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<p>In the context of Renormalized Pertubation Theory Peskin Schröder says:<br />
The Lagrangian
$$
\mathcal{L}=\frac{1}{2} (\partial_\mu\phi_r)^2-\frac{1}{2}m^2\phi_r^2-\frac{\lambda}{4!}\phi_r^4 + \frac{1}{2} \delta_Z(\partial_\mu\phi_r)^2 -\frac{1}{2}\delta_m^2\phi_r^2-\frac{\delta_\lambda}{4!}\phi_r^4
$$
gives the following set of Feynman rules:<br />
------------>------------ = $\frac{i}{p^2-m^2+i\epsilon}$<br />
------------X------------ = $i(p^2\delta_Z-\delta_m)$<br />
and the two 4-vertices.<br /><br />
The question is: Why look the Feynman rules for the first and the fourth term of the Lagrangian look so different? I believe the answer is connected to the fact that one has to bring the kinetic term of the Lagrangian to its canonical form $\frac{1}{2} (\partial_\mu\phi_r)^2$ and has to interpret everything else as (possibly momentum dependent) vertices. How does this look in formulae?</p>
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] |
<p>Burning wood emits smoke and black. Provided more oxygen or whatever required, can wood be practically burnt fully like petroleum gasses that emits a blue flame and little smoke and little black.</p>
| 10,105 |
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<p>Over the past hour or so I've been following one of my standard physics-based, wanders-through-the-internet. Specifically, I began by reviewing some details of dark energy theory but soon found myself pondering a totally unassociated topic..</p>
<p>One of today's tweets by #arxivblog concerns a speculative (IMO) paper about the possibility that "Mini Black Holes Could Form Gravitational Atoms". I read the article on it (<a href="http://www.technologyreview.com/blog/arxiv/26726/" rel="nofollow">here</a>) and a few of the comments and soon became perplexed by how it contradicted something I already knew. Namely, small black holes don't last very long so how-on-earth (or in space) could one "of about 10 to 1000 tonnes" exist long enough to capture a passing particle into a quasi-stable orbit? (I was recalling something I'd read about possible micro black holes, being produced at the LHC, and subsequently dissapearing in a tiny fraction of a second, due to Hawking radiation.) </p>
<p><strong>~</strong> My first question is therefore do these micro black holes decay in a <em>particle-like</em> fashion or does Hawking radiation theory have significant consequences on the possible decay channels?</p>
<p>A little more digging and I soon found that the lifetime of a black hole is $ t_{l}=M^{3}/3K $</p>
<p>Where $ K=h .c^{4}/30720.\pi^{2}.G^{2}=3.98\times10^{15} kg^{3}/s^{-1} $</p>
<p>Giving, for a mini black hole of mass, $M=100\times10^{3}kg$, $ t_{l}=0.084s $</p>
<p>Which, I'm sure you'll agree, is clearly too short to allow anything that could meaningfully be described as a gravitational atom. (Incidentally, my calculation for the lifetime of an LHC black hole gave something of order $10^{-94}s$!!) </p>
<p>Just then, the resolution to this apparent contradiction dawned on me - <em>the above formula is derived from the radiated energy given off by a black hole via Hawking Radiation. Conversely, the paper in question is written from an initial, unstated, assumption that Hawking Radiation does not exist.</em></p>
<p><strong>~</strong> This led me wonder, to what extent is Hawking radiation accepted amongst professional theorists? (I'm aware that it has not yet been directly prooven, so consideration of the implications of it's non-existence must surely be worthwhile, and interesting, IMO.)
I realise this question is somewhat vague but I am hoping for some elucidation on how firm-a-foundation Hawking first derived this phenomena and, given it is based on QFT <em>bolted-onto</em> a curved spacetime, has modern developments (in for example, String theory) corroborated the possibility of energy loss over the event horizon in this way?</p>
<p><strong>~</strong> A quick glance at the wiki article on primordial black holes (<a href="http://en.wikipedia.org/wiki/Primordial_black_hole#String_theory" rel="nofollow">here</a>) informed me that the Fermi Gamma-ray Space Telescope (GLAST experiment), launched in 2008, is hoping to find evidence of primordial black holes:</p>
<blockquote>
<p>"If they observe specific small interference patterns within gamma-ray bursts (GRB), it could be the first indirect evidence for primordial black holes and string theory."</p>
</blockquote>
<p>The reference to string theory appears only to concern the longer predictions of the primordial back hole's lifetimes; based on the extra, 'rolled-up' spatial dimensions posited by some/all(?) string theories (AFAIK due to gravity being able to propagate in these extra-dimensions).</p>
<p><strong>~</strong> Finally, therefore, if GLAST does/has find/found certain characterists in GRB data, how strongly does it (will it) rate as evidence for or against: the existence of primordial black holes, the existence of Hawking radiation and indirectly as evidence for String theory? I have had a look at papers relating to recent GRB data from the LAT experiment (<a href="http://arxiv.org/find/astro-ph/1/ANDNOT+OR+abs%3a+AND+Fermi+gamma+ti%3a+AND+Fermi+gamma+au%3a+OR+Abdo+Collaboration/0/1/0/past/0/1" rel="nofollow">here</a> for eg) but, as a non-specialist, it is very unclear to me whether the data has any implications to the questions posed above.</p>
| 10,106 |
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<p>I learned programming as a child, for fun. Now I am working as a programmer, even though I got a business major degree.</p>
<p>I wonder if there are career paths for doing physics <em>other</em> than becoming a researcher in a University or a professor, that are not <em>all</em> about credentials?</p>
| 10,107 |
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] |
<p>Are photons <a href="http://en.wikipedia.org/wiki/Electromagnetic_radiation" rel="nofollow">electromagnetic waves</a>, quantum waves, or both?</p>
<p>If I subdivide an electromagnetic field into smaller <a href="http://en.wikipedia.org/wiki/Electromagnetic_field" rel="nofollow">electromagnetic fields</a>, should I eventually find an electromagnetic wave of a <a href="http://en.wikipedia.org/wiki/Photon" rel="nofollow">photon</a>? </p>
<p>How can individual quantum waves combine to form the macroscopic observable of an electromagnetic field?</p>
| 304 |
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<p>You buy one of those remote control toy helicopters. You bring it into an elevator. The elevator goes up. Does the helicopter hit the floor or does the floor of the elevator push the air up into the bottom of the helicopter so that it maintains altitude relative to the elevator? The same question can be reversed. If the elevator goes down will it hit the cieling? If the elevator is air-tight, would it make a difference?</p>
| 743 |
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<p>This question is an attempt to complement <a href="http://physics.stackexchange.com/questions/9421/strange-modulation-of-radiactive-decay-rates-with-solar-activity">this other question</a> about fluctuations in radiactive decay.</p>
<p>This question is completely experimental though: in general, suppose i have certain sample of a radiactive substance. My goal is to gather experimental data about fluctuations in net momenta of all decay products. I certainly would have to set the sample temperature near to zero Kelvin in order to reduce or eliminate thermal contributions to net momenta. However, what is the experimental setup one would have to have?</p>
<p>Should one only consider individual events where all decay products have been detected and their individual momenta obtained?</p>
<p>How do you make sure that a set of decay product events are associated to the same event? time correlation? does this mean that the sample size needs to be small enough so overlapping decay events are percentually few and can be filtered out?</p>
<p>what other considerations are required?</p>
| 10,108 |
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<p>Yesterday I was standing by the campfire. I used to think that campfire heat carried to me only by air. It was heating my face too much, so I blocked it with my hand just like blocking the sun. Then area on face which is shadowed by my hand stopped getting heat from campfire. Then I thought if it were getting transferred only by air it would pass by around my hand, and something that travels in straight line must be transferring most of the heat. But yet the campfire does not look so bright. Is it some kind of radiation, invisible light?</p>
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<p>Howdy, I'm building a simulation for looking at the light field underwater. In order to verify my simulation, I'm looking for some data showing the far-field intensity that comes from single scattering from many small particles in suspension. I suspect Mie theory plays a part here, but I'm having a hard time finding some results, rather than doing all the derivations myself.</p>
<p>In other words, I want to know the power distribution on a plane after a beam of light has been scattered by a bunch of small particles through a volume. I know Oregon Medical has a nice online simulation that produces scattering phase functions (http://omlc.ogi.edu/calc/mie_calc.html), but that doesn't give me the power on a plane - only the scattering profile from individual particles. I'm fine with only a single scattering result.</p>
<p>I want to do initial verification using a fixed particle size. Having a hard time finding a reference with this data. Help? </p>
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<p>The chemical potential is defined as:
$$
\mu = -T\frac{\partial{S(N,V,E)}}{\partial{N}}
$$
It seems to me that this is completely independent of where I put the reference point of energy, because only the difference of entropy is relevant (and also temperature is defined as a difference in entropy). </p>
<p>However the Fermi-Dirac distrbution is:
$$<n_r> = \frac{1}{exp(\beta(\epsilon_r-\mu))+1}$$</p>
<p>But if I change the value of the reference point of energy, the value of $<n_r>$ changes, which causes a contradiction. In my book about statistical mechanics, they state though that $\epsilon_r-\mu$ is independent of the reference point of energy, but I see not why, because $\mu$ seems to be independent, while $\epsilon_r$ doesn't seem independent of the reference point.</p>
| 10,111 |
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<p>Looking through this AP Physics question, I was struck by how the 'collision' between a photon and electron looks so much like a macroscopic collision. Is this even physically possible?</p>
<p>Look at the last page of this pdf: <a href="http://lodischool.tripod.com/dovesol/DOVE02SOL.pdf" rel="nofollow">http://lodischool.tripod.com/dovesol/DOVE02SOL.pdf</a></p>
<p>EDIT: Some more questions:</p>
<p>How could a photon <em>collide</em> with an electron, when their positions cannot be determined exactly? Also considering how very small the electron is, I doubt that it is even possible to make the two collide; and if it is, how could you possible detect that? It also seems as though the photon and electron are acting as particles, which seems to me not to be the whole story.</p>
