Real-Time Thermal Green's Functions

We study perturbative techniques for evaluating real-time Green's functions of relativistic fields in thermal equilibrium. Our primary conclusion is that a consistent perturbation theory for real-time thermal Green's functions must include contributions from all segments of the time -integration contour C and three commonly used procedures for calculating real-time thermal Green's functions, which derive the same two-component finite-temperature Feynman rules, are incorrect. These three procedures are: the factorization prescription of Niemi and Semenoff; finite -temperature canonical quantization combined with a limiting procedure; and thermo field dynamics. We examine each of the three procedures and exhibit its flaw and we show why the two-component finite-temperature formalism these procedures yield is incorrect. After presenting the correct real-time thermal Feynman rules, we calculate the finite-temperature effective potential and the finite -temperature effective mass, both to one-loop order.