# Tag Info

5

I) The main point is that we usually only consider tensor products $V \otimes W$ of vector spaces $V$, $W$; but groups (say $G$, $H$) are often not vector spaces. If we only consider tensor products of vector spaces, then the object $G \otimes H$ is nonsense, mathematically speaking. With further assumptions on the groups $G$ and $H$, it is sometimes ...

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The submaximal dimension of the group of isometries of a Pseudo-Riemanniann manifold of dimension $n$ with $n\ge4$ and $n\neq5$ is $$\frac{1}{2}n(n-1)+ 2 .$$ However, a result proved here(Theorem 3.2) shows that a spacetimes with that amount of isometries in dimension $4$ must be Minkoswki spacetime. Hence, the maximal number of Killing vectors you can ...

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I'm as confused as you by the boxed equation. At best the author is making that all-too-common mistake of reordering the expressions in a transitive equals relation, making the equation nonsensical when read left to right. However, it is not quite a tautology to prove what I think this is trying to prove: If $E$ and $\psi$ satisfy $H \psi = E \psi$, ...

1

The Weyl system $\exp\left[\frac{i}{\hbar} Q \hat{p}\right]$ and $\exp\left[\frac{i}{\hbar}P\hat{q}\right]$ comprise two "presentation" elements of the Heisenberg group. To the extent $\hat{p}$ is a derivative with respect to position q, Q is the shift amount q in any function of it is translated by the action of $\exp\left[\frac{i}{\hbar} Q ... 1 This answer outlines how the defining matrix representation of the symplectic group Sp(2m,R) is ray optics, whilst the infinite-dimensional unitary rep of Sp(2m,R) carried on the space of wavefunctions is diffractive optics in the Fresnel approximation. The outline is for Sp(2,R) (cylindrical lenses) but the generalization to Sp(2m,R) is reasonably ... 1 I wanted to complement the answers above. For (1)$so(4) = so(3) \times so(3)$, one$so(3)$is from the geometric 3D symmetry of the Hamiltonian, and the other$so(3)$is from the potential term of$\frac{k}{r}$. For (2). the second$so(3)$symmetry is a dynamic symmetry and only holds when potential term is inversely proportional to$r\$. One has to do ...

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Actually there are analogies or generalisations of results which reduce to Noether's theorems under usual cases and which do hold for discrete (and not necesarily discretised) symmetries (including CPT-like symmetries) For example see: Anthony C L Ashton (2008) Conservation Laws and Non-Lie Symmetries for Linear PDEs, Journal of Nonlinear Mathematical ...

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I'll start with describing the best known instance of center symmetry breaking: The center of the gauge group is significant in lattice gauge theories because the expectation value of the Polyakov loop is only invariant under a central transformation if it is zero. Hence, although the lattice action is invariant under a central transformation, this ...

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There is no general algorithm for doing so, and even figuring out how many conserved quantities a system has can be difficult. A famous example is the Toda lattice, a system which was originally proposed by Toda in 1967 and was believed to be chaotic, but was in fact proven to be integrable (to have too many conserved quantities to be chaotic) in 1974 by ...

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