<p>What if I put the electrons behind a double slit apparatus, and treat individual photons as particles? Based on this "compton scattering," it's possible for the photon to be deflected any which way. I could claim that the diffraction pattern observed in the double-slit experiment is due to compton scattering, among other factors. Prove me wrong! </p>
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<p>Consider the dimensional regularized integral</p>
<p>$$ \int d^{d}k (k^{2}-m^{2}+i\epsilon)^{-\lambda} $$</p>
<p>For positive $ \lambda $ this integral has a pole at $ k=m $. Is this the reason we we insert the $ i \epsilon $ part?</p>
<p>Using Shotkhotsky's theorem
$$(k^{2}-m^{2}+i\epsilon)^{-\lambda}= -i\pi \delta ^{\lambda}(k^{2}-m^{2})+PVC(\lambda) (k^{2}-m^{2}+i\epsilon)^{-\lambda}$$
but how do we regularize the IR divergence at $ k_0=m $ for every $\lambda$? Do we simply ignore this pole $ k=m$ and compute the integral?</p>
<p>Can we compute the integral by defining a parameter $ b^{2}=-m^{2} $ and compute the integral for every $b$ and then take the limit $ b\to-im$?</p>
| 10,113 |
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<p>As we know time began with the big bang. Before that there was no time, no laws, nothing. Mathematically how can an event take place when no time passes by? How did the big bang took place when there was no time?</p>
<p>Note my question is not about weather big bang took place or not, my question is about is this a mathematical anomaly? Thanks</p>
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<p>An acquaintance of mine, while being home alone, saw that the light bulb in the room which was hanging from the ceiling with wires having a pendulum motion which was more than noticeable. He says that that was a ghost but i know better than that. </p>
<p>I just can't find an explanation. Maybe you can help.</p>
<p>The light bulb was ON at the time. Some windows in the flat could have been open but the person said he was in the room at that time and there was no draft.</p>
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<p>It's logical to think that the time it takes a microwave to heat the food would be proportional to the mass heated. But since a microwave is based on dielectric heating, I think that if you increase the mass of food there will be more water, which will heat the food faster (due to thermalization). Is this reasoning right?</p>
<p>Is there an optimal quantity of food to heat and the time it takes?</p>
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<p><a href="http://www.nasa.gov/centers/goddard/news/topstory/2007/antimatter_binary.html" rel="nofollow">Being said</a> that the antimatter - matter reaction is faster than that of a fission and fusion, what if the antimatter cloud found at the center of our galaxy could really able to react with matter from the stars around it and cause a chain reaction of annihilation that spreads through the galaxy?</p>
| 10,117 |
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<p>The Einstein-Cartan theory is a generalisation of General Relativity insofar as the condition that the metric affine connection is torsion-free is dropped. In other words, the space time is a Riemannian manifold together with the datum of a metric affine connection (which may differ from the Levi-Civita connection by a suitable contorsion tensor).</p>
<p>In this case, geodesics (paths that locally extremalise the length, and which are given by a variational principle) generally differ from auto-parallels. As far as I know, the trajectories of spin-less particles in Einstein-Cartan theory are usually assumed to be geodesics (rather than autoparallels) so they don't feel the difference between the given connection and the Levi-Civita connection. (By the way, is there a good reference for this statement?)</p>
<p>My question is, how classical particles with spin are supposed to behave? Will they also travel along geodesics with the only difference that their spin direction will evolve according to the contorsion tensor (viewed as a so(1,3)-valued one-form)?</p>
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] |
<p>Batteries have a circuit which looks like this : </p>
<p><img src="http://i.stack.imgur.com/1DJrq.png" alt="enter image description here"></p>
<p>The electrons go around the circuit and then return through the battery where they get charged again and flow around. </p>
<p>My issue is, what about alternating current? For example main power, simply it looks like this (pretend the globe is in your house) </p>
<p><img src="http://i.stack.imgur.com/JkMho.png" alt="enter image description here"></p>
<p>Do the electrons return or go to the Earth?
And how does it work when there is a short-circuit? Why does it go to the Earth instead of the neutral? Is it because of the lower electronegativitiy?</p>
| 10,119 |
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<p>How would pressure of an ideal gas be distributed over the inside of a capsule (a cylinder with semi-spheres on the ends)? What about the strain on the material? Is there a general formula for how much?</p>
| 10,120 |
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<p><img src="http://i.stack.imgur.com/qJfUU.png" alt="enter image description here">
"Fermi level" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. Why does the <a href="http://en.wikipedia.org/wiki/Topological_insulator" rel="nofollow">Fermi level for the bulk of topological insulator</a> fall within the bulk band gap? There is no energy level in the band gap.</p>
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<p>What is the meaning of the electron probability cloud? </p>
<p>I understood it to mean that the electron has a probability to be found in a certain postion before measurement, but now after reading experiments involving Schrödinger's cat type states (with bonding and antibonding gaps in $\mathrm{H_2}$ and molecule and squid measurements where the wavefunction interferes and separates out the bonding band from the antibonding band) and that this physical effect of the supercurrent moving in both directions across the squid junction proves the electron is really everywhere in the cloud, smeared out and not just a probability cloud, which interpretation is right? </p>
<p>Does the above experiement prove the electron is everywhere and smeared out in the wavefunction?</p>
| 10,122 |
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<p>This is a physics/calculus/computer science problem, but I think I'll get better results in the Physics SE.</p>
<p>I have a fun little project I've been working on (hobby, not homework/production), that models the flight of a projectile in various environments (2 Dimensional). Everything is nice and shiny with gravity and no air resistance. But of course, I've run into the classical Quadratic Air Resistance problem. </p>
<p>Now I have no problem with just accurately calculating various position: I have a 4th Order Runge-Kutta Method which works just fine for that. What I would like is, given an (x,y) coordinate pair, find the Required Angle needed to hit that coordinate, as well as the time needed to get there.</p>
<p>Currently I'm working on an algorithm that starts by finding two angles, one which overshoots, and another that undershoots, then using a binary search to narrow the range between them by plotting using the Runge-Kutta method, and increasing the stepsize until I've reached an acceptable accuracy.</p>
<p>Unfortunately I fear this will be much slower than I like, not to mention that one angle might overshoot one time, then when I increase the stepsize, could undershoot it, which would just make things crazier than I think they should be.</p>
<p>I've googled this quite a bit, but sadly couldn't find anything that, you know, <em>helps</em>.
If you guys know of any algorithms that might be helpful, I would greatly appreciate it.</p>
<p>Even better would be a way to rework the Differential Equations to simplify the search, but sadly I am no where near good enough at Calculus for that to be a viable option.</p>
<p>Thanks,
Jacob</p>
| 10,123 |
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<p>I know a photon has zero rest mass, but it does have plenty of energy. Since energy and mass are equivalent does this mean that a photon (or more practically, a light beam) exerts a gravitational pull on other objects? If so, does it depend on the frequency of the photon?</p>
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<p>A example of amplitude in Relativistic Quantum Mechanics or specifically in QFT is the amplitude of a field configuration on a space-like hyper-surface of space-time to "lead" to another field configuration on another space-like hyper-surface of space-time. In the path-integral picture one simply integrates over all possible field configurations on the interior, giving each a weight in the normal way. Now if one wants to generalize this to finite closed boundaries, we would get an amplitude for each field configuration on a finite closed boundary of space-time. but how would we interpret this? This question relates to interpretations of quantum mechanics, has anyone investigated this line ? </p>
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<p><em>I know that all planets and stars have a gravitational pull but does a simple much smaller object have a gravitational pull for example a pebble?</em></p>
| 10,125 |
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] |
<p>I was studying the <a href="http://www.scholarpedia.org/article/Center_manifold" rel="nofollow">centre manifold theory</a>. It says (see Kuznetsov page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right. </p>
<p>$
\begin{equation}
\label{}
\left\lbrace
\begin{array}{lcr}
\dot{u} \: & = & \: Bu + g(u,v)\\
\dot{v} \: & = & \: Cv + h(u,v)
\end{array}\right. \Rightarrow
\left\lbrace
\begin{array}{lcl}
\dot{u} \: & = & \: Bu + g(u,V(u))\\
\dot{v} \: & = & \: Cv
\end{array}\right.
\end{equation}
$</p>
<p>Here $v=V(u)$ is the center manifold.</p>
<blockquote>
<p>I'm interested in understanding things from an heuristic point of view but i can't figure out why in the second system $h(u,v)=0$ and what this means.</p>
</blockquote>
<p>Other books don't tell anything about the second equation of the second system. They mainly say that the most important information are embedded in the first equation of the second system. What if i want to graph the second system?
I also searched Carr's book for a demonstration but i did not find anything useful.</p>
| 10,126 |
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<p>Is it possible that from the same initial mass different black hole radius will be created due to different mass distribution during black hole creation? If mass is concentrated more on the outside bigger event horizon will be created? If mass is concentrated more in the center smaller black hole will be created? </p>
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<p>Apparently there has been a paper in nature claiming the finding of a new particle?
