Complex-time path integrals beyond the stationary-phase approximation: Decay of metastable states and quantum statistical metastability

The complex-time path-integral expression for $\mathrm{Tr}{\mathrm{}(E-H)\mathrm{}}^{-1\mathrm{}}$ is reexamined and applied to barrier-penetration effects in quantum mechanics and field theory. Shortcomings of the conventional method for low-lying states are analyzed and resolved. This is achieved by an appropriate treatment of quasisymmetry modes occurring in the path integral and by computing certain contributions to the Fourier transform of the path integral without a stationary-phase approximation. We obtain agreement with the results of the instanton method for the ground state. For growing quantum numbers our results smoothly approach those of the standard WKB approximation. We also consider a scalar field theory with a false vacuum and compute the decay rates ${\Gamma }_n$ of $n$-meson states built thereon. Final emphasis is placed on the quantum-statistical metastability of a grand canonical ensemble of these mesons at temperature ${\beta }^{-1\mathrm{}}$.