>Similarly, I can follow arguments that pseudovectors behave differently in "mirrors" than vectors. But my response to this is always: Okay, so what? When would I ever "do physics" in a mirror?</p>
<p>The usefulness eludes me. I'd like to gain a better understanding of the importance of this difference.</p>
<ul>
<li>When is it useful for an <em>experimental physicist</em> to distinguish between the two?</li>
<li>When is it useful for a <em>theoretical physicist</em> to distinguish between the two?</li>
</ul>
<p>I believe symmetry is important to at least one of these, but would appreciate a practical rather than abstract argument of when one has to be careful about the distinction.</p>
| 8,009 |
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<p>Electromagnetic waves travel mostly in vacuum medium, in outer space, but sometimes in gaseous media, such as in gaseous atmospheres involved in nebulae. If electromagnetic-waves come anywhere in the vicinity of a black-hole, the whole of it enters the black-hole. Can we suppose, that the event-horizon of the black-hole acts in a way like an interface, separating two optical media with different refractive indices <strong>:</strong> vacuum (the medium from which EM waves are "incident"), and the unknown medium beyond. If, this supposition holds true, then, we would know one thing at least regarding this unknown medium, that, EM waves travel in this medium with a velocity > c, as this medium is rarer, than vacuum itself.</p>
<p>(Question of an amateur)</p>
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] |
<p>Question inspired by a question thread <a href="http://www.reddit.com/r/askscience/comments/ik4jj/during_spiral_galaxy_formation_how_does_gas/" rel="nofollow">here</a>.</p>
<p>So when there's lots of dust in a galaxy, the galaxy tends to collapse into a spiral galaxy (to maintain angular momentum and to minimize gravitational potential energy). Is this the same thing that happens in the inner regions of the solar system? The outer regions have less dust so the orbits of minor planets "out there" tends to be more elliptical.</p>
<p>And could this perhaps mean that the orbits of planets tend to be more coplanar around stars of higher metallicity?</p>
| 8,011 |
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<p>Random question that popped into my mind after a 4-hours power outage. Let us assume that I am eating an extra dessert (250 kcal) and that I am using a bike and a generator to power my laptop (it consumes 10W while idle). How much time can I use the laptop?</p>
<p>I computed this as follows:
$$
t = \frac{E}{P} = \frac{250 kcal}{10W}=\frac{250 * 4,18KJ}{10J/s} = 1.045 * 10^5s=30h
$$</p>
<p>The above calculation assumes 100% efficiency. But even if we assume 10% efficiency, I would still be able to power my laptop for $3h$ and all this by eating just one extra dessert.</p>
<p>Questions: Are my calculations correct? Are my assumptions reasonable? Why don't we see a proliferation of bike-laptop devices?</p>
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<p>As the question states, what is our current best machine for translating falling gravitational potential energy, such as a large weight, into launching a smaller projectile vertically? A lever? A trebuchet? What sorts of efficiency levels have been achieved?</p>
<p>What I'm looking for is the best way to translate:</p>
<p>$$ g * h * m_{\textrm{weight}} * \textrm{efficiency} = \frac{1}{2} * m_{\textrm{projectile}} * v^2. $$</p>
<h1>Edit</h1>
<p>Here's an example: I want to launch a 10kg mass at 100m/s using only gravitational energy (no chemical rockets, no rail guns, etc.). What's our most efficient machine to do this, and what percentage of the energy of the falling weight will actually be transferred to the projectile? This would get me an idea of the size of the weight and how much it has to fall to accomplish this. The only machine I know of that does this on a large scale and for decent size masses is a trebuchet, so this might be a good starting point unless there are others I'm unaware of.</p>
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<p>The scenario is the following, I am given 2 loops with the same radius, r, a distance of d, and same current of I. In the left loop the current goes counter clockwise, in the right loop the current is clockwise. The two loops centers lie on the same axis which are perpendicular to the plane of the loops. I am asked to find the magnetic flux of the left loop due to the current on the right loop. </p>
<p>I know that the magnetic flux of a loop is $\phi=B\pi r^2$, where $B=\dfrac{ \mu_0I}{2R}$ So how exactly do I find the Total magnetic flux on the loop due to the magnetic flux on the other? Am I going to add or subtract? I get confused at this part</p>
| 8,014 |
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<p>This was an experiment I saw in my son's workbook. It said to mark out the top of your forehead and the bottom of your chin on a mirror using a whiteboard marker. Then slowly move backwards, and investigate what happens to the size of the reflection subjective to the two marks made. It actually got me quite flabbergasted. I always thought the reflection would get smaller as you moved away from the mirror.</p>
<p>Why is this?</p>
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<p>My question started out as finding the maximum speed of a go kart, taking into account only the drag forces as force opposing the motor. </p>
<p>I've done some investigation to find:</p>
<p>$$
F_{drag}=\; \frac{\rho v^{2}\ C_{x} A_{f}}{2}
$$</p>
<p>Where $C$ = drag coefficient, and $\rho$ is the density of air. </p>
<p>Further, with $ F_{motor}=\dfrac{P_{motor}}{v} $ and the condition that at maximum speed, the acceleration will be 0, net force will be 0 as well, so $F_{motor}=F_{drag}$. </p>
<p>Combining equations, $F_{drag}=\dfrac{P_{motor}}{v}$, so $\dfrac{P_{motor}}{v}=\; \dfrac{\rho v^{2} C A_{f}}{2}$</p>
<p>Simplifying gives $P_{motor}=\; \dfrac{\rho\cdot v^{3}C\cdot A_{f}}{2}$</p>
<p>Solving for $v$ yields: </p>
<p>$$v=\; \sqrt[3]{\dfrac{2P_{motor}}{\rho C A_{f}}}$$</p>
<p>Converting to units of miles per hour for $v$, and horsepower for $P_{motor}$:</p>
<p>$$v=\; 2.2 \sqrt[3]{\dfrac{HP_{motor} \cdot 745\cdot 2}{\rho\mbox{C}A_{f}}}$$</p>
<p>From <a href="http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCIQFjAA&url=http%3A%2F%2Fdidattica.uniroma2.it%2Ffiles%2Fscarica%2Finsegnamento%2F149462-Machine-Design%2F20685-Go-kart-drag-paper&ei=0SrlU7_lN8PeoATRioGwCQ&usg=AFQjCNGBKMBbB8nqQCbTAJ-4XoqkWvDQBQ&sig2=trMMnDfg5ad8F9vZaM_bQA&bvm=bv.72676100,d.cGU" rel="nofollow">this pdf</a>, I've found values of $\rho = 1.2 \frac{kg}{m^{3}}$, $C=.8$, and $A_{f}=.57$.</p>
<p>So! Questions: </p>
<ol>