<a href="http://www.sciencedaily.com/releases/2012/04/120413160004.htm" rel="nofollow">http://www.sciencedaily.com/releases/2012/04/120413160004.htm</a></p>
<p>Does that mean a new fundamental particle has been found with a special design of "common materials"? How does that relate to the standard model?</p>
<p>I hope someone can clarify the meaning of that announcement.</p>
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<p>We are supposed to give a matrix representation of $L\cdot S$ for an electron with $l=1$ and $s=\frac{1}{2}$. </p>
<p>I read $L\cdot S$ as $L \otimes S$. Is this correct? Then we would have e.g. for </p>
<p>$L\otimes S (|1,1\rangle \otimes |1/2,1/2\rangle) = L |1,1\rangle \otimes S|1/2,1/2\rangle $
$= \sqrt{2} \hbar |1,1\rangle \otimes \sqrt{\frac{3}{4}} \hbar |1/2,1/2 \rangle = \sqrt{\frac{3}{2}}\hbar^2 |1,1\rangle \otimes |1/2,1/2\rangle $.</p>
<p>Is this correction correct? In that case should I proceed in this way with all the other basis vectors and write the eigenvalues down the diagonal in a matrix?</p>
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<blockquote>
<p>Rubbing a glass rod with silk causes charges to be exchanged and consequently both objects get charged.</p>
</blockquote>
<p>Why do the objects have to be "rubbed"? I get that one has a stronger pull on the electrons than the other, but shouldn't just allowing the objects to make contact be enough.?
I would appreciate a "visualization" of whats happening.</p>
<p>Similar questions: <a href="http://physics.stackexchange.com/questions/4617/why-two-objects-get-charged-by-rubbing">Why two objects get charged by rubbing?</a></p>
<p><a href="http://physics.stackexchange.com/questions/44486/how-does-rubbing-cause-the-transfer-of-electrons-from-one-object-to-the-other">How does rubbing cause the transfer of electrons from one object to the other?</a></p>
<p><strong>Neither question addresses why the objects need to be rubbed instead of just making contact.</strong></p>
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<p>Charge is carried by electrons moving. The protons are always stationary.</p>
<p>The answer I found online is the protons are stuck in the nucleus so they can't move ("strong nuclear force").</p>
<p>But why can't the whole positively charged atom move? </p>
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-0.007615353912115097
] |
<p>I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a unit circle. However, I don't know how <a href="http://en.wikipedia.org/wiki/Solid_angle" rel="nofollow">solid angles</a> are actually quantified. </p>
<p>Are solid angles just a single number, the describes this fraction of the area? It's confusing to me since often times, I've seen integrals that integrate over a sphere using solid angles, which seems to imply that solid angles are multi-dimensional quantities (e.g. when integrating using spherical coordinates, the solid angle would have to consist of the azimuthal and polar angles covered by the differential solid angle).</p>
<p>Following from this, how would you write down a solid angle that covers the entire surface of a unit sphere?</p>
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<p>We are searching for a software program that simply helps us to visualize in 3D the interference of planar waves of <em>multiple</em> sources (frequencies) and as sepctrum on a wall. Something like a well known visualisation of the famous double split, but instead of the splits just havnig the possibility to have multiple sources with different frequencies.</p>
<p>I would be thankful for any hint/reference.</p>
<p>Many thanks</p>
| 10,133 |
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<p>If we add 10 spherical bubbles then there will be some change in their area. Is there any way to find out change in temperature of the liquid from the change of its area?</p>
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<p><strong>Partial measurement and partial trace</strong></p>
<p>There is a connection between a measurement of a part of a system and tracing this subsystem out. Say, we have a system composed of subsystems $A$ and $B$ in a pure state $|\psi\rangle = \sum_{i,j}a_{ij}|i\rangle_A|j\rangle_B$ and do a projective measurement on part $B$, projecting it on a state $|k\rangle_B$. The resulting state of subsystem $A$, conditioned on the measurement result, will be (up to normalization)
$$
|\psi^{(k)}\rangle_A = \sum_i a_{ik}|i\rangle_A.
$$
Taking partial trace over the subsystem $B$ then corresponds to taking a statistical mixture of all possible measurement outcomes (this can be shown by a simple explicit calculation),
$$
\mathrm{Tr}_B(|\psi\rangle\langle\psi|) = \sum_{k}|\psi^{(k)}\rangle_A\langle\psi^{(k)}|.
$$</p>
<p><strong>Gaussian states</strong></p>
<p>When dealing with Gaussian states in quantum optics [i.e., states with Gaussian phase space representation, where the phase space is spanned by quadrature operators $x = a+a^\dagger$, $p = i(a^\dagger-a)$], one often uses the description by first and second statistical moments. The first moment is, as far as entanglement is concerned, irrelevant and one then works just with the covariance matrix, i.e., the second moment. It can then be shown that a partial measurement affects the covariance matrix of the remaining subsystem in such a way that the resulting covariance matrix does not depend on the particular measurement result.</p>
<p><strong>The question</strong></p>
<p>Does this mean that, when dealing with covariance matrices of Gaussian states, partial measurement and partial trace are effectively the same? This is something one could intuitively (and perhaps naively) expect from the observations above. However, there is a difference in the purity of the state after partial measurement (pure for pure initial state) and partial trace (generally mixed). For Gaussian states, state purity is directly connected to determinant of the covariance matrix (the purity scales as $P\propto 1/\sqrt{\det\gamma}$, where $\gamma$ is the covariance matrix). Therefore, there must be a difference between covariance matrix after a partial measurement and partial trace. What am I missing?</p>
| 10,135 |
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<p>At the end of this nice video, she says that <a href="http://en.wikipedia.org/wiki/Electromagnetic_radiation">electromagnetic wave</a> is a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward.</p>
<p>I wonder where the <a href="http://en.wikipedia.org/wiki/Photon">photon</a> is in this explanation. What is the relation between electromagnetic wave and photon?</p>
| 599 |
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<p>My question is about the field theoretic version of Noether's theorem. I am deeply troubled by one of the hypotheses of the theorem.</p>
<p>As it is the standard textbook for Lagrange mechanics, I'll follow Goldstein's account (starting p. 588 in the second edition of "Classical Mechanics").</p>
<p>I have no problem with condition 1 since I work in Minkowski space. I am completely okay with condition 2, which amounts to asking that the equations of motion be the same for two observers who use different systems of coordinates to describe the same spacetime and different functions to describe the same fields.</p>
<p>However, I can't make any sense of condition 3. I don't see what its physical meaning can be. I haven't seen it explained convincingly anywhere, and can't seem to figure it out for myself.</p>
<p>For those who don't have any access to Goldstein's book but feel they might be able to help, condition 3 is the requirement that the action integrals be equal for the two aforementioned observers.</p>
<p>I hope someone has some fantastic insight on this! :-)</p>
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] |
<p>In the field of fluid mechanics, what is the momentum flux tensor? Is there an easy explanation for how it "works"?</p>
| 10,137 |
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<p>So I understand the the Superposition Principle states that all the forced oscillations, as determined by multiple external forces, are to be added up in order to get the entire solution.</p>
<p>However, I'm at a loss as to go about proving this - do I start with the general solution of a simple driven oscillatory system, $x(t)=x_o \cos(\omega t - \phi)$? To be honest, I'm quite confused about how to go about this.</p>
<p>EDIT: this problem in particular is what i'm trying to prove,</p>
<blockquote>
<p>Prove the superposition principle for inhomogenous linear equations of motion used in deriving the motion of a driven oscillator. Will it still apply if the force on an oscillator was $-kx^2$ instead of $-kx$?</p>
</blockquote>
<p>Like I mentioned before, i'm stumped about how to even attempt it.</p>
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<p>I'm doing a research for my stats class in high school and I chose quantum mechanics as my subject. I narrowed down to electron localization in an atom and radial probability distribution. However, I can't find any data to support my claims that probability and statistics are very important in quantum mechanics. Is there any data that I can analyse to prove my claims that is suitable for me?</p>
<p>I introduced the double slit experiment, Schrodinger's equations, wave function and superposition.</p>
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<p>Assume a static metric with (known) components $g_{\mu\nu}$. I'd like to know what is the gravitational pull $g$ of a test particle placed on an arbitrary point $X$.</p>
<p>The gravitational pull being defined as the acceleration the particle suffers as measured by an observer sitting in a reference frame fixed at the origin. <strong>What are (theoretically speaking) the steps one needs to take to find this acceleration?</strong> <em>(no need to actually calculate for the general case, just list the general steps)</em></p>
<p>Furthermore, <strong>what is the result for the simple case of the Schwarzschild metric?</strong><br>
In other words, what is the gravitational pull of a star or black hole? Will it matter if the test particle is moving or is static?</p>
| 587 |
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<p>We're all familiar with basic tenets such as "information cannot be transmitted faster than light" and ideas such as information conservation in scenarios like Hawking radiation (and in general, obviously). The Holographic Principle says, loosely, that information about a volume of space is encoded on its two-dimensional surface in <a href="http://en.wikipedia.org/wiki/Planck_area" rel="nofollow">Planck-sized bits</a>. </p>
<p>In all these contexts, I can take "information" to mean predictive or postdictive capability, i.e. information is what enables us to state what the outcome of a measurement was or will be (locally). But what <em>is</em> information, exactly? Do we have any kind of microscopic description of it? Is it just a concept and, if so, how can we talk about transmitting it?</p>
<p>I suspect this is probably as unanswerable as what constitutes an observer/measurement for wave function collapse, but I'd love to know if we have any formulation of what information is made of, so to speak. If I'm talking nonsense, as I suspect I may be, feel free to point this out.</p>
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<p>I'm wondering what some standard, modern references might be for continuum mechanics. I imagine most references are probably more used by mechanical engineers than physicists but it's still a classical mechanics question. </p>
<p>This came up in a conversation with my father (who is a mechanical engineer). I was curious to see the types of mathematics they use in stress analysis. Complex analysis gets used to study 2-dimensional isotropic material here:</p>
<blockquote>
<p>N.I. Muskhelishvili. <em>Some basic problems of the mathematical theory of elasticity</em>.
