Localization via Automorphisms of the CARs. Local gauge invariance

TL;DR

This paper demonstrates how automorphisms of the CAR algebra can be used to achieve localization and gauge invariance in quantum field theory, revealing that only trivial gauge-invariant observables exist.

Contribution

It introduces a novel approach to localize quantum fields via automorphisms of the CAR algebra, connecting classical restriction maps to quantum invariance groups.

Findings

The module action induces automorphism groups on the CAR algebra.

Localization of the algebra corresponds to invariance under subgroup automorphisms.

Only trivial gauge-invariant observables are present in the algebra.

Abstract

The classical matter fields are sections of a vector bundle E with base manifold M. The space L^2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^\infty(M) on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra A, of the canonical anticommutation relations on L^2(E), with which we can perform the analogous localization. That is, the net structure of the CAR, A, w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps. As a corollary, we prove a well-known "folk theorem," that the algebra A contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E.