<li><p>Are my equations correct? I'm particularly concerned about the third root portion, given that drag is a quadratic. Doesn't connect in my mind, so perhaps my substitutions are faulty?</p></li>
<li><p>Are the values I have for those constants for ideal values, (i.e. perfectly enginneered kart, very unlike what I'd be able to construct myself). If so, by what general percent will my values be different?</p></li>
<li><p>Finally, is it unrealistic to call all forces other than drag insignificant? If so, by what general percent will my values for speed be too high?</p></li>
</ol>
<p>For reference, this calculation has a 5 hp engine at a max speed of 52 mph, 15 hp at 75 mph, and 35 hp at 100 mph.</p>
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<p>I am puzzled why we always see the same side of the Moon even though it is rotating around its own axis apart from revolving around the earth. Shouldn't this only be possible if the Moon is not rotating?</p>
| 263 |
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<p>The title might be misleading, but my question is in regard to what happens when we reach temperatures close to absolute zero (Kelvin). I've found different quotes as to what happens on the low end of the scale:</p>
<blockquote>
<p>At about 10 micro degrees Kelvin, Rubidium atoms move at about 0.11
mph (0.18 km/hr) — slower than a three-toed sloth, says physicist Luis
Orozco of the University of Maryland.</p>
</blockquote>
<p>But what happens if we turn it around, and (hypothetically) increase the speed of the atoms to speeds close to c?</p>
| 8,017 |
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<p>First, I apologize for being a mathematician and having no scientific background in physics. The following questions on color came up in a discussion at lunch and I would be very happy to get some answers or corrections of my understanding.</p>
<ul>
<li>The first question is, what a color the human eye can see is in the first place. According to <a href="http://en.wikipedia.org/wiki/CIE_1931_color_space#Tristimulus_values" rel="nofollow">this wikipedia article</a>, the human eye has three cone cells $L$, $M$ and $S$ responsible for color vision which can be stimulated each in a one dimensional way, i.e. ''more'' or ''less''. Combining the stimulation of all three, one gets a $3$-dimensional space called the <a href="http://en.wikipedia.org/wiki/LMS_color_space" rel="nofollow">LMS color space</a>. Is a color the human eye can see just a point (in a specific range) in this LMS-space? A negative answer could be that time plays a role (switching between e.g. two points of the LMS-space with a certain frequency defines a color). The model is made in the way that only one point in the space at a time represents a state of the eye at that time and not many.</li>
</ul>
<p>For the following questions, I suppose that the answer to the above question is positive, e.g. a color the human eye can see is defined by a specific point in the LMS-space. By experiments, one could possibly determine the body in the LMS-space, the human eye can see (this is what I meant by the ''specific'' point). Of course, this may differ from person to person but perhaps one can average over some and get an average body $AB$ in the LMS-space. Points inside are now exactly the colors the human eye can see.</p>
<ul>
<li><p>Now there is this <a href="http://en.wikipedia.org/wiki/CIE_1931#CIE_xy_chromaticity_diagram_and_the_CIE_xyY_color_space" rel="nofollow">CIE color space</a> which is vizualized by this $2$-dimensional picture looking like a <a href="http://en.wikipedia.org/wiki/File%3aPlanckianLocus.png" rel="nofollow">sole of a shoe</a>. As far as I understand, there is a bijection (continuous?) from the LMS-space above to this CIE-space where two axes describe the so called <a href="http://en.wikipedia.org/wiki/Chromaticity" rel="nofollow"><strong>chromaticity</strong></a> and one axis describe the so called <a href="http://en.wikipedia.org/wiki/Luminance" rel="nofollow"><strong>luminance</strong></a>. What is the figure shoe sole now exactly? Is it a certain hyperplane in the $3$-dimensional CIE color space to which the luminance-axis in perpendicular? (Perhaps one can say something like ''it is the color at full luminance''. Note, that black isn't in here.)</p></li>
<li><p>The <a href="http://en.wikipedia.org/wiki/RGB_color_space" rel="nofollow">RGB color space</a> has the drawback of having a <a href="http://en.wikipedia.org/wiki/Gamut" rel="nofollow">gamut</a> not filling the whole visible range. Why doesn't one use the frequencies of the $L$, $M$ and $S$ cone cells as the primary colors for technical devices like screens? Couldn't one describe the whole visible range then (i.e. wouldn't the gamut be the whole shoe sole then)?</p></li>
<li><p>My last question concerns this old painters belief, that every color can be obtained by mixing red (R), yellow (Y) and blue (B) or the belief that every color can be obtained by mixing cyan (C), magenta (M) and yellow (Y). Both models, the RYB and the CMY model, are subtractive. When one mixes these colors, doesn't one ignore the luminance? I.e. don't the painters want to say that they can obtain each <strong>chromaticity</strong> by mixing these colors? (Of yourse, if they want to say this, it is also not really true since the gamut of these three colors determine a certain region in the shoe sole plane from the previous questions which is not the complete sole.)</p></li>
</ul>
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<p>I think <a href="http://mathoverflow.net/q/165038/14414">this</a> can revolutionize the quantum world! Any ideas on how to impress physicists to get a full fledged funding for research?</p>
| 8,019 |
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<p>As title says, how does a system interact with environment? I realize that this interaction can lead to interference terms and non-diagonal terms in density matrix being reduced (quantum decoherence). But what exactly is this interaction of system and environment? What would be the example of this interaction and when does this interaction occur? </p>
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<p>A vehicle moves from rest to a certain distance. Would the distance time graph always be curved ? Why cannot it be a straight line ? </p>
<p>For example, a car moving from rest to 100m, with a constant velocity of 10m/s . So, if i draw a distance time graph, the line will start from s=0 and end at s=100 ( with t=0 to t=10 ). Would this graph always be curved ? Why can't it be a straight line ?</p>
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<blockquote>
<p>Nonlinear field theories contain a large number of localized solutions.</p>
</blockquote>
<p>I have found this text in a article. What I don't understand is "what is localized?". Is it refer defining position of a particle or a wave?
