Factorial Representations of Compact Lie Groups, Wigner Sets and Locally Invariant Quantum Fields
Abstract
The fibre bundle construct defined in our previous work continues to be the context for this paper; quantum fields composed of fibre algebras become liftings of; or sections through; a fibre bundle with base space a subset of curved spacetime. We consider a compact Lie group such as SU(n) acting as a local gauge group of automorphisms of each fibre algebra A(x). Compact Lie groups, represented as gauge groups acting locally on quantum fields, are key elements in electroweak and strong force unification. In our recent joint work we have focused on the translational subgroup of the Poincare group as the generator of local diffeomorphism invariant quantum states. Here we extend those algebraic nonperturbative approaches to address the other half of unification by considering the existence of quantum states of the fibre algebra A(x) invariant to the action of compact nonabelian Lie groups. Wigner sets are complementary to little groups and we prove they have the finite intersection property. Exploiting this then allows us to show that invariant states are common in the sense that the weakly closed convex hull of every normal (density matrix) state contains such an invariant state. From these results and our related research emerges the existence of a locally invariant density matrix quantum state of the field.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.10601
 arXiv:
 arXiv:1710.10601
 Bibcode:
 2017arXiv171010601M
 Keywords:

 Mathematical Physics;
 General Relativity and Quantum Cosmology
 EPrint:
 11 pages