  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.