Can someone give me an elaboration with example?</p>
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<p>In the book <em>String Theory and M-Theory</em> by K. Becker, M. Becker and J.H. Schwarz:</p>
<ol>
<li><p>Why is the potential for moduli given by eq (10.168):
$$\tag{10.168 }V(T,K) ~=~ \frac1{4\mathcal{V}^3} \Big( \int_{CY_4} F \wedge \star F - \frac16 \chi T_{M2} \Big)?$$<br>
Maybe the answer of this question is trivial, but I cannot see how $T$ and $K$ enter this picture and get $V(T,K)$ as their potential.</p></li>
<li><p>How can one arrives at eq (10.181):
$$\tag{10.181} V~=~e^{\mathcal{K}} \Big( G^{a \bar{b}} \mathcal{D}_a W \mathcal{D}_\bar{b} \bar{W} - 3|W|^2 \Big)?$$</p></li>
</ol>
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<p>This is a rather soft question, but I would like to know how physicists would approach a problem which seems to be hard from the mathematical prospective. </p>
<p>The Grimmett percolation model is defined as follows. Consider the $\mathbb{Z}^2$ lattice and randomly assign an orientation for every bond of the lattice in the following way. Let $p$ be the percolation parameter and for every horizontal bond orient it rightward with probability $p$ and leftwards otherwise. The same rule applies for vertical bonds: a bond is oriented upwards with probility $p$ and downwards otherwise. Thus we get an oriented graph supported on the whole lattice $\mathbb{Z}^2$. </p>
<p>Grimmett conjectures that for all $p \neq 1/2$ there is an infinite directed path from the origin to infinity. By coupling with classical bond percolation it is not hard to show that at $1/2$ the system does not percolate - so presumably $1/2$ is a critical point. On the other hand, the conjecture is quite old and seems to be very far from resolution mainly because all the known methods used for other percolation models fail to work. </p>
<p>I am wondering if this model has any meaning in statistical mechanics and if so how a physicist would approach Grimmett's conjecture?</p>
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<p>Often times in quantum field theory, you will hear people using the term "vacuum expectation value" when referring to the minimum of the potential $V(\phi )$ in the Lagrangian (I'm pretty sure every source I've seen that explains the Higgs mechanism uses this terminology).</p>
<p>However, a priori, it would seem that the term "vacuum expectation value" (of a field $\phi$) should refer to $\langle 0|\phi |0\rangle$, where $|0\rangle$ is the physical vacuum of the theory (whatever that means; see <a href="http://physics.stackexchange.com/questions/75834/the-vacuum-in-quantum-field-theories-what-is-it">my other question</a>).</p>
<p>What is the proof that these two coincide?</p>
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<p>Would you outlive everyone?</p>
<p>I'm coming from the point of view that time would be experienced more slowly (although not from your point of view) the denser the gravity gets.</p>
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<p>Hans de Vries (who happens to be a no-longer-active physics.SE user) has an online book (referenced below) in which ch. 6 is a presentation of an object he calls the Chern-Simons current, electromagnetic spin density, or Chern-Simons electromagnetic spin:</p>
<p>$ C^a = \frac{\epsilon_0}{2} \epsilon^{abcd} A_b F_{cd} = \epsilon_0 \epsilon^{abcd} A_b \partial_c A_d $ .</p>
<p>He has a long and detailed presentation of this thing, including graphs and examples. Unfortunately I'm not having much luck extracting from this what he claims is the interpretation of it, whether his interpretation is standard, and whether it has a classical interpretation. He references Mandel and Wolf (which I don't have access to), but what they apparently present is a different expression, $\epsilon_0 \textbf{E}\times\textbf{A}$, and refer to it as the intrinsic angular momentum of the electromagnetic field. De Vries says that $C^a$ is the natural way of making this tensorial. It seems hard to check whether what he's saying is standard, since he <a href="http://www.physicsforums.com/showpost.php?p=1535058&postcount=1" rel="nofollow">says</a>, "The derivations (which I had to do myself since somehow one can't find these anywhere) and many details can be found in my paper..." (linking to a paper that duplicates the material in the book).</p>
<p>The expression is manifestly classical, so I don't really see how it's to be interpreted as the intrinsic or spin contribution to the field's angular momentum. The classical/quantum interpretation is also obscured because factors of $\hbar$ start appearing at de Vries' eq. 6.6, but these equations are supposed to be justified somewhere later on.</p>
<p>It seems odd to me that this is written as a product of the four-potential and a derivative of the four-potential. This makes it not manifestly gauge-invariant. If I was going to write down a density of angular momentum for the electromagnetic field, I would start from the stress-energy tensor, which is a product of $F$ with $F$, and therefore independent of gauge.</p>
<p>It's not at all obvious to me what one would even mean by a spin density for the electromagnetic field. I guess for a classical fluid of electromagnetic radiation in equilibrium (e.g., the kind of environment we had during the early universe), I would define a comoving frame and look at the amount of angular momentum $L^{ab}=r^ap^b$ in a small volume element. But that clearly isn't going to work for the electromagnetic field in general, since you can't define a comoving frame for, e.g., an electromagnetic plane wave.</p>
<p>It does make sense that the expression is manifestly translation-invariant, since if there is some sensible way to split the angular momentum into spin and orbital parts, only the orbital part should depend on one's choice of axis.</p>
<p>De Vries, Understanding Relativistic Quantum Field Theory, <a href="http://www.physics-quest.org/" rel="nofollow">http://www.physics-quest.org/</a> , ch. 6</p>
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<p>I am practicing for an exam in my Physics $2$ course.</p>
<p>One of a previews exam questions described a plate capacitor and asked
to calculate the initial energy $U_{0}$, then a dielectric table
was inserted between the plates and I was asked to calculate the energy,
$U_{1}$, at this state.</p>
<p>I got that there was less energy in the final state than there was
at the initial state, i.e
$$
\triangle U=U_{1}-U_{0}<0
$$</p>
<p>The question asked if the table is attracted to the plates or repealed
from it, and the answer claimed that since there was less energy at
the final state then it means that the table is attracted to the plates.</p>