3rd edition, Moscow-Leningrad. 1949. (Translated by J. R. M. Radok. Noordhoff. 1953.) </p>
</blockquote>
<p>but that's quite an old reference and my library does not have it. Any favourite more modern and easy-to-find references? </p>
<p>I'm fine with mathematical sophistication (I'm a mathematician) but I'm not particularly seeking it out. I'm looking for the kind of references that would be valued by physicists and engineers. References "for math types" are fine too but that's not really what I'm after. </p>
| 907 |
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<p>Ignoring the quantum zeno effect (if possible?), can we observe in real-time the transformation of one element to another? I'm talking about an amount visible to the naked eye where one could see obvious changes in colour, reflectivity, phase, surface finish etc. occurring in say, seconds or minutes.</p>
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<p><em>I haven't studied much about this, so if I am mistaken about something please correct me.</em></p>
<p>From what I have seen around the Internet, a force applied to a object takes time to propagate through the object because it has to interact with the molecules around the ones that the force was applied to. That means that for the force to get to the other side of the object it will take time especially if it is over a large distance. To my understanding the displacement of those molecules propagating through the rest of the object can be called a deformation wave. This deformation wave travels at the speed of sound through the medium (the object). </p>
<p>What if the object that the force was being applied to was a wheel with a bar from the top of the wheel to the bottom that crosses the center? The bar is made out of the same material as the wheel and is bonded to the wheel (there are no gaps between the bar and the wheel). If a force was applied to the top of the wheel would the deformation wave travel half the circumference of the wheel or the diameter bar to get to the other side? Basically what I am asking is:</p>
<ol>
<li>Is the information presented in this question accurate, or am I mistaken about someting?</li>
<li>Does the deformation wave move at the speed of sound through the medium, or is it even necessary for the answer to this question? </li>
<li>And does the deformation wave find the fastest path to the opposite side of the wheel (the bar)? Does it go in all directions? Or would it just go the direction that the force was applied to (the outside of the wheel)?
</li>
</ol>
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<p>I have a s****d question, how to calculate the central charge of $bc$ conformal-field theory in Polchinski's string theory, Eq. (2.5.12)?
For a $bc$ CFT given by</p>
<p>$$S=\frac{1}{2\pi } \int d^2 z \,\,b \bar{\partial} c $$</p>
<p>where $b$ and $c$ are anticommuting fields,
define normal ordering as
$$:b(z_1) c(z_2): = b(z_1) c(z_2) - \frac{1}{z_{12}}. \tag{2.5.7} $$</p>
<p>Given the energy-momentum tensors
$$ T(z) = : (\partial b) c: - \lambda \partial ( : bc : ), \tilde{T}(\bar{z})=0 $$
The $TT$ Operator Product Expansion (OPE)
$$ T(z) T(0) \sim \frac{c}{2 z^4} + \frac{2}{z^2} T(0) + \frac{1}{z} \partial T(0) $$
has central charges,
$c=-3 (2 \lambda -1)^2+1 $ and $$\tilde{c}=0. \tag{2.5.12}$$</p>
<p>For my understanding I should compute the cross-contraction to find the central charges. First I construct the relation
$$:\mathcal{F}::\mathcal{G}: =\exp\left(\int d^2 z_1 d^2 z_2 \frac{1}{z_{12}} \frac{ \delta }{\delta b(z_1)} \frac{\delta }{\delta c(z_2)} \right) :\mathcal{FG}: $$
then apply it to $T(z) T(0)$.</p>
<p>My question starts from the very beginning, about $\partial ( : bc : )$ in $T(z)$, does it stand for $\partial_{z_1} ( : b(z_1) c(z_2):)$ or ? if it means $\partial_{z} ( : b(z) c(z):)$
, the right hand side of (2.5.7) is singular..</p>
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<p>In a picture or video of a numerical relativity simulation, such as a neutron star merger into a black hole, how do they set up their coordinate system? Lets take the point in a video corresponding to x=10km, y=20km, z=30km, t=1ms. Spacetime itself is distorted, in a very complex way, so how do you make sense of these numbers?</p>
<p>Website to find some nice videos:
<a href="http://numrel.aei.mpg.de/images" rel="nofollow">http://numrel.aei.mpg.de/images</a></p>
<p>Just to clarify: There are simulations in which space-time is fixed to the a well defined metric (e.g. Kerr black hole accretion disk MHD simulation with no disk self-gravity). But for true numerical relativity, in which the shape of space-time itself has to be simulated, there is no "clean" metric.</p>
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] |
<ul>
<li><p>Did <a href="http://en.wikipedia.org/wiki/Charles-Augustin_de_Coulomb" rel="nofollow">Charles-Augustin de Coulomb</a> know:</p>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Coulomb%27s_constant" rel="nofollow">Coulomb's constant</a></li>
<li>Coulomb (as a unit)</li>
</ul></li>
</ul>
<p>if not then what was the first time it was measured?</p>
| 10,144 |
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<p>It seems to me that there is no such thing as time. There is only movement in the universe and we compare our own movement to a different object to have a sense of time. It can be a clock or a atomic vibration.</p>
<p>Does this view of time work within the current framework of phsics?</p>
<p>Do physicists have an explanation/proof about time's existence?</p>
| 813 |
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<p>For some time now I've been confused about heat capacity. The way I understand it, if I put in an amount of heat energy into the system, $dQ$, its temperature will change by $CdT$. But now, everywhere I look I only find $C_p$ and $C_V$, the heat capacities in isobaric/isochoric processes. But for an arbitrary process, how do I calculate its heat capacity, in other words how do I find $$\frac{\partial Q}{ \partial T}, \mbox{ with } p=p(V)$$</p>
<p>where $p=p(V)$ is the equation of this process.</p>
<p>My question: I wish to calculate the heat capacity in a polytropic process with exponent $n$, so the equation is $pV^n=const$, and so $p=\frac{p_1 V_1 ^n}{V^n}$.</p>
<p>How do I get from here to actually calculating the heat capacity, that is the partial derivative?</p>
<p>P.S. I apologize if I'm mixing or misunderstanding notions here, but the way I was taught thermodynamics was horrible, just learning how to solve particular problem types without understanding the physics of it at all, and I hadn't yet had time to look for some source to correct all these misconceptions. Would be greatful if someone could recommend a mathematically rigorous text on thermodynamics that doesn't make unexplained simplifications and helps intuitively grasp the physics of it.</p>
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<p>Suppose we are given a stick and a stone tied to the stick by a string. Now if we rotate the stone around the stick the stone rises in height (see picture below). My question is which force accounts for this rise in height? </p>
<p><img src="http://i.stack.imgur.com/0oySD.jpg" alt="enter image description here"></p>
<p>According to me (please correct me if I am wrong) this happens due to centrifugal force. The centrifugal force is directed along the string outwards, so we can resolve it into two components, one is the horizontal component and the other is the vertical component. If the mass of the stone is m, its velocity is v, the length of the string is r, and the angle the string makes with the horizontal is θ, then the total centrifugal force is mv^2/r, the horizontal component of the centrifugal force is mv^2/r * cos(θ), and its vertical component is mv^2/r * sin(θ) (see picture below). Can anyone tell me if I am correct? The directions of the forces are given in the picture, but I am not sure whether the directions are right or not. Also note that in the second picture for sake of clarity I have not drawn the direction of rotation of the stone.</p>
<p><img src="http://i.stack.imgur.com/Mx16W.png" alt="enter image description here"></p>
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<p>I'm having a bit of trouble understanding Balmer lines, is this correct, and the questions are in brackets:</p>
<p>1) A star needs to be hot enough such that electrons are in the n=2 state <em>(why does the temperature cause the electrons to be in n=2 state?)</em></p>
<p>2)As light of a specific frequency passes through the star's atmosphere it excites these electrons in the n=2 state raising them to higher energy levels <em>(why can't it excite electrons not in n=2?)</em></p>
<p>3) The electrons the de-excite and drop back down to n=2 <em>(why can't say and electron originally in n=3, drop down to n=2 without being excited in point 2)</em></p>
<p>4) The lose of energy is emitted as photons (E=hf) in random directions, and possibly in multiple steps <em>(what is the correct phrasing for this, I mean perhaps two photons will be emitted from n=4 to n=3 to n=2 and so won't be the same frequency as the specific frequency in point 2 which correlates to Balmer lines)</em></p>
<p>5) this results in reduced intensity at the specific that correlate to Balmer absorption lines. </p>
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<p>I've gotten interested in physics recently due to the many educational channels on YouTube such as sixtysymbols and minutephysics. They talk about quarks sometimes, and I was wondering if there is anything smaller than a quark. I'm not at that stage in school yet, the smallest we have discussed in class have been neutrons, protons, and electrons, so I am just curious.</p>
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<p>For a homogeneous material of length $L$, we can write the heat equation as $$\rho c\frac{dT}{dt}=k\frac{du^2}{dx^2}, \text{ } x\in (O,L)$$ where $T$ is the temperature, $\rho$ is the thermal conductivity, $c$ is specific heat, and $k$ is the thermal conductivity. </p>
<p>To non-dimensionalize it, I know that I can define a new variable for space $\hat{x}=\frac{x}{L}$, time $\hat{t}=\frac{kt}{L^2\rho c}$ and temperature $\hat{T}=\frac{T}{T_0}$ where $T_0$ is the initial temperature, using the chain rule, obtain the simplified form $$\frac{d\hat{T}}{dt}=\frac{du^2}{dx^2}.$$</p>
<p>When the material is heterogeneous, we often assume that the thermal conductivity varies in space and is modeled by the function $k=k(x)$. When this is the case, I'm not quite sure what is the standard approach non-dimensionalization. </p>
<p>I conjecture that the process should still be similar even though $k(x)$ is not a constant. Should I choose a reference thermal conductivity $k$ from the range of values $k(x), x\in (0,L)$ follow the same proceedure? Is there an alternative methodology to non-dimensionalize in the case of spatially varying material parameters?</p>
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<p>Can you explain me some of the mathematical details of such concept as symmetries?
