@bolbteppa ok finally wrote all this up. maybe youd be interested in this chat on subj hope you can drop by with open mind. @Secret + @DanielSank think youd find it of interest + relevant too :) chat.stackexchange.com/transcript/message/56826061#56826061
Neumann's principle states that the physical properties of a crystal must have the same symmetry as the crystal itself. I have only seen this applied to macroscoping properties such as currents and electromagnetic fields. Does anyone know of an application to internal properties? Specifically I want to consider the exchange coupling between localized spins.
The reason I suspect it is different, is that in contrast to currents and electromagnetic fields, I expect the exchange coupling to be positional dependent, not only depend on direction.
@Slereah I also read about this in a paper about Einstein-Cartan theory. But what does the shortest curves mean if the metric is pseudo-Riemann, non-positive definite?
if the affine connection is not Riemann, then the corresponding curvature doesn't indicate the same for the geometry as the Riemann curvature does, right?
@Slereah I have seen that kind of separation in papers which illustrate the teleparallel equivalent to general relativity.
$R\eta=R^{\{\}}\eta+xxx$, then a teleparallel equivalent to GR is formulated by the Lagrangian $-xxx$, where $R^{\{\}}$ is the Riemann curvature.
I found the papers about teleparallel gravity is of a great diversity.
some formulate teleparallel gravity by the Lagrangian with the fundamental variables the tetrad and affine connection and others formulate it by the Lagrangian with the fundamental variable the translational gauge potential.
If we're looking at $\phi^4$ loop corrections, and we get an integral (which we've Wick rotated) that looks like: $$\int \tilde d^4k_E \frac{1}{k_E+m^2}$$ for which we then impose a cutoff at $\Lambda$ and note that at large $k_E$, $k^2_E+m^2\sim k^2$. Apparently the integral becomes $$\int^\Lambda \frac{|k_E|^3 d|k_E|}{|k_E|^2},$$ but how come the measure changes here?
what I basically don't see is why $d^4k_E\rightarrow |k_E|^3 d|k_E|$ is true (not the absolute value signs, just why this reduces to a one dimensional integral)
If we're looking at $\phi^4$ loop corrections, and we get an integral (which we've Wick rotated) that looks like: $$\int \tilde d^4k_E \frac{1}{k_E+m^2}$$ for which we then impose a cutoff at $\Lambda$ and note that at large $k_E$, $k^2_E+m^2\sim k^2$. Apparently the integral becomes $$\int^\Lambda \frac{|k_E|^3 d|k_E|}{|k_E|^2},$$ but how come the measure changes here?
in the teleparallel gravity, if the nonmetricity vanishes, a global frame which is achieved by parallelly transporting a frame at a certain point has a constant metric, so this is like an Euclidean metric, but if the nonmetricity doesn't vanish, the metric is not constant, the global frame doesn't look like Euclidean, but the paper says we can choose the global frame to be an orthonormal frame, which has a constant metriic, so why is there such contrdiction?
another puzzle I encounter is that whether there is really a torsion $T^\alpha$ in a teleparallel gravity. If I choose a global frame to be the covariant constant frame, that is, a frame on which $\Gamma^\alpha{}_{\beta\gamma}=0$, then $T^\alpha=D\vartheta^\alpha=d\vartheta^\alpha$, but if $\vartheta^\alpha$ is a global coframe, then it can be chosen to be a holonomic frame globally $\vartheta^\alpha=dx^\alpha$, then $T^\alpha=D\vartheta^\alpha=d\vartheta^\alpha=ddx^\alpha=0$.
of course I can choose another coordinate system so that $\vartheta^\alpha$ is not holonomic, but the torsion is a tensor, so if it vanishes in a coordinate system, it should vanish in all coordinate systems.
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Quick question, the logarithmic divergence of the one-loop correction in $\phi^4$ theory, doesn't this have two divergences, one as $\Lambda\rightarrow 0$ and one as $\Lambda\rightarrow \infty$? Does this just mean it shows IR and UV divergence?
@ACuriousMind since your the most mathematical physics person I know. I was wondering if you knew anything about Lieb Robinson bounds? and how would someone with little math background (like me) gain the technical expertise?
Reading through the wiki page would be a good start, then check the links they're giving, if it's a fairly obscure topic there may not be that many standard texts that are very comprehensive, you might just end up reading papers
I actually through non-rel qm was entirely non-causal, don't the amplitdues just decay exponentially outside the lightcone
Background
So basically there is a dictionary/correspondence between first and second quantised quantum mechanical theories. When I suddenly turn on a potential in potential in first quantised theories I can use the sudden approximation and find the probability of the final state. I can use the d...
there's not necessarily a deep reason that a specific choice of gauge is possible - you just have to look at the transformations and figure out whether or not you can solve the equations for the transformed potentials being equal to the desired values in your gauge
Right. So $\mathbf A'=\mathbf A +\nabla \lambda$ and $V'=V-\frac{\partial \lambda}{\partial t}$. Choose $\lambda$ so that $\frac{\partial \lambda}{\partial t}=V$?
@bolbteppa the motivation I read from papers of my MSc supervisor is that the energy-momentum 3-form or even the angular momentum and center-of-mass moment 3-form of the gravity formulated by teleparallel theory is a tensor, rather than psuedotensor.
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