<p>I lack intuition on this and I don't understand how the conclusion
was made.</p>
<p>I tried to think it with terms of work done by the electrical field:
If I assume that the table is attracted to the plates then I think
that the work is negative (since I don't have to do actual work because
there is an attraction),</p>
<p>and since there was negative work done $W<0$ I have $$U_{1}=U_{0}+W<U_{0}$$
The other case leads to $U_{1}>U_{0}$ and so I get that the first
is the one that occurs.</p>
<p>But I am not to sure about this argument since $$W=\int F\cdot dl$$
and since there is an attraction the direction of the path is the
same as the direction of the force and so $W>0$.</p>
<p>Can someone please help me understand why there is an attraction ?</p>
| 8,028 |
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<p>I have a question in <a href="http://www.damtp.cam.ac.uk/user/tong/string/bq4.pdf" rel="nofollow">David Tong's Example Sheet 4</a>
Problem 5b, how to verify the last equation (*) on p.2? (<a href="http://www.damtp.cam.ac.uk/user/cdab3/teaching/StringSoln3.pdf" rel="nofollow">There is a solution for example sheet 3</a>, but seems to be no solution for example sheet 4.)</p>
<blockquote>
<p>Problem 5b:</p>
<p>Show that the equations of motion arising from the Born-Infeld action are equivalent to the beta function condition for the open string,<br>
$$\beta_\sigma\left(F\right)=\left( \frac{1} {1 -F^2} \right)^{\mu \rho}\partial_\mu F_{\rho\sigma }=0 $$
<strong>Note:</strong> To do this, it will prove very useful if you can first show the following results:<br>
$$∂_μ\left[\operatorname{tr} \ln(1 − F^2)\right] = −4 ∂_\rho F_{μ\sigma}\left(\frac{F}{1-F^2}\right)^{\sigma \rho }
$$ </p>
<p>which requires use of the Bianchi identity for $F_{\mu \nu }$ and </p>
<p>$$ \tag{*}
\begin{align}
\partial _\mu \left( \frac{ F}{1-F^2} \right)^{\mu\nu}
&=
\left( \frac{ F}{1-F^2} \right)^{\mu\rho} \partial_\mu F_{\rho\sigma} \left( \frac{ F}{1-F^2} \right)^{\sigma \nu}
\\ &\qquad+ \left( \frac{1} {1 -F^2} \right)^{\mu \rho} \partial_\mu F_{\rho\sigma}\left( \frac{ 1 }{1-F^2} \right)^{\sigma \nu} \end{align}$$ </p>
</blockquote>
<p>In addition, as given in question 5a </p>
<blockquote>
<p>$$F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} $$</p>
</blockquote>
<p>My attempt to prove the problem:</p>
<p>LHS
$$\partial_{\mu} \left( \frac{ F}{1-F^2} \right)^{\mu\nu} = \partial_{\mu} \left[ F^{\mu}_{\alpha} \left( \frac{1}{1-F^2} \right)^{\alpha \nu} \right] = \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + F^{\mu}_{\alpha} \partial_{\mu} \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} \tag{1} $$</p>
<p>Using the formula in <a href="http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf" rel="nofollow">matrix cookbook</a> for the derivative of inverse matrix, Eq. (53) </p>
<p>Eq. (1) becomes
$$\left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 F^{\mu}_{\alpha} \left[ \frac{1}{1 -F^2} \left( \partial_{\mu} F \right) F \frac{1}{1-F^2} \right]^{\alpha \nu} $$<br>
$$= \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + 2 \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} \tag{2} $$</p>
<p>The second term in Eq.(2) cancels the second term in the RHS in the problem sheet equation.
We then need to show
$$ \left( \partial_{\mu} F^{\mu}_{\alpha} \right) \left( \frac{1}{1 -F^2} \right)^{\alpha \nu} + \left( \frac{F}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{F}{1-F^2}\right)^{\sigma \nu} - \left( \frac{1}{1 -F^2} \right)^{\mu \rho} \left( \partial_{\mu} F \right)_{\rho\sigma} \left(\frac{1}{1-F^2}\right)^{\sigma \nu} =0 \tag{3} $$</p>
<p>then I didn't find a way to show Eq. (3) hold. I tried to combine the second and third term, and rearrange them, but didn't got a simple expression.</p>
| 8,029 |
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<p>I am posing this question with condensed matter systems in mind. <strong>Is it, in principle, possible to obtain emergence using the renormalization group (RG)?</strong> I read in X.-G. Wen's book (Quantum Field Theory of Many-Body systems) that "<em>we cannot use the renormalization group approach to obtain the emergence of qualitatively new phenomena</em>" (Sec. 3.5.4 on page 110). It seems to me that the statement should have been something like RG, <strong>in practice</strong>, cannot be used to capture emergence. Or, more precisely, local perturbative RG schemes are not suited for capturing emergence.</p>
<p>Here is what confuses me: if emergence is nothing but the statement that the low-energy degrees of freedom may not be the same as the high-energy, microscopic degrees of freedom in a theory, then <strong>in principle</strong> RG should be able to capture emergence, right?</p>
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] |
<p>For helping with judging <a href="http://en.wikipedia.org/wiki/Noctilucent_cloud" rel="nofollow">NLC</a> candidates (are they NLC or not) I have a set of formulas to calculate the minimum altitude (in km) of the candidate given an observed altitude (in degrees) of the candidate (using known stars, like <a href="http://en.wikipedia.org/wiki/Capella_%28star%29" rel="nofollow">Auriga</a>, as reference points) and the date/time (giving the Sun's altitude). </p>
<p>A typical observed altitude is 10 degrees, largely unaffected by refraction and a typical result is 70 km (ruling out any other type of cloud). The formulas are limited to only work in the horizontal direction (azimuth) of the Sun (that is under the horizon), but this is not a problem in practice.</p>
<p>Reasonable assumptions are used, except one: no atmospheric effects like refraction and scattering are accounted for.</p>
<ol>
<li><p>Are NLCs only visible to the naked eye if directly exposed to sunlight (the Sun directly visible by the NLC)? Or is scattered light enough to make them visible?</p></li>
<li><p>What about refraction? Light coming to the NLC goes through near vacuum when it is close to the NLC so the refraction must be less than for an observer on the ground, but what is the exact figure (when the Sun is at the horizon as seen by the NLC)?</p></li>
</ol>
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<p>In <em>BRST Symmetry in the Classical and Quantum Theories of Gauge Systems</em>, Henneaux says the Fock representation is not applicable to an odd number of constraints. Then he goes on to say that the Kugo-Ojima quartet requires the constraints to be in pairs. For BRST theories, when are they not in pairs? </p>
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<p>We all know farady effect is observed in linearly polarized wave when it passes through a dielectric medium and magnetic field is along the direction of propagation. Is this phenomenon observable in circularly polarized waves?