In physics, we have some manifold, and fields are functions on this manifold.</p>
<p><em>On the one hand</em>, we have <strong>symmetries of fields</strong>: for example, in the simplest case of spontaneous symmetry breaking group $\mathbb{Z}_{2}$ acts on the algebra of functions on the manifold. But coordinates remain untouched.</p>
<p><em>On the other hand</em>, there are <strong>spatial symmetries</strong> in physics, for example Poincare symmetry. In this case, as I understand, Poincare group acts on our manifold (in the mathematical sense: <a href="http://en.wikipedia.org/wiki/Group_action">Group action</a>) and <strong>somehow induces an action</strong> on the algebra of fields (How? Can you explain this construction strictly?).</p>
<p>Also we can consider a situation with both kinds of symmetry:</p>
<p>$$
S=\int \left[\left(\partial_{a} \phi_{1} \right)^{2}+ \left(\partial_{a} \phi_{2} \right)^{2}+ \frac{m^2 \phi_{1}^{2}}{2} + \frac{m^2 \phi_{2}^{2}}{2}\right]
$$</p>
<p>In this case we have internal symmetry group $O(2)$: $S[\phi(x)]=S[M\phi(x)]$, $M^{T}M=1$.<br>
It was said to me that in this case physicists write $P\otimes O(2)$, where $P$ is Poincare group, $O(2)$ is acting only on fields, but Poincare group on both space manifold and fields.</p>
<p><strong>So my questions are:</strong></p>
<p><strong>What is the relation between these symmetries and what is the exact construction here? How group acting on space manifold induces action on fields?</strong></p>
<p>Can you explain this strictly in mathematical sense?</p>
| 10,149 |
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<p>I'm trying to get to grips with the Schrödinger equation by looking at a free particle. I'm certain at some point I massively misunderstood something.</p>
<p>According to a textbook and a lecture the free particle moving in positive x direction can be described by</p>
<p>$$
\Psi(x,t) = A e^{i(kx - \omega t)} = \psi(x) \cdot e^{-i\omega t}
$$</p>
<p>Classically, I would expect the particle to move along its path with some constant velocity $v = (x-x_0)/t$, so I would like to determine the probability $P(x,\Delta x,t)$ to find the particle between $x$ and $\Delta x$ at time $t$ in order to compare it with the classical location $x(t) = x_0 + v\cdot t$.</p>
<p>Since I'm looking for a location, I have to use the (trivial) location operator $\hat{r} = r$ and I get:</p>
<p>$$
P(x,\Delta x,t) = \int_x^{x+\Delta x}\Psi(x,t)^* \hat{\Psi(x,t)} dx\\
= \int_x^{x+\Delta x}\Psi(x,t)^* \Psi(x,t) dx\\
= A^2\int_x^{x+\Delta x} {e^{i(kx - \omega t)}}^* {e^{i(kx - \omega t)}} dx \\
= A^2\int_x^{x+\Delta x} dx = A^2 \Delta x
$$</p>
<p>Which doesn't make sense to me at all, since it doesn't depend on either $x$ or $t$. The particle is <em>not</em> at all locations along the x-axis with the same probability at all times; I would rather expect it to move along the axis with the velocity $v$ but regardless which constants I shove into $A$ by using normalisation constraints, $P$ will never depend on $t$. But according to my understanding it should.</p>
<p>Obviously, my understanding is wrong and/or I made some mistakes in my calculation. Where am I going wrong?</p>
| 10,150 |
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<p>Ok so while there have been many discussions regarding the solution(say using the Fourier transform or the lattice Green's function approach) to the two-point resistance of an infinite-square resistive lattice my doubt is actually related to one thing common to almost all the approaches to the problem i.e how does having applied a unit current and removed the same current between the points on which one wants to determine the effective resistance between, imply that the currents at the other nodes are zero.Moreover in order to obtain the difference equation (discretized Laplacian) one makes use of the Kirchoff's current law(which talks about the current in the branches connecting neighboring nodes).</p>
<p>So say i applied a unit current, $I_{m,n}=\delta_{m,mo}\delta_{n,no}- \delta_{m,0}\delta_{n,0}$ at the $(m,n)$th point on the lattice. In order to use my KCL i would need talk about the current in the branches which are incident at the point/node (m,n).So this means that the current emanates from point $A$ and reaches $B$, $C$, $D$ and $E$. But then as soon as it reaches those points does it become zero??</p>
<p>But this doesn't make sense right since we are removing the same unit current from another point(different from the m,nth point on the lattice) in this case from node $O$.</p>
<p>Could someone resolve the same for me.</p>
<p><img src="http://i.stack.imgur.com/qwNjh.png" alt="enter image description here"></p>
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<p>Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as:</p>
<p>$j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation)</p>
<p>, but I'm a little unsure how to interpret the introduction of the modulus operation ($|...|$) and couldn't easily find any examples. </p>
<p>I'm assuming you apply the modulus to any expression which would otherwise yield a negative value for $j$? </p>
<p>I'd appreciate a nod from someone in the know :-). </p>
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<p>Is there any way of achieving effective thrust in space without using conventional fluid or gas propellants such as rocket-fuels or supercooled material? </p>
<p>For instance: gaining thrust through solar energy conversion or pressure pulses? (If that is even possible.)</p>
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<p>It is common when evaluating the partition function for a $O(N)$ non-linear sigma model to enforce the confinement to the $N$-sphere with a delta functional, so that
$$
Z ~=~ \int d[\pi] d[\sigma] ~ \delta \left[ \pi^2 + \sigma^2 -1 \right]
\exp (i S(\phi)),
$$
where $\pi$ is an $N-1$ component field. Then, one evaluates the integral over $\sigma$, killing the delta functional. In my understanding, this gives rise to a continuous product of jacobians,
$$
\prod_{x=0}^L \frac{1}{ \sqrt{1 - \pi^2}} ~=~
\exp \left[- \frac{1}{2} \int_0^L dx \log (1 - \pi^2) \right]
$$
(where I have now put everything in one dimension). Now, obviously this is somewhat non-sense, at the very least because there are units in the argument of the exponential. The way I actually see this written is with a delta function evaluated at the origin,
$$
\exp \left[- \frac{1}{2} \int_0^L dx \log (1 - \pi^2) \delta(x-x) \right].
