The Schwinger function, confinement and positivity violation in pure gauge QED

The lattice regularized pure gauge compact U(1) theory is an ideal laboratory to explore how confinement is realized as its phase diagram has a confined and a deconfined phase that depends on the value of the coupling constant, i.e. on $\beta$. Herein, the connection between confinement and positivity violation through the Schwinger function associated with the Landau gauge photon propagator is investigated. The simulations reported show a very clear link between the realization of confinement and positivity violation of the photon Schwinger function and, therefore, of the photon K\"all\'en-Lehmann spectral density. Furthermore, a mass scale that characterizes the decay of the Schwinger function for small time separations is computed and used to distinguish the two phases of the theory.