<img src="http://i.stack.imgur.com/q0H18.png" alt="enter image description here"></p>
<p>Circularly polarized wave travel through the medium with different phases. But as soon as they are out of the medium they retain their polarization. So basically a circularly polarized wave 's polarization is not affected but only it's phase. Whether we split circularly polarized wave into two linearly polarized wave and take the equation of form:
E(z)=E.e^[i{(2πz/λ)+ϕ}]----------- eq.1 (I am unable to add equations from MS word equation object)</p>
<p>Frequency and wavelength remain same for the em wave when it is entering and when it is leaving the dielectric medium. Say the em wave is split into two linearly polarized waves- one is sin wave along yz plane and the other is cos wave along xz plane. Both waves rotated by same angle due to faraday rotation because rotation is not dependent on plane polarization or phase of the wave. When we combine them again only the phase(ϕ) of circularly polarized wave changes. First question is this theory correct?</p>
<p>Second question is how do we calculate the change in phase. For example, a linearly polarized wave's change in polarization is given by experimental formula of becquerel equation(change in angle of polarization(β)=Verdet constant(V) X Magnetic field(B) X Length of dielectric(d), see the figure to get an idea ). Is there any experimental formula for circularly polarized waves? </p>
<p><strong>Edit</strong>: I guess the change in phase of circularly polarized wave follows different equation as seen in the becquerel equation. I am unable to find any experimental data on this. </p>
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<p>I've just started reading Feynman and Hibbs path integrals and Quantum mechanics after a decade hiatus from my undergraduate math degree (including a few semesters of physics for engineers). It would be tremendously helpful to see the step by step solutions to some of the first set of problems (p27-28, #2.1 and on)
As many as folks are willing to post! Or if anyone knows of the solutions being available on-line that would also be helpful. </p>
<p>The specific concept question is finding the extremum of functional that is the integral of the lagrangian of a system (classical action).
The easiest example is 2.1, show that for L=m/2 (dx/dt)^2, the extremum is m/2*(Xb-xa)^2/(tb-ta). I have tried manipulating the integral by replacing 'partial of L with respect to x' with 'd/dt (partial L with respect to dx/dt) but haven't gotten there. I know it's the easiest problem but I think if I see an example, I will be able to apply what I learn to the harder subsequent problems </p>
<p>I hope that's enough to come off "hold"
Thanks!</p>
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<p>Just wondering when water undergoes electrolysis why does oxygen goes towards cathode and hydrogen towards anode why not the other way round</p>
| 8,035 |
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<p>Consider two parallel-plate capacitors $C_1, C_2$ in series. For the "equivalent" circuit, clearly
$$ Q_{equiv} = C_{equiv}V$$
should hold, where V is the total voltage drop between input and output. It is also obvious that
$$ Q_1 = Q_2 $$
where $Q_1$ and $Q_2$ are the charges (plus and minus on opposite sides) accumulated on each of the two sets of plates. Obviously in general,
$$ V_1 \neq V_2 \qquad \text{}$$</p>
<p>However, just because we have two capacitors with equal charge inside the circuit why is it true that for the equivalent circuit</p>
<p>$$Q_{equiv} = Q_1 = Q_2$$</p>
<p>I see no obvious reason why this should be true, and this is <em>assumed</em>, though not explained in the discusssion below: <a href="http://farside.ph.utexas.edu/teaching/302l/lectures/node46.html" rel="nofollow">http://farside.ph.utexas.edu/teaching/302l/lectures/node46.html</a></p>
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<p>So, when wavefunction collapses, there is a spike occuring. Does this mean that there are parts with the continuous probability of 0? (For example, x position from -9 to -3 has probability of 0, while from -3 to 3, the probability of 100% (1) and so on)</p>
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<p>We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, "with a twist") and topological order (which covers a variety of exotic properties in certain quantum many-body ground states). The topological insulators are clearly "topological" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not "robust" in the same way as topologically ordered matter.</p>
<p>My question is this: Topological order is certainly the more general and intriguing situation, but the notion of "topology" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this?</p>
<p>Perhaps a starting point might be, can we imagine a "topological insulator in Fock space"? Would such a beast have "long range entanglement" and "topological order"?</p>
<p><strong>Edit:</strong></p>
<p>While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the "standard definitions" of (symmetry protected) topological insulators and topological order and why they are very different phenomena.</p>
<p>However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, "Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable" and they seem quite happy with this.</p>
<p>On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in "TQFTs" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the <em>geometric</em> properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?</p>
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] |
<p>In David Tong's QFT lecture notes (<a href="http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf" rel="nofollow"><em>Quantum Field Theory: University of Cambridge Part III Mathematical Tripos,</em> Lecture notes 2007, p.8</a>), he states that</p>
<blockquote>
<p>We can determine the equations of motion by the principle of least action. We vary the path, keeping the end points fixed and require $\delta S=0$,
$$
\begin{align}
\delta S
&=
\int d^4x\left[\frac{\partial \mathcal L}{\partial\phi_a}\delta\phi_a+\frac{\partial \mathcal L}{\partial(\partial_\mu\phi_a)}\delta(\partial_\mu\phi_a)\right]
\\&=
\int d^4x\left[\frac{\partial \mathcal L}{\partial\phi_a}
-\partial_\mu\left(\frac{\partial \mathcal L}{\partial(\partial_\mu\phi_a)}\right)
\right]\delta\phi_a
+\partial_\mu\left(\frac{\partial \mathcal L}{\partial(\partial_\mu\phi_a)}\delta\phi_a\right)
\tag{1.5}
\end{align}
$$
The last term is a total derivative and vanishes for any $\delta\phi_a(\vec x,t)$ that decays at spatial infinity and obeys $\delta\phi_a(\vec x,t_1)=\delta\phi_a(\vec x,t_2)=0$. Requiring $\delta S=0$ for all such paths yields the Euler-Lagrange equations of motion for the fields $\phi_a$,
$$
\partial_\mu\left(\frac{\partial \mathcal L}{\partial(\partial_\mu\phi_a)}\right)
-\frac{\partial \mathcal L}{\partial\phi_a}
=0
\tag{1.6}
$$</p>
</blockquote>
<p>Can someone explain a little more that why the last term in equation (1.5) vanishes?</p>
| 8,039 |
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<p>We all know that liquid will take a shape of container in which its filled, but What will be the shape of liquid if there is no gravitational force?</p>
| 264 |
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<p>I am a little confused about the workings of the Hubble Redshift. I do understand the classical Doppler-effect, however in special relativity the velocity of light c is a natural velocity limit. So how does Hubble Redshift come about?</p>
| 768 |
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] |
<p>The <a href="http://en.wikipedia.org/wiki/Metric_signature" rel="nofollow">signature</a> of Minkowski spacetime is 2, as is explained here: <a href="http://physics.stackexchange.com/questions/24495/metric-signature-explanation">metric signature explanation</a>. The signature is related to the form the fundamental equations take, but I'm not totally clear on the link here. Furthermore I believe there is a link between second order PDEs and the metric of spacetime. I was wondering if anyone could clarify the links here, or could provide more details of how it goes.