$$
I see that this makes the units work, but what does that really mean? How do people know to put it there? I know $\delta(0)$ can sometimes be understood as the space-time volume. However, in this case, it clearly has units of $1/L$, so is presumably more like a momentum-space volume. In one dimension, does that mean I can just replace it with $1/L$ (up to factors of $2$ or $\pi$)?</p>
<p>In particular, I have noticed this in the following papers:</p>
<ul>
<li><p><em>Renormalization of the nonlinear $\sigma$ model in $2+\epsilon$ dimensions.</em> Brezin, Zinn-Justin, and Le Guillou. <a href="http://dx.doi.org/10.1103/PhysRevD.14.2615" rel="nofollow">Abstract page</a>.</p></li>
<li><p><em>Perturbation theory for path integrals of stiff polymers.</em> Kleinert and Chervyakov. <a href="http://arxiv.org/abs/cond-mat/0503199" rel="nofollow">Abstract page</a>.</p></li>
</ul>
<p>Kardar does something similar in his Statistical Physics of Fields book, but he simply calls it $\rho$.</p>
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<p>In special relativity, I know you can prove that the derivative with respect to a contravariant 4-vector component transforms like a covariant vector operator by using the chain rule, but I can't work out how to prove the inverse, that the derivative with respect to a covariant 4-vector component transforms like a contravariant vector operator.</p>
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<p>Water boils at positive temperatures when put into a vacuum. Is this the case with all liquids, e.g. mercury?</p>
| 10,156 |
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<p>The Conformal group contains the Poincare group. Typically, if you take a representation of a group and then look at it as a representation of a subgroup, the representation will be reducible. I often hear that CFT's cannot have particles, and I have some understanding of this since $P_\mu P^\mu $ is not a Casimir of the conformal algebra. However, I would think reps of the conformal group should still be reducible representations of the Poincare group, and thus have some particle content. </p>
<p><strong>Is it known how to decompose representations of the conformal group into reps of Poincare? Can we understand it as some sort of integral over masses that removes the scales of the theory? Are there any significant differences between reps of Poincare appearing in reps of the conformal group and the usual representations we are familiar with from QFT?</strong></p>
<p>I'd appreciate any information or a reference that treats this thoroughly from a physics perspective.</p>
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<p>I've gone through an intermediate classical mechanics course, and in solving the two-body problem, we reduce it to a one-body between a larger stationary mass, and a smaller reduced mass.</p>
<p>Most solutions I've seen of the hydrogen atom simply neglect the movement of the nucleus. Some solutions replace the electron mass with the reduced mass, for a more accurate answer.</p>
<p>Since the reduced mass is a result derived in classical mechanics, is it valid to apply it in quantum mechanics? Is it possible to derive the reduced mass within a quantum mechanical framework?</p>
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<p>I've been asked to study on any phenomena related to Weather and Climate. While thinking about this, I noticed that each day, there was a small difference on sunrise and sunset. Like, at one day it was 5:58 AM and at the second day, it was 5:40 AM etc. Does this have anything to do with weather or harmful/useful effects on humans/Earth ? Is this change significance for us, if yes, how? (if no, why?) </p>
<p>There are many questions that arise in my mind and I've been studying a lot about Sunrise and Sunset on internet but I can not find any link or reference where I can find the answers of my doubts. </p>
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<p>If I switch on a generator connected to an open circuit, are the charges oscillating along wires? Where has the energy gone in the open circuit?</p>
| 10,160 |
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<p>Motivated by a previous <a href="http://physics.stackexchange.com/q/79296/6316">question</a>, consider bosonic creation/anihilation operators $a, a^+$ such that $[a, a^+]=1$, and $N = a^+a$.</p>
<p>Is there an analytic expression for the following commutators: </p>
<p>$[e^{za}, e^{wN}]$ and $[e^{za^+}, e^{wN}]$</p>
<p>where $z$ and $w$ are complex (belong to $\mathbb{C}$). </p>
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<p>We use <strong>stimulated emission</strong> and not <strong>spontaneous emission</strong> to produce lasers. Why is this? Can't we produce lasers by the spontaneous emission method?</p>
| 10,162 |
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<p><em>Is it necessary for work to be done on a body that the agent of the force remains in contact with the body?</em> For example, if I hit a football with my foot with a small amount of force and it moves a certain distance on the ground, then what could we say about the 'work' in this case if there would be no friction and the air resistance to stop the ball?</p>
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<p>Why and when do we need to place oil over the sample to achieve higher optical resolution ? Is this idea is valid for the enhancement of all optical microscopy techniques and magnification scales ? </p>
| 10,164 |
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<p>Before anything, I'm sorry for being an outsider coming to opine about your field. This is almost always a stupid decision, but I do have a good justification for this case. I've been reading about <a href="https://en.wikipedia.org/wiki/Superdeterminism">superdeterminism</a> and it bothered me that most of you <a href="http://motls.blogspot.com.br/2013/10/superdeterminism-ultimate-conspiracy.html">treat it as a joke</a>. Well, before anything, I think at this point in history we should've already learned the lesson: ridiculous sounding ideas often end up becoming standards science, and that form of idealogical bullying should never exist. But, if at least the arguments were convincing, one could make an excuse for the author. Except not, his arguments sound just like someone with total lack of understanding of idea who rants against it just because he doesn't like it. <em>"Superdeterminism would mean nature is sending magic agents to control our brains and conspire against us!"</em> Is this even serious?</p>
<p>Yes, as a computer scientist I do think superdeterminism is the most natural approach to how the universe works. On my view the idea of "free will" - as in, something coming from outside the universe, interacting with the information stored in our brain and determining what we do physically - is the sketchy approach, not the opposite. I do think true randomness is the one that requires a lot of magical intervention, while superdeterminism is much more solid. And I even think <a href="http://physics.stackexchange.com/questions/77730/what-is-the-actual-significance-of-the-amplituhedron">recent insights</a> support it without need for any "magical agents". But that is not the reason I'm mad - after all, I'm an outsider and my views doesn't matter a dime. It is because such an important idea with huge implications is regarded as a joke and not even considered just for matters of taste, with no actual proof or evidence of the case. And this is something I do have the right to complain, regardless of the field.</p>
<p>So, please, could anyone justify this prejudice?</p>
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<p>I have recently found my copy of Hawkings' <em>A Brief History of Time</em> which I have never finished. This time I'm determined to read it all the way through. However, the book is now almost 25 years old. Is the book still relevant, or has any recent discoveries invalidated parts of it?</p>
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<p>I am learning how to physically render images in computer graphics. I just saw that the area that gets light is given by the <a href="http://en.wikipedia.org/wiki/Lambert%27s_cosine_law" rel="nofollow">Lambert's cosine law</a>.
In my head it makes perfect sense the relation but once I see the drawing I just cannot see the relation.</p>
<p>Can someone help me with a draw where I can see where does it come from or any suggestion on how to see it on a picture?</p>
<p><strong>EDIT</strong></p>
<p>After some thinking, I get to the following image according to the answer. Can someone confirm if this is ok?</p>
<p><img src="http://i.stack.imgur.com/u1FVc.jpg" alt="enter image description here"></p>
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<p>We have a uranium-236 nucleus that fissions into two equal fragments, and I'm supposed to find the electrical potential energy just as the two fragments split apart. No other information is provided. </p>
<p>I am very confused about this problem for two reasons. First, U-236 is electrically neutral, with exactly 92 protons and 92 electrons. After it splits into two equal fragments, why would there be a charge on the resulting pieces? Secondly, electric potential energy of two point charges is inversely proportional to the distance r between the particles. But since the question asks for the potential energy immediately after fission, r=0 and potential energy goes to infinity.</p>
<p>I'm appreciate any guidance!</p>
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<p>Lets have the scalar Klein-Gordon field interacting with EM field:</p>
<p>$$
L = \partial_{\mu}\varphi \partial^{\mu}\varphi - m^2\varphi \varphi^{*} - j_{\mu}A^{\mu} + q^{2} A_{\mu}A^{\mu}\varphi \varphi^{*} - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}. \qquad (1)
$$
I heard that the normalization of Klein-Gordon field in a theory $(1)$ is invariant under gauge transformations. What normalization is meaned? Does it refer to the factor $\frac{1}{\sqrt{2(2 \pi)^{3} E_{\mathbf p}}}$? How to prove it?</p>
<p>An edit.</p>
<p>It was the invariance of condition $\int j^{0}d^{3}\mathbf r = q$ under $U(1)$ local gauge transformations. $j^{0} = \frac{q}{2m}(\psi^{*}\partial^{0}\psi - \psi \partial^{0}\psi^{*}) - \frac{q^2}{m}A^{0}|\Psi |^{2}$.</p>
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<p>What will be the velocity of a comet is falling to the Earth from infinity at the time of impact if Earth had no atmosphere? the comet is falling radially towards the earth.</p>
<p>Will this velocity be different for comets with different masses or same?</p>
| 10,170 |
