/sci/ - Science & Math

>>11550816
Ehrenfest's theorem states that expectations of observables satisfy classical laws of motion. Why did you think Coulomb stops working at the quantum scale? That's absurd.
>>11551359
None. Symmetry indices are indices corresponding to a representation space [math]V[/math] where [math]\pi: G\rightarrow GL(V)[/math]. For Lorentz [math]G[/math] happens to be the homogeneous isometries [math]\operatorname{Isom}^+ M \cong SO(1,3)[/math] of the tangent bundle [math]V = TM[/math].

>>11504004
>Is this in any way physically meaningful?
I don't know physically but it's mathematically meaningless. Generally irreps of semisimple Lie algebras form Verlinde algebras over [math]\mathbb{Z}[/math]. Projection into [math]\mathbb{Z}_n[/math] makes you lose information about certain irreps so the entire Verlinde algebra structure breaks, which makes the whole representation theory of Lie algebras via Verma modules meaningless. In particular the non-zero character of [math]\mathbb{Z}_n[/math] leads to non-unique singular vectors in the Verma modules which makes your vacuum sector degenerate and zero-point energy blow up, even for finite-dimensional Lie algebras such as [math]\mathfrak{su}(2)[/math], which is absurd.

>>11333684
So what I have been saying, again and again, is that as soon as [math]L^2[/math] is an integral of motion, there exists a point, called the CoM, in the object such that every other point rotates about it under rigid body rotation. This will happen whenever the motion is free.

>>11265350
Families of sup-seminorms [math]N_C^\infty(x) = \sup_{y \in C}|y(x)|[/math] over compact sets [math]C\subset Y[/math] on locally convex spaces [math]X[/math] generates the Mackey topology [math]\tau(X,Y)[/math], so we can consider the "balls" obtained from t[math]N_C^\infty[/math].
By Mackey-Arens, any dual topology [math]\mathscr{T}[/math] is both coarser than the weak-operator topology [math]\sigma(X,Y)[/math] and finer than the Mackey topology [math]\tau(X,Y)[/math], which is in turn finer than the strong-operator topology [math]\rho(X,Y)[/math] (due to Bourbaki-Alaoglu/"balls are compact" theorem). We may wish, then, to investigate if we can still generate dual topologies with seminorms as we tune its coarse-ness between weak- and Mackey. Perhaps we need to change what the [math]C[/math]'s are in [math]Y[/math] or consider [math]N_C^p[/math] for [math]p<\infty[/math]. I'm not sure, but it certainly can be interesting.
>>11265377
Let's investigate this possibility in QM. Note that w- and s-closures of the algebra of observables coincide via von Neumann's bicommutant theorem. Since states are given by linear [math]*[/math]-irreps, weak and strong notions of convergence can only differ due to the topology on the rep-space, not that on the algebra.
Now when dealing with rigged reflexive Hilbert spaces with Frechet topology, such as tempered distributions, however, one can prove that closed bounded subsets in [math]\mathcal{H}'[/math] is compact, hence [math]\text{weak}\xrightarrow{\text{Frechet}}\text{Mackey} \rightarrow \text{strong}\rightarrow \text{weak}[/math]. So w- and s-convergence again coincide on Frechet spaces. In particular, since Stone-von Neumann tells you that there is essentially only one unitarily-equivalent irrep of the Weyl algebra, and Groenewold-van Hove quantization lands your rep-space in a maximal Poisson subalgebra of [math]C^\infty[/math] (which has Frechet topology), the strong-weak distinction basically never happens in QM.

>>11248345
The equality is after integration [math]\int_M d^4p \delta(p_0 - E) = \int_{\Sigma_E} d^3p[/math], where [math]\Sigma_E = \{p\mid p_0 = E\}\subset M[/math] is a constant-energy surface.

https://www.cambridge.org/core/books/quantum-theory-of-fields/22986119910BF6A2EFE42684801A3BDF
https://arxiv.org/abs/1105.5289v3
https://www.worldscientific.com/worldscibooks/10.1142/3708