I have been reading from the following essay: <a href="http://www.fqxi.org/data/essay-contest-files/Callender_FQX.pdf" rel="nofollow">http://www.fqxi.org/data/essay-contest-files/Callender_FQX.pdf</a></p>
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<p>As I am talking about 'smallest' can I expect that it should be a quantum system? I understand that we use quantum chaos theory instead of perturbation theory when the perturbation is not small. For example when quantum numbers are very high for an EM field interacting with an atom, we cannot use perturbation theory. In this case we will use the theory of quantum chaos.</p>
<p>My question is what is the lower boundary of the size of the system or more specifically of the quantum number in case of a EM-field atom interaction for which we must use quantum chaos theory?</p>
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<p>This is a question which has been puzzling me quite a while.</p>
<p>There was a news report saying that the farthest galaxy spotted so far is at 13.3 billion light-years away, and it is said that the galaxy can show us how the universe at about 500 million years old was looked like.</p>
<p>It is commonly accepted that the universe was started from a single point some 13.8 billion years ago, and continuously expanding since then. Therefore, the new discovered farthest known galaxy should be much closer to us, or, at least to the materials forming our galaxy, at 13.3 billion years ago.</p>
<p>It implies, if the universe has been expanding in a constant speed, then we must be traveling away from the farthest known galaxy at 13.3/13.8, or 96%, of the light speed!</p>
<p>If the universe's expanding speed is not constant (it is said that the universe's expanding is accelerating due to the dark energy), then we must traveling away from the farthest known galaxy at an even greater speed.</p>
<p>But we know, it is not possible.</p>
<p>So, what is wrong here? Is my calculation (96% c) wrong, or the assertion of that the farthest known galaxy reflecting the universe in 13.3 billion years ago wrong?</p>
| 10 |
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<p>When discussing stars, theorists tend to use effective temperature $T_\text{eff}$ and luminosity $L$ (on logarithmic scales). Observers, on the other hand, usually talk about observed colours and magnitudes (e.g. $B$ vs. $B{-}V$). These two sets of parameters are obviously related. Temperature corresponds to colour; luminosity to magnitude.</p>
<p>How does one transform between them? That is, given $L$ and $T_\text{eff}$ (from which one can derive $R$ through $L=4\pi R^2\sigma T_\text{eff}^4$), what are $B$ and $B{-}V$, or any other colour/magnitude? Alternatively, given $B$ and $B{-}V$, what are $L$ and $T_\text{eff}$? I figure $(L,T_\text{eff})\to(B,B{-}V)$ is easier.</p>
<p>I'll start trying to answer it myself just by presuming the star is emitting a blackbody spectrum and integrating against the $B$ and $V$ filter functions. I realise this will be a major simplification but it's a start. I'm hoping it's been done (and published) before, maybe with better treatment of reddening etc.</p>
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<p>I have been taught that water levels will always equal out. However, now I find that sumps and some other setups allow for water not to become equal. What other arrangements allow for the water levels of two containers not to be equal? Also, and more importantly, why does this occur? I hope someone is able to explain this for an aspiring physicist. </p>
<p>Edit: Another example I have found is a phenomena called <a href="http://en.wikipedia.org/wiki/Heron%27s_fountain" rel="nofollow">Heron's fountain</a>, how does the water level rise instead of all of it pooling up in the lowest container?</p>
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] |
<p>What is Mathematical formulation of <a href="http://en.wikipedia.org/wiki/Holographic_principle">Holographic principle</a> </p>
<blockquote>
<p>The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon.</p>
</blockquote>
| 8,044 |
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<p>This is an engineering question, but it is adressed to physicists who build accelerators. This question: <a href="http://physics.stackexchange.com/questions/35375/an-electromagnetic-space-elevator">An electromagnetic space elevator?</a> notices that a NbSn superconducting ring around the equator will launch itself into orbit. The question is whether there is enough He to cool such a massive structure.</p>
<p>What is the minimum amount of He required to maintain a stable 4-degree temperature safely throughout a long cylindrical wire? Is is necessary to use a fluid at all, or can one stably and safely maintain a stable He-range temperature in a very long wire without a liquid coolant?</p>
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<p>I'm doing a project about Moore's Law, one of the subtopics I've come to is photolithography. The way I understand it is that the MOSFET transistors are currently printed on a silicon wafer by projection printing (mostly with 193nm UV light).</p>
<p>With this technique your minimum feature size (CD) will be:</p>
<p>$CD = k_1 \cdot \frac{\lambda}{NA}$</p>
<p>but you're limited by your depth of focus ($D_f$)</p>
<p>$D_f = k_2 \cdot \frac{\lambda}{NA^2}$</p>
<p>From what I've read immersion lithography (increasing the NA) is the most advanced form of photolithography, but it's reaching its limit. Other lithography techniques are being researched; smaller wavelengths (EUV, X-ray lithography, electron beam lithography), each with their own issues and too expensive as of yet for large scale production.</p>
<p>My question is, since I'm looking into the physical bariers Moore's law is going to face, is there a calculable limit to the minimum feature size possible with photolithography and/or next-generation lithography.</p>
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<p>I have some difficulty considering the relative size of each and the meaning behind the shape of Higgs boson. I ask relating to the <em>structures</em> of both the Higgs field and quarks. How is it that the structure of a Higgs boson flows into that of, for instance, a bottom-antibottom quark pair? </p>
<p>Essentially I am asking (or at least think I am asking): If the interactions for the field to exist occurred at some point in the universe's past, the particle is expressing it's shape in relation to the field, etc, etc.. Does this mean {when viewing some of the type of symmetries seen in readouts of the possible Higgs boson decay} quarks themselves are further expressions of the same field's shape or instead some manner of deformation?</p>