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<p>My question is based on the paper <a href="http://arxiv.org/abs/hep-th/0702146" rel="nofollow">Split states, entropy enigma, holes, halos</a>. </p>
<p>What are the scaling solutions discussed on page 49 of the paper ?</p>
<p>It is stated that the equations ${\sum_{j, i\neq j}\frac{I_{ij}}{r_{ij}} = \theta_{i}}$ always have solutions os the form $r_{ij}= \lambda I_{ij}$. why is that true? </p>
<p>I don't understand this as some of the I's may be negative and then a single $\lambda$ can cannot give such a solutions as the distance will be negative in such cases. </p>
<p>I would greatly appreciate an answer explaining the proper meaning of such solutions and what are the conditions for their existence. </p>
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<p>$$m_i\mathbf{r}_i\times\frac{\mathrm{d}^2\mathbf{r}_i}{\mathrm{d}t^2} = \frac{\mathrm{d}}{\mathrm{d}t}\biggl(m_i\mathbf{r}_i\times\frac{\mathrm{d}\mathbf{r}_i}{\mathrm{d}t}\biggr)$$</p>
<p>I do not understand this. How did the $\mathrm{d}/\mathrm{d}t$ go out?</p>
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<p>Lets say we have a tank with a fixed mass of liquid at atmospheric pressure and room temperature. How do we influence the temperature when we exert pressure (e. g., with a piston) on the liquid? Are pressure-enthalpy diagrams the key for that question?</p>
<p>If it is incompressible, temperature would not change, since the applied force does no work, right?</p>
<p>Somehow this collides with my understanding of the microscopic realm, since I associate higher pressure with more microscopic movement and more energy per particle.</p>
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<p>Consider the imaginary time Greens function of a fermion field $\Psi(x,τ)$ at zero temperature</p>
<p>$$ G^τ = -\langle \theta(τ)\Psi(x,τ)\Psi^\dagger(0,0) - \theta(-τ)\Psi^\dagger(0,0)\Psi(x,τ) \rangle $$</p>
<p>It is well known that we can obtain the retarded Greens function by performing Fourier transformation into frequency space and performing the analytic continuation $iω \to ω + i\eta$.</p>
<p>What I would like to do is to perform the analytic continuation directly in the form $iτ \to t$, but I don't know how to deal with the $\theta(τ)$ terms.</p>
<blockquote>
<p>How to perform the analytic continuation $iτ \to t$ of the step function $θ(τ)$?</p>
</blockquote>
<p>In my case, I am dealing with a chiral Luttinger liquid, giving something like</p>
<p>$$ G^τ(x,τ) = -\left[\theta(τ)\frac i{iλ + ivτ - x} - \theta(-τ)\frac i{iλ - ivτ - x}\right] $$</p>
<p>where $λ \approx 0$ is an infinitesimal but important regularization. Of course, the analytic continuation into the time domain is going to look something like</p>
<p>$$ \frac1{iλ + vt - x} $$</p>
<p>but I'm interested in the precise form.</p>
<p>Also, I'm ultimately interested in the spectral function, so I don't mind if analytic continuation gives me yet another variant of a Greens function, but I would like to obtain it precisely from the imaginary time Greens function without going through a tedious Fourier transform. For instance, Giuliani and Vignale's book <a href="http://books.google.com/books?id=kFkIKRfgUpsC" rel="nofollow">"Quantum Theory of the Electron Liquid"</a> uses the Greens function $G_{>}(x,t)$ to great effect (equation (9.133)).</p>
| 10,174 |
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<p>For a probability distribution $P$, Renyi fractal dimension is defined as</p>
<p>$$D_q = \lim_{\epsilon\rightarrow 0} \frac{R_q(P_\epsilon)}{\log(1/\epsilon)},$$
where $R_q$ is Renyi entropy of order $q$ and $P_\epsilon$ is the coarse-grained probability distribution (i.e. put in boxes of linear size $\epsilon$).</p>
<p>The question is if there are any phenomena, for which using non-trivial $q$ (i.e. $q\neq0,1,2,\infty$) is beneficial or naturally preferred?</p>
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<p>Essentially I am wanting to evaluate $$\langle j\, m \mid a^\dagger(\mathbf{k}, \lambda) \mid 0 \rangle \,,$$ where $\lambda$ indicates the circular polarization (about $\mathbf{k}$). We have that $\mathbf{J}= \mathbf{L} + \mathbf{S}$. It's straightforward to show that circular polarization corresponds to definite spin projections along the $\hat{\mathbf{k}}$ axis (the direction of propagation). However, I don't yet know how to find the $\mid \ell\, m_\ell \rangle$ projections. I don't know how how to express $\mid \ell\, m_\ell \rangle$ states in terms of Fock states, for example. With $\hbar = c = 1$,</p>
<p>$$\mathbf{L} = \frac{1}{4\pi}\int \mathop{d^3r} \sum_i E_i\left(\mathbf{r} \times \boldsymbol{\nabla} \right)A_i $$
$$\mathbf{S} = \frac{1}{4\pi}\int \mathop{d^3r} \mathbf{E} \times \mathbf{A} = -i \int \mathop{d^3k} \mathbf{a}^\dagger\left(\mathbf{k}\right) \times \mathbf{a}\left(\mathbf{k}\right)\,. $$</p>
<p>The above uses the following definition $$\mathbf{a}\left( \mathbf{k}\right) = \sum_{\lambda=\pm1} \boldsymbol{\epsilon}_\lambda\left( \mathbf{k}\right) a_\lambda\left( \mathbf{k}\right)$$</p>
<p>So, any help determining the angular momentum of a photon would be appreciated (incuding mention of references dealing with this subject). I have been using the Coulomb gauge.</p>
| 10,176 |
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] |
<p>I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently.</p>
<p>Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy?</p>
| 325 |
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<p>I want to calculate the total electrostatic energy of the Cavendish Experiment (two concentric spheres of radii $R_{1,2}$ which are connected, outer one gets charged, then removed, then after removing the outer sphere and measuring the charge of the inner one, it's zero).</p>
<p>The formula for this is: </p>
<p>$E_{tot} = \frac{1}{2} \sum_{i,j = 1,2} E_{ij}$ with
$E_{ij} = \frac{\sigma_i \sigma_j}{4 \pi \epsilon_0} \int_{S_i} d^2r \int_{S_j} d^2 r' \frac{1}{|\vec{r} - \vec{r}'|} e^{-\mu |\vec{r}-\vec{r}'|} $ where $S_{i,j}$ denotes integrating over the sphere i or j.</p>
<p>I have the sample solution for this exercise and they massively confuse me. They simply state:</p>
<p>"We calculate: $|\vec{r}-\vec{r}'| = \sqrt{r^2 + r'^2 - 2\vec{r}\cdot\vec{r}'} = \sqrt{R_i^2 + R_j^2 - 2R_i R_j cos \theta'}$" Now everything would be fine here if $cos \theta '$ would simply be the angle between $\vec{r}$ and $\vec{r}'$. But they actually go on calculating the integral then and use $\theta '$ as the same angle as the polar angle in the $r'$ spherical coordinatesystem. How is this true? This greatly simplifies the integral, such that we can immediately solve $\int_{S_i} d^2r$ to give $4 \pi R_i^2$ and with the substitution $x = cos \theta'$ the integral over sphere $j$ gets easy aswell.</p>
<p>But I can't make sense of why that formula is true. If I directly compute the scalar product of $\vec{r}'$ and $\vec{r}$ this gives me:</p>
<p>\begin{pmatrix} R_j sin(\theta') cos (\phi')\\ R_j sin (\theta') sin(\phi') \\ R_j cos(\phi') \end{pmatrix} scalar produced with \begin{pmatrix} R_i sin(\theta) cos (\phi)\\ R_i sin (\theta) sin(\phi) \\ R_i cos(\phi) \end{pmatrix} $= R_i R_j (sin(\theta') cos (\phi'))sin(\theta) cos (\phi) + sin (\theta') sin(\phi')sin (\theta) sin(\phi)+ cos(\phi')cos(\phi))$
This does never equal simply $R_i R_j cos \theta'$</p>
<p>Where did I go wrong here? I really hope someone can clear this up for me</p>
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<p>A book which I referenced for Electrical Machinery states that the voltage induced in a conductor inside a magnetic field is given by</p>
<p>$$ \mathcal{E}=(\mathbf{v} \times \mathbf{B})\cdot \mathbf{l}$$</p>
<p>Since all three are vectors, and the cross multiplication inside the brackets results in another vector and that value is being dot product with another vector, the last result should be a scalar. But how can the last result (induced voltage) have a direction then?</p>
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<p>Sean Carroll has a new popularization about the Higgs, The Particle at the End of the Universe. Carroll is a relativist, and I enjoyed seeing how he presented the four forces of nature synoptically, without a lot of math. One thing I'm having trouble puzzling out, however, is his treatment of gravity as just another gauge field. First let me lay out what I understand to be the recipe for introducing a new gauge field, and then I'll try to apply it. I'm not a particle physicist, so I'll probably make lots of mistakes here.</p>
<p>Translating the popularization into a physicist's terminology, the recipe seems to be that we start with some discrete symmetry, expressed by an $m$-dimensional Lie algebra, whose generators are $T^b$, $b=1$ to $m$. Making the symmetry into a local (gauge) symmetry means constructing a unitary matrix $U=\exp[ig \sum \alpha_b T^b]$, where $g$ is a coupling constant and the real $\alpha_b$ are functions of $(t,x)$. If $U$ is to be unitary, then in a matrix representation, the generators must be traceless and hermitian. If you already have an idea of what the matrix representation should look like, you can determine $m$ by figuring out how many degrees of freedom a traceless, hermitian matrix should have in the relevant number of dimensions. For each $b$ from 1 to $m$, you get a vector field $A^{(b)}$, which has <a href="http://physics.stackexchange.com/questions/31143/counting-degrees-of-freedom-of-gauge-bosons">two actual d.f. rather than four</a>.</p>
<p>Applying this recipe to electromagnetism, the discrete symmetry is charge conjugation. That's a 1-dimensional Lie algebra, so you get a single field $A$, which is the vector potential from E&M, and its two d.f. correspond to the two helicity states of the photon.</p>
<p>Applying it to the strong force, the discrete symmetry is permutation of colors. That's going to be represented by a 3x3 matrix. The traceless, hermitian 3x3 matrices are an 8-dimensional space, so we get 8 gluon fields.</p>
<p>So far, so good. Now how the heck does this apply to gravity? Carroll identifies the symmetry with Lorentz invariance, so the symmetry group would be SO(1,3), which is a 6-dimensional Lie algebra. That would seem to create <em>six</em> vector fields, which would be 12 d.f. Does this correspond somehow to the classical description of gravitational waves? If you express a gravitational wave as $h^{\alpha\beta}=g^{\alpha\beta}-\eta^{\alpha\beta}$, with $h$ being traceless and symmetric, you get 6 d.f...?</p>
| 10,179 |
