@WilliamBulmer Ah, the trace must be an actual energy density because the whole EM stress-energy tensor must be massless. The trace of the full tensor is $\rho +\mathrm{tr}(\sigma) = 0$
@WilliamBulmer skimmed wikipedia re stress tensors Cauchy/ Maxwell. alas not familiar with this area. are you the 1st to make a correlation? seems interesting on the face. is the "Cauchy Stress tensor of Classical Field Theory" a thing somewhere?
@WilliamBulmer suggest dont give up on dimensional analysis despite other "advice", it is downplayed or dismissed by some theorists but some others have used it to great effect, had this debate with slereah weeks back, could dig up a link of it being used by serious scientist. think there is some key analogy of dynamics of spacetime that is yet to be made (but right on the verge) & it will open up a lot of new vistas.
@BernardMeurer Well, they tried that with personalized tickets a while back but legal difficulties in actually forbidding people to tamper with those and banning resellers led to them not doing that anymore
@dmckee dimensional analysis is very useful for showing "patterns" of energy dynamics throughout different physical systems where the math is equivalent but the milieus are or seem different.
@vzn The thing about dimensional analysis is that you only know it was right in retrospect: after you done the full calculation or run the test. It's a powerful tool that has occasional silent flops. And the sometimes non-signalling nature of the flops is the reason to treat it with caution.
@JohnDuffield agree the connection is interesting or even far more than merely interesting. in the way the tip of an iceberg is "interesting". this basic analogy is based on dimensional analysis
On the other hand, it's a great way to make short plausibility argument when you don't want to deal with a complete treatment. I got a lot of those from profs in graduate school.
@dmckee its different than solving eqns but goes "hand in hand". theres still lots of hard work with eqns. its a sort of parallel or shadow system of physics/ dynamics analysis. it helps devise eqns. aka einstein "imagination is more important than knowledge." arguably many important eqns from physics were devised wrt dimensional analysis techniques. imho it could be said of eg schroedinger eqn.
@vzn That cauchy stress tensor is the spatial part of the stress-energy tensor as ACuriousMind said. It appears in classical field theory, and is part of the Cauchy Momentum Equation, which we use to describe non-relativistic momentum transform in a continuum
@vzn I don't think dimesnional analysis ever suffices as a valid physical argument for why something should be true. As I had said, I think it is more helpful as a compass for finding deeper physical/mathematical relationships
@WilliamBulmer agreed! its not any kind of proof of anything. only an xkcd cartoon would be so naive etc... but it can be something like a semiformalized intuition
@WilliamBulmer it seems that there is some small scale eqn of spacetime distortion that might apply/ generalize at larger scales, still yet to be uncovered, and that this is part of the difficulty of merging QM+GR
@vzn I personally have much of a distaste for even using it as intuition for why something should be true. It's a bit like reasoning that Antonio Banderas is Hispanic because his name sounds hispanic
@ACuriousMind all great new physics starts as intuition... are you interested or not? (rhetorical! usually so far the answer is not) :P
@ACuriousMind theres a reason very precise software engineering full of copious amts of airtight logic is also called development... also in crisp photography etc
@vzn Physical intuition is different from blind guessing or prophecies that rival horoscopes in their specificity. I might be interested in - although sceptical of - the former, but I'm not interested in that latter.
@ACuriousMind try a real comic/ art instead, just watched the movie, its amusing, it has feynman playing bongos at beginning & think it might have been filmed partly at caltech, is anyone else gonna see it? phdcomics.com/comics.php
I read a paper of Ed Witten where he pointed out, that the Hamiltonian for a particle moving on a riemannian surface is equal to the Laplace-Beltrami Operator.
@ACuriousMind the top physicists in the world expect a GR+QM unification and of course lose no time or see no contradiction sniffing at/ criticizing einstein for seeking it for nearly ~½ his life without publishable results.
@vzn There is a concrete reason to expect a unification of GR and QFT: both describe our world at different scales, and reductionism has so far served us well in physics - by inference, it is reasonable - though not certain - to expect both of these theories can be reduced to a more fundamental theory of which they'll both be a part. What is your reason for expecting a "small scale eqn of spacetime distortion" and what does that even mean?
@ACuriousMind it means nothing until someone figures out what it means, as we both are well aware, still working on that, & thx so much for all the help :P
@Ocelo7 But if you want to study (mathematical) twistors, you maybe like these videos (about supersymmetric gauge theories from a mathematical point of view):
@ACuriousMind look, am willing to take a small stab at this/ go out on a limb (for you to immediately saw off...) a lot of eqns point that spacetime itself has a dynamic/ elastic property, and lightwaves are apparently just the small scale distortion of it whereas gravity is the large scale distortion of it. so then, the zen question, how do these two mesh?
@vzn I don't know what "elastic property of spacetime" means, or why you would call lightwaves a "small scale distortion" of spacetime when spacetime is not dynamical in the classical or quantum theorie sof electromagnetism that describe light waves.