<p><img src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/CMS_Higgs-event.jpg/650px-CMS_Higgs-event.jpg" alt="Now well known shape of a Higgs Boson, computer generated from Wikipedia"></p>
<p>This now fairly well known image from Wikipedia is a computer generated Higgs boson demonstrating simulated decay trajectories. This has often given me some considerations and can hopefully serve to slightly illustrate the structures I'm inquiring. (Knowing this is neither the boson or the quarks themselves)</p>
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<p>I was looking at old photographs of the nuclear tests on the <a href="http://io9.com/secret-history/" rel="nofollow">bikini atoll</a>. It dawned on me that you don't want to run film through airport x-rays, as it exposes the film. I've been told that a nuclear explosion emits all energy on the spectrum (IR to Gamma). If this includes x-rays, why wasn't the film ruined? Is it because:</p>
<p>1) I'm an amateur radio operator. One of the things I'm familiar with is that RF signal strength starts at a point, and changes by a sum of squares (correct me if I'm wrong PLEASE!) as you move away from the point. If all energy behaves similarly in this respect, is this just a, "not enough x-rays at this distance to effect the film discernibly?"</p>
<p>2) The body of the camera must be taken into account (maybe?). Most cameras of this time period were metal (I believe). Most of the film curtains/iris' we're made of metal too. The only point of entry might have been during exposure?</p>
<p>3) A combination of 1+2</p>
<p>4) Some other awesome physics thing that I can't wait to learn about?</p>
<p>5) Pics of nuclear blasts are all photoshopped productions (I'm kidding of course)?</p>
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<p>Say I have two quantum systems $A$ and $B$ I can look at the joint (composite) system $AB$ which is given by $H_{AB} \in H_A \otimes H_B$</p>
<p>Measuring a subsystem with respect to a collection of measurement matrices $\textbf{M} = \{M_i\}_{i \in I} \in Meas_{I}(H_A)$ acts as measuring $AB$ with respect to $\textbf{M} \otimes \mathbb{I}_B = \{M_i \otimes \mathbb{I}_B\}_{i \in I}$</p>
<p>Q: Can I do this with entangled states? </p>
<p>I know that making a measurement causes the entangled state of say 2 qubits to decompose in to two states. If I understand correctly, they are not entangled anymore. Now because of this we can seperate the states in to a sum of the product states $|AB> = \sum \alpha_j|j> \otimes |\psi_j>$ over all basis states $|j> \in S(H_A)$. What this says to me is that we can some how distinguish between these qubit states (hence they are not entangled anymore). Do I have this right?</p>
<p>This leads me in to the difference between $S(H_A) \otimes S(H_B)$ vs. $S(H_A \otimes H_B)$. Now the first case is not entangled and we have the "product states" of two wavefunctions/state vectors , but in the second case we have some sort of combination (composite?) of states? I would guess this is when the states are entangled and after measurement they decompose in to say $|\psi> \otimes |\phi> \in S(H_A) \otimes S(H_B)$. </p>
<p>Now if I look at an entangled 2-qubit state that I know is entangled $|\Phi> = |\Phi^+> = \frac{1}{\sqrt{2}}(|0>|0> + |1>|1>) \in H_A \otimes H_B$ (as written out in some notes) I see that we can have an entangled state embedded somehow in the tensor of two state vectors in Hilbert space. I don't know if this is supposed to be $S(H_A \otimes H_B)$ or $S(H_A) \otimes S(H_B)$ or something else completely. </p>
<p>Edit: I removed some tensor math that was incorrect. I had initially thought that if some mixture of states could be decomposed and have just a $\otimes$ and nothing such as an addition or subtraction operator it. I know that is very rudimentary but from my basic understanding of product states: <a href="http://en.wikipedia.org/wiki/Product_state" rel="nofollow">http://en.wikipedia.org/wiki/Product_state</a> I see that if a probability density can be written as the tensor product of two different probability densities, then we have a non-quantum correlation, although in the wiki article above, this is neither quantum or classical in nature. I however also see that a mixed state such as $\rho_{AB}=\frac{1}{2}(|0_A0_B⟩⟨0_A0_B|+|1_A1_B⟩⟨1_A1_B|)$ only has classical correlations. Furthermore I have found that Two states $\rho, \sigma$ are called $\textbf{perfectly distinguishable}$ if there exists a measurement $M \in Meas(H) \text{ with } I = {0,1,...}$ such that $p_0(M,\rho) = 1 = p_1(M,\sigma)$. Now from my reading, in the case of a von Neumann measurement $M = \{M_i\}, p_i$ simplifies to $p_i = tr(M_i \rho)$ and in the case of a complete von Neumann measurement $M = \{ |i \rangle \langle i | \}$, $p_i$ simplifies to $p_i = tr(|i \rangle \langle i | \rho) = \langle i | \rho |i \rangle$</p>
<p>So basically, I just have to take the traces of the respective measurements on probability densities, if they are equal (hence perfectly distinguishable) then the system of, say qubits, are not entangled, otherwise they are? Is this correct?</p>
<p>Thank you,</p>
<p>Brian</p>
| 8,049 |
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<p>Inpired by <a href="http://music.stackexchange.com/questions/7064/physiological-responses-to-different-rhythms">http://music.stackexchange.com/questions/7064/physiological-responses-to-different-rhythms</a>, I'm wondering if the human body has a strong resonant frequency. I guess the fact that that it's largely a bag of jelly would add a lot a damping to the system, but is that enough to dampen it entirely?</p>
<p>What models for resonance might be used to model the human body? (eg. weight-on-a-spring, with legs as springs?). What about individual, semi-independent body parts, like legs, or lung cavity (acoustic resonance?).</p>
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<p>I have some problems to demonstrate the non affine geodesic equation from Euler-Lagrange's equations. I start defining the Lagrangian $L=\sqrt f$, but then I'm not able to find the <a href="http://en.wikipedia.org/wiki/Christoffel_symbols" rel="nofollow">Christoffel symbol</a>'s expression. Can anybody help me?</p>
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<p>The Bloch sphere is an excellent way to visualize the state-space available to a single qubit, both for pure and mixed states. Aside from its connection to physical orientation of spin in a spin-1/2 particle (as given by the Bloch vector), it's also easy to derive from first principles mathematically, using the Pauli spin operators $$\mathcal S = \bigl\{\mathbf 1, X,Y,Z \bigr\} = \Bigl\{\mathbf 1,\; \sigma^{(x)}\!,\; \sigma^{(y)}\!,\; \sigma^{(z)}\Bigr\}$$ as an operator basis, and using only the facts that $\mathbf 1$ is the only element in $\mathcal S$ with non-zero trace, and that $\def\tr{\mathop{\mathrm{tr}}}\tr(\rho) = 1$ and $\tr(\rho^2) \leqslant 1$ for density operators $\rho$.</p>
