Mathematical structure of the temporal gauge

TL;DR

This paper analyzes the mathematical structure of the temporal gauge in QED, revealing how different formulations impact symmetries, vacuum states, and the existence of certain operators, with implications for quantum field theory consistency.

Contribution

It provides a detailed comparison of positive and indefinite formulations of the temporal gauge, highlighting their mathematical and physical differences and implications.

Findings

Positive case exhibits non-regularity and vacuum degeneracy.

Indefinite case maintains gauge-invariant theta-vacua with a KMS structure.

Functional integral representations differ based on the formulation.

Abstract

The mathematical structure of the temporal gauge of QED is critically examined in both the alternative formulations characterized by either positivity or regularity of the Weyl algebra. The conflict between time translation invariance and Gauss law constraint is shown to lead to peculiar features. In the positive case only the correlations of exponentials of fields exist (non regularity), the space translations are not strongly continuous, so that their generators do not exist, a theta vacuum degeneracy occurs, associated to a spontaneous symmetry breaking. In the indefinite case the spectral condition only holds in terms of positivity of the energy, gauge invariant theta-vacua exist on the observables, with no extension to time translation invariant states on the field algebra, the vacuum is faithful on the longitudinal algebra and a KMS structure emerges. Functional integral representations are derived in both cases, with the alternative between ergodic measures on real random fields or complex Gaussian random fields.