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Feynman's functional formulation of statistical mechanics is used to study general-relativistic quantum field theories at finite temperature. The techniques are then applied to gauge theories. The partition function Tre^- is discovered to be a gauge-dependent quantity which is meaningless in most gauges. Instead, we define a physically meaningful partition function which is gauge-invariant and only equal to Tre^- in certain "physical" gauges. Feynman rules for this partition function and for finite-temperature Green's functions are derived for a general gauge.
Claude W. Bernard (Sat,) studied this question.