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<p>I am a physicist. I always heard physicists used the terminology "<a href="http://www.encyclopediaofmath.org/index.php/Symmetric_operator" rel="nofollow">symmetric</a>", "Hermitian", "<a href="http://en.wikipedia.org/wiki/Self-adjoint_operator" rel="nofollow">self-adjoint</a>", and "essentially self-adjoint" operators interchangeably.</p>
<p>Actually what is the difference between all those operators? Presumably understandable by a physicist.</p>
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<p>I'm trying to wrap my head around how geodesics describe trajectories at the moment.</p>
<p>I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must move along a timelike geodesic. And a timelike geodesic can be defined as a geodesic that lies within the light cone.</p>
<p>I want to know why exactly null geodesics define the light cone. Or, why null geodesics define the path of light.</p>
<p>Also, if there's a better explanation why matter follows timelike geodesics, that would also be welcome.</p>
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<p>I always heard the eigenfunctions of a <a href="http://en.wikipedia.org/wiki/Self-adjoint_operator">self-adjoint operator</a> form a complete basis. Where can I find a proof in infinite dimension space? Presumably readable for physicists.</p>
| 10,182 |
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<p>Can anyone tell me how to solve this problem: </p>
<blockquote>
<p>Alpha Centauri is $4.4$ light years away from Earth. What speed $u$
would a spaceship headed towards Alpha centauri had to have in order
to last $t' = 10 \text{ years}$ for a passanger onboard?</p>
</blockquote>
<p>The only thing i know how to do is to determine the proper time which is the one measured on a spaceship... </p>
<p>$$\boxed{t'\equiv \tau}$$</p>
<p>I don't know how to calculate any other variable if i don't know $\gamma$ or $u$. I think i should get a system of 2 equations and bust $u$ out of there... But i just cant solve this after 4 hours of trying...</p>
<hr>
<p><strong>EDIT:</strong> </p>
<p>I know equations for time dilation, length contraction: </p>
<p>\begin{align}
\Delta t &= \gamma \Delta t' \xleftarrow{\text{time dilation}}\\
\Delta x' &= \gamma \Delta x \xleftarrow{\text{length contraction}}
\end{align}</p>
<p>and Lorenz transformations:</p>
<p>\begin{align}
\Delta x &= \gamma(\Delta x' + u \Delta t')\\
\Delta x' &= \gamma(\Delta x - u \Delta t)\\
\Delta t&= \gamma\left(\Delta t' + \Delta x' \frac{u}{c^2}\right)\\
\Delta t'&= \gamma\left(\Delta t - \Delta x \frac{u}{c^2}\right)
\end{align}</p>
| 10,183 |
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<p>Is <a href="http://en.wikipedia.org/wiki/Potential_energy" rel="nofollow">potential energy</a> and "<a href="http://en.wikipedia.org/wiki/Work_%28physics%29" rel="nofollow">work</a> done" the same thing?</p>
<p>If they are not one and the same thing then why is potential energy always associated with "work done"?</p>
<p>Could you explain me with some examples?</p>
| 10,184 |
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<p>Suppose I want to write down an interaction term for an action for spin 1/2 fermions that is $SU(2)$-symmetric. </p>
<p>I start from the most naive general form of such an action:
$$S_{int} ~=~ \int_{4321} \sum_{\alpha \beta \gamma \delta} \bar \psi(4)_\alpha \bar \psi(3)_\beta \psi(2)_\gamma \psi(1)_\delta V(4,3,2,1)_{\alpha \beta \gamma \delta}$$</p>
<p>where the indices $1$ to $4$ stand for momenta and frequencies of my fermions.</p>
<p>Now I want to find the form $V$ must have in order to be $SU(2)$ symmetric. By transforming the fermion fields and demanding that the action must stay invariant under that, I can show that $V$ must transform as
$$V_{\alpha' \beta' \gamma' \delta'} ~=~ \sum_{\alpha\beta\gamma\delta} R^\dagger_{\alpha \alpha'} R^\dagger_{\beta \beta'} R_{\gamma \gamma'} R_{\delta \delta} V_{\alpha \beta \gamma \delta}$$
where $R \in SU(2)$.</p>
<p>Well, and now I'm stuck continuing from here. Using some handwaving I think I could argue that $V$ must preserve total spin and also total spin in $z$-direction I could probably argue that $V$ can only scatter triplets to triplets, singlets to singlets, and also can't change the $z$-component of the triplet, but I would rather use a more rigorous approach.</p>
<p>Which will probably involve irreducible representations? I could probably get to the singlet/triplet statement above by noting that $SU(2)$ will transform multiplets into the same multiplet, so the singlet would be invariant under $SU(2)$ and the triplets would somehow mix. But why is it appropriate then to look at an "ingoing" singlet or "ingoing" triplet formed by indices $\gamma$ and $\delta$ as opposed to forming such states with, e.g., indices $\alpha$ and $\gamma$?</p>
<p>ADDENDUM:
Well, I guess I can also start with the spins in a different basis: Assuming that I can put the two "ingoing" and the two "outgoing" spins into either a singlet or one of three triplets, I guess I can write the action as
$$S \sim \int_{1234} \sum_{jm j'm'} (\bar \psi(4) \bar \psi(3))_{jm} (\psi(2) \psi(1))_{j'm'} V(4,3,2,1)_{jm;j'm'}$$
Then I can first argue that due to conservation of total spin we require $j = j'$. And then for $V$ I can look at singlet-singlet scattering and triplet-triplet scattering separately:
For $j = j' = 0$, $m$ must be $0$ and so $V$ is a scalar, invariant under $SU(2)$, But for $j = j' = 1$, the states with $m = 0, \pm 1$ transform into each other in some way, and thus I must work a bit harder to get the symmetry right. I'll think about this, but in the meantime I'm open for more suggestions.</p>
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<p>Is there a fundamental difference in the definition of entropy when considering the classical thermodynamic picture vs. the quantum mechanical picture, or are they both fundamentally equivalent?</p>
| 10,186 |
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<p>I am trying to find the following dipole moment matrix element $(|n,\ell,m\rangle)$.</p>
<p>$$e\langle1,0,0|\vec r|2,0,0\rangle$$
I believe that I can say this matrix element is zero because of parity. The wavefunctions have parity $(-1)^\ell$ and seeing as each has $\ell =0$, they are even parity. Then r is odd, as it sends $\vec r \rightarrow-\vec r$. This means the entire expression is odd, therefore the matrix element is 0.</p>
<p>Is my reasoning sound?</p>
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<p>Do all forms of energy have a mass? We know by $E=mc^2$ that mass and energy are directly proportional, but there are massless forms of energy such as electro-magnetic waves. I am also told that there are different forms of mass, such as invariant mass, virtual mass, and relativistic mass. This electro-magnetic wave seems to be a kind of quantum mechanical wave, that collapses into a particle dubbed a photon. When it interacts with matter, it gives that matter energy, which gives it more mass. Particles with an invariant mass also can be described by probabilistic waves, and behave in a similar way as photons. That is, when its wave function is disturbed in some way, it too collapses into a particle of some form, depending on what it is. The wave function must not be able to exceed the speed of light, because it has an invariant mass.</p>
<p>So my question is, what happens to the energy of an electromagnetic wave, when it transforms into an energy that has a form of mass, and where is that mass from, and what is the difference between the wave function of an invariant mass, verses the wave function of a photon without mass? Okay lots of question here, sorry.</p>
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<p>Consider a point-mass $m$ having constant velocity but undergoing influence from two forces, $F_1$, $F_2$, having equal magnitude but opposite directions. Because the forces' magnitudes are equal, I would expect no net acceleration of $m$, but if $m$ is moving, the forces are doing work on $m$ ($F_1$'s work being the inverse of $F_2$'s work), whereas if $m$ is stationary, no work is done by either $F_1$ or $F_2$; why is that? In neither case is there net acceleration, and in neither case is there net work, but why is work calculated differently in one case than in the other?</p>
<p>Note: This is a conceptual question, so I'm afraid that simply appealing to definitions, i.e., "because work is defined that way", will be unsatisfactory.</p>
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<p>I know that matter can be converted to energy through E=mc^2.</p>
<p>I also know that engery can be and has been converted to information through Landauer's principle (with Maxwell's demons).</p>
<p>Does this mean that I can take a brick and covert it into information? (It is irrelivent if there is no known process yet)</p>
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<p>A Cessna 150 aircraft has a lift-off speed of approximately 125 $kmh^{-1}$.
What minimum constant acceleration does this require if the aircraft is to be airborne after a take-off run of 129 m?</p>
<p>So I wanted to use the formula that $x=\cfrac{(v_{final}^2 - v_{initial}^2)}{2a}$</p>
<p>To solve for the constant acceleration but I don't have a final velocity from this problem? My initial velocity is 125 $kmh^{-1}$ and my $x$ (displacement) would be 129 meters but how do I find the constant acceleration?</p>
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<p>I have done a physics experiment (setup below). And was asked to determine the experimental and theoretical acceleration. </p>
<p><img src="http://i.stack.imgur.com/xW1il.png" alt="enter image description here"></p>
<p>The data I've got </p>
<p><img src="http://i.stack.imgur.com/oqn75.png" alt="enter image description here"></p>
<hr>
<p>Ok, am I right to say </p>
<p>Experimental acceleration = $2(s_f - s_i) / t^2$</p>
<p>Theoratical acceleration = $m_2 \times 0.98 / m_1$</p>
<p>Then </p>
<p>Percentage discrepancy = $\frac{|(Experimental - Theoretical)|}{Theoretical} \times 100$%</p>
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<p>I had asked a similar question about a calculation involving the winding number <a href="http://physics.stackexchange.com/questions/30997/winding-number-in-the-topology-of-magnetic-monopoles">here</a>. But i haven't got a satisfactory response. So, I am rephrasing this question in a slightly different manner. What is the winding number of a magnetic monopole solution? Why is it a topological invariant? How is it connected to the degree of a map and the vector potential? While answering please could you bear in mind the fact that I have some very little knowledge of point-set topology, and no knowledge at all of algebraic topology.</p>
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