Mathematical structure of the temporal gauge in quantum electrodynamics

Abstract

The conflict between Gauss’ law constraint and the existence of the propagator of the gauge fields, at the basis of contradictory proposals in the literature, is shown to lead to only two alternatives, both with peculiar features with respect to standard quantum field theory. In the positive (interacting) case, the Gauss’ law holds in operator form, but only the correlations of exponentials of gauge fields exist (nonregularity) and the space translations are not strongly continuous, so that their generators do not exist. Alternatively, a Källen–Lehmann representation of the two point function of Ai satisfying locality and invariance under space–time translations, rotations and parity is derived in terms of the two point function of Fμν; positivity is violated, the Gauss’ law does not hold, the energy spectrum is positive, but the relativistic spectral condition does not hold. In the free case, θ-vacua exist on the observable fields, but they do not have time translationally invariant extensions to the gauge fields; the vacuum is faithful on the longitudinal field algebra and defines a modular structure (even if the energy is positive). Functional integral representations are derived in both cases, with the alternative between ergodic measures on real random fields or complex Gaussian random fields.