Local Quantum Constraints

TL;DR

This paper develops a C*-algebraic framework for local quantum field theories with constraints, exemplified by Gupta-Bleuler electromagnetism, ensuring the usual axioms are satisfied without indefinite metrics.

Contribution

It introduces 'weak' Haag-Kastler axioms for constrained theories and demonstrates their validity through a detailed C*-algebraic analysis of Gupta-Bleuler electromagnetism.

Findings

Established 'weak' Haag-Kastler axioms for constrained quantum field theories.

Constructed the physical algebra without indefinite metric representations.

Connected the 'weak' and usual spectral conditions in the context of Gupta-Bleuler.

Abstract

We analyze the situation of a local quantum field theory with constraints, both indexed by the same set of space-time regions. In particular we find ``weak'' Haag-Kastler axioms which will ensure that the final constrained theory satisfies the usual Haag-Kastler axioms. Gupta-Bleuler electromagnetism is developed in detail as an example of a theory which satisfies the ``weak'' Haag-Kastler axioms but not the usual ones. This analysis is done by pure C*-algebraic means without employing any indefinite metric representations, and we obtain the same physical algebra and positive energy representation for it than by the usual means. The price for avoiding the indefinite metric, is the use of nonregular representations and complex valued test functions. We also exhibit the precise connection with the usual indefinite metric representation. We conclude the analysis by comparing the final physical algebra produced by a system of local constrainings with the one obtained from a single global constraining and also consider the issue of reduction by stages. For the usual spectral condition on the generators of the translation group, we also find a ``weak'' version, and show that the Gupta-Bleuler example satisfies it.