Factorial Representations of Compact Lie Groups, Wigner Sets and Locally Invariant Quantum Fields

TL;DR

This paper explores the algebraic structure of quantum fields with compact Lie group symmetries, proving the existence of invariant quantum states using fibre bundle and Wigner set techniques, advancing non-perturbative unification approaches.

Contribution

It extends algebraic non-perturbative methods to demonstrate the existence of locally invariant quantum states under compact Lie group actions.

Findings

Wigner sets have the finite intersection property.

Invariant states are dense in the convex hull of normal states.

Existence of a locally invariant density matrix quantum state.

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 space-time. 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 non-perturbative 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 non-abelian 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.