@ACuriousMind did you see JDs eqn/ analogy or not? speed of light, permissivity/ permittivity of free space etc... nearly same as what WB is "poynting" at :P
@ACuriousMind (its always so fun arguing with you because you are nearly always a perfect standin/ representative for The Physics Establishment™)
@vzn I am not saying they are distortions of space time. I'm currently interested in why the eigenvalues of any stress tensor should have units in energy density, other than mathematical coincidence. @ACuriousMind answered part of it, I think
@vzn Oh yes, I've seen that "analogy" from JD many times until I began to ignore him. "Elasticity" is a well-defined notion in continuum mechanics of materials, but it's not a well-defined notion w.r.t. spacetime, at least not in GR. It doesn't actually mean anything to say that "spacetime is elastic", and it doesn't mean anything to say that light is "distortion of spacetime" because there are no equations that would allow that interpretation.
@ACuriousMind he just posted (some of) the equations, the equations exist but they havent been analyzed/ recognized fully for what they are, agreed that work (both analytical/ interpretive) remains to be done
I mean: let Lie group $G$ act on symplectic manifold $M$ to preserve the symplectic form. Then why must there be a momentum map $\mu:M \to \mathfrak{g}^*$? That is, why must it be possible to associate to every element $X$ of $\mathfrak{g}$ a function $H$ so that the vector field $\tilde{X}$ on $M$ induced by $X$ satisfies $\omega(X,-)=dH$?
@ACuriousMind surely easily an entire paper could be written expanding on the "analogy" that you just dismissed. easily in the sense there is that much material but very hard that almost nobody alive could do it or has done it... based on convention
@vzn SR is not deforming spacetime, and it's clear that the technical meaning of "distortion of spacetime" is a change in the metric from a "standard metric" understood in the context. What is the technical meaning of "elasticity of spacetime"?
@ACuriousMind all those theories/experiments/ideas which about the elasticity/permeability/etc of spacetime It would be cool if someone would explain this as I'm sure it could be interesting or new and might shed light into string theory.
If $H^1(M)=0$ then you can just integrate (some vector field relating to $\tilde{X}$) along $M$ to get $H$. So you're guaranteed a momentum map in this case. But $H^1(M)=0$ is pretty strong a condition. Is there a weaker one?
@ACuriousMind you just said distortion of spacetime is not defined, and now you say its clear what the technical meaning is. sorry am getting mixed msgs here :(
@ACuriousMind it appears the math has been long telling us that light waves themselves are ripple in spacetime.... think its the tip of iceberg of large new theory also merging GR+QM. it will take a genius on the level of einstein to work out the details... o_O
@Alyosha There always must be a momentum map. But, sorry, I am not so familiar with symplectic geometry (and Hamiltonian group actions related to your question).
@0celo7 it seems light and space are not actually different, its a flaw in our (human, that is) habitually dualistic thinking. light is rippling space.
@PhysicsGuy You only get the momentum map if you're given a Hamiltonian group action. The question is when such an action can exist (I don't know the answer, unfortunately).
@vzn "etc."? What is this et cetera that no one but you and JohnDuffield have access to? Stop weaseling around and write down a single equation substantiating any of the claims you made.
@ACuriousMind lol some of the striking thing is that maybe most of the eqns are already written down... (but ofc nobody will believe it until something new is derived from the "new" assumptions)
I mean, surely I can take a Hamiltonian and do stuff like cavity QED with photon wave functions, perfectly well with just regular first quantization. And I think I can call them photons as well.
I'm just tired to see this room filled with not only one but more people who clearly have no idea what they are talking about yet try to deceive others into believing physics is a great conspiracy.
@MikaelKuisma I think Gennaro's comment is supposed to mean that you haven't really a reason to connect those "photons" with the electromagnetic field.
Aside from it working as an effective theory, that is
Which is a good reason to use it as an effective theory, but not really satisfactory when used as an answer to what a "photon" is. But maybe it just illustrates that the concept of "photon", just as the concept of "particle" in general, becomes rather hard to nail down when you dig deep enough
@ACuriousMind good question. there is an answer. the photon is a localized/ probabilistic measurement of the (spacetime) wavefront. by an atom (or other particle)... the only known spacetime-rippling measuring device that we can interface with
@MikaelKuisma I think it's actually a good reminder to all us QFT purists that there are such "first quantized" approaches that work rather well in their contexts ;)
I have recently become interested in quantization of fields for practical scientific purposes related to plasmons, I think physics SE is a good sandbox for that.
@MikaelKuisma Yeah just be careful because a lot of the really experienced and knowledgeable people here mix relativity into QFT without telling you they're doing that.
It's really hard to ask questions about quantized fields here without everyone immediately getting into complexities that don't exist in non-relativistic situations.
@ACuriousMind It was certainly a big surprise to me to find that out (like a month ago, when reading some cavity QED paper). I have not studied any QFT and always thought it as super mystical stuff, and that you 'actually have particles' there. But then, it is just plain old boring wave functions it seems (at least for such sand box model I presented).
Right now I am wondering about longitudinal photons. As far as I understand, they are under constraints defined completely by the electrons. So they differ very little from stuff familiar to me from TDDFT, such as obtaining the Hartree-potential by solving the Poisson-equation. So that, the dynamical degrees of freedoms (to get the 'harmonic oscillator') to longitudinal photons comes purely from the quantization of electrons then. Hmm... I think I need to read a book or a few.
