  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.
