Simple rules for discontinuities in finite-temperature field theory

Abstract

The discontinuity, or imaginary part, of the self-energy function at $T\ensuremath{\ne}=0$ is found to be of the form $\ensuremath{\Gamma}={\ensuremath{\Gamma}}_{d}\ensuremath{\mp}{\ensuremath{\Gamma}}_{i}$ for bosons and fermions, respectively. The generalized decay rate ${\ensuremath{\Gamma}}_{d}$ and inverse decay rate ${\ensuremath{\Gamma}}_{i}$ are recognizable as integrals over phase space of amplitudes squared, weighted with certain statistical factors that account for the possibility of particle absorption from the medium or particle emission into the medium. Nonequilibrium statistical mechanics shows that $\ensuremath{\Gamma}$ gives precisely the rate at which the single-particle distribution function approaches the equilibrium form.