When to carry out analytic continuation?
  This paper discusses the analytic continuation in the thermal field theory by using the theory of $\eta-\xi$ 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 $\eta-\xi$ spacetime plays important role in the discussion. 
Jian Zuo Yuan-Xing Gui