Analytic continuation of temperature-dependent diagrams and unitarity conditions at finite temperatures

A study is made of the analytic properties of the quantities defined in the temperature-dependent diagram technique and their analytic continuation from the region of discrete imaginary frequencies to the real axis. Two examples – the kinetic equation for a Fermi system and the equation for the self-energy part of the Green's function – are used to illustrate the analytic continuation of equations. The unitarity conditions for various verbs parts at finite temperatures are then discussed. It is shown that these conditions differ from the unitarity conditions at $T=0$ only by the form of the statistical weight of the intermediate states; this weight has the form of standard combinations of Fermi or Bose distribution function.