When to carry out analytic continuation?

Abstract

This paper discusses the analytic continuation in the thermal field theory by using the theory of $η-ξ$ spacetime. Taking a simple model as example, the $2\times 2$ matrix real-time propagator is solved from the equation obtained through continuation of the equation for the imaginary-time propagator. The geometry of the $η-ξ$ spacetime plays important role in the discussion.