<p><strong>Question.</strong> Using either the Pauli operators as a matrix basis, or some other decomposition of Hermitian operators / positive semidefinite operators / unit trace operators on $\mathbb C^4 \cong \mathbb C^2 \otimes \mathbb C^2$, is there a simple presentation of the state-space of a two-qubit system — or more generally, a spin-3/2 system in which we do not recognise any tensor-product decomposition of the state-space (but might use some other decomposition of the state-space)?</p>
<p>My interest here is that the representation be <em>simple</em>.</p>
<ul>
<li><p>The representation doesn't have to be presented visually in three dimensions; I mean that the constraints of the parameter space can be succinctly described, and specifically can be presented in algebraic terms which are easy to formulate and verify (as with the norm-squared of the Bloch vector being at most 1, with equality if and only if the state is pure).</p></li>
<li><p>The representation should also be practical for describing/computing relationships between states. For instance, with the Bloch representation, I can easily tell when two pure states are orthogonal, when two bases are mutually unbiased, or when one state is a mixture of two or more others, because they can be presented in terms of colinearity/coplanarity or orthogonality relationships. Essentially, it should be a representation in which linear (super-)operations and relationships on states should correspond to very simple transformations or relationships of the representation; ideally linear ones.</p></li>
<li><p>The representation should be <em>unable</em> to represent some Hermitian operators which are not density operators. For instance, there is no way to represent operators whose trace is not 1 in the Bloch representation (though non-positive operators can be represented by Bloch vectors with norm greater than 1). In fact, the Bloch representation essentially is that of the affine space of unit trace in the space of Hermitian matrices, centered on $\mathbf 1/2$. A simple and concise geometric description of density operators as a subset of the affine plane of unit-trace Hermitian operators centered on $\mathbf 1/2 \otimes \mathbf 1/2$, <em>i.e.</em> a <em>generalization</em> of the Bloch sphere representation (but not necessarily using the Pauli spin basis), would be ideal.</p></li>
</ul>
<p>If there is such a representation, does it generalize? How would one construct a similar representation, for instance, for qutrits (spin-1 systems) or three-qubit states (spin-5/2 systems)? However, these questions should be understood to be secondary to the question for two qubits.</p>
| 8,052 |
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<p>I have a problem verifying the following equation (in three dimensions)</p>
<p>$$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$</p>
<p>where $R$ is the Ricci scalar and $R^{bc}$ is the Ricci curvature</p>
<p>Attempt at a solution:</p>
<p>$$\epsilon_{abc} e^a\wedge R^{bc}=\epsilon_{abc} e_\mu^ae_\alpha^be_\beta^c R^{\alpha\beta}_{\nu\rho} dx^\mu\wedge dx^\nu\wedge dx^\rho$$</p>
<p>Now the idea is that the number of dimensions and the Levi-Civita tensor and the antisymmetry of the three-form forces the set $\{\alpha,\beta\}=\{\nu,\rho\}$. This will give the expression</p>
<p>\begin{align}\epsilon_{abc} e^a\wedge R^{bc}&=\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{12} dx^0\wedge dx^1\wedge dx^2+\\&\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{12} dx^0\wedge dx^1\wedge dx^2+({\rm cyclic\,permutations})\end{align}</p>
<p>The problem now is that the Ricci scalar is $R^{12}_{12}+R^{21}_{21}+({\rm cyclic\,permutations})$, so when counting the number of terms I obtain $2\sqrt{|g|}Rd^3 x$ which is wrong by a factor of 2. Can anyone see where I made a mistake?</p>
| 8,053 |
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] |
<p>Take the Mach-Zehnder interferometer as an example. A photon passes through a beam splitter, is reflected off mirrors, and interferes with itself at another half-silvered mirror. No measurements or disturbances are made in between. The preferred basis of the photon is given by which path it takes after passing the second beam splitter. The problem is, this preferred basis is nonlocal in between.</p>
<p>Is there a problem with locality here, or is silly ol' me just plain confused? </p>
| 8,054 |
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<p>I know how the unit vectors are defined in cylindrical coords. If I have a point P, how do I express it as a combination of the unit vectors u<sub>ρ</sub>, u<sub>φ</sub> and u<sub>z</sub>. In the case of Cartesian coordinates this combination is linear. But what about cylindrical coords? Does such a combination exist for them?
And, BTW, could you suggest a simple Physics problem where the use of cylindrical coordinates is convenient or, in general, the reason to choose them?</p>
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<p>I have seen <a href="https://physics.stackexchange.com/questions/53148/in-qft-why-does-a-vanishing-commutator-ensure-causality">this question</a> and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the <em>anticommutator</em> $[\phi,\phi^\dagger]_+$ to turn zero. The answer given to the question above does not seem to address this.</p>
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<p>There is another paradox that I need to resolve:</p>
<p>The Berezin integration rules for Grassmann odd variables give the same result as differentiation:</p>
<p>If $f=x+\theta\psi$ is a superfunction, the integral</p>
<p>$$\int d\theta(x+\theta\psi)=\psi$$</p>
<p>gives the same result as differentiation</p>
<p>$$\frac{d}{d\theta}(x+\theta\psi)=\psi.$$</p>
<p>How is this supposed to work in supersymmetry, where the Grassmann coordinates carry mass dimension -1/2? If I integrate, I expect a result to drop mass dimension by 1/2, whereas differentiation would lead to a gain. In the above example, I end up with the same object. What is its mass dimension?</p>
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<p>What does one get if the take the tensor product of a bra and a ket, for instance, $\langle\uparrow \rvert \otimes \lvert \downarrow\rangle$?</p>
<p>What I mean it, what is this object? What does it act on? Or does it get acted on? Is it meaningful, like the tensor product of two kets is, etc?</p>
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<p>Reading some science history, Werner Heisenberg and Bohr created the Copenhagen interpretation, but what I didn't get is how can we connect this interpretation to Schroedinger's cat and the double slit experience? Are they confirming Heisenberg and Bohr's work?</p>
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