Equations of motion as covariant Gauss law: The Maxwell–Chern–Simons case

Abstract

Time-independent gauge transformations are implemented in the canonical formalism by the Gauss law which is not covariant. The covariant form of Gauss law is conceptually important for studying the asymptotic properties of the gauge fields. For QED in 3 + 1 dimensions, we have developed a formalism for treating the equations of motion (EOM) themselves as constraints, that is, constraints on states using Peierls’ quantization.1 They generate spacetime dependent gauge transformations. We extend these results to the Maxwell–Chern–Simons (MCS) Lagrangian. The surprising result is that the covariant Gauss law commutes with all observables: the gauge invariance of the Lagrangian gets trivialized upon quantization. The calculations do not fix a gauge. We also consider a novel gauge condition on the test functions (not on quantum fields) which we name the “quasi-self-dual gauge” condition. It explicitly shows the mass spectrum of the theory. In this version, no freedom remains for the gauge transformations: EOM commute with all observables and are in the center of the algebra of observables.