New time contour for equilibrium real-time thermal field theories.

A new time contour is used to derive real-time thermal field theory. Unlike previous path integral approaches, no contributions to the generating functional are dropped. PACS number: 11.10-z In the path integral approach to equilibrium thermal field theories, one is calculating thermal expectation values, typically ΓC(τ1, τ2, ..., τN) = Tr{eΦ−βHTCφ1(τ1)φ2(τ2)...φN(τN )}/Tr{eΦ−βH}. (1) Here Tc indicates that the fields are path-ordered with respect to the order along a path, C, of their complex time arguments {τ} [1, 2] with appropriate sign changes when fermionic fields are involved . Throughout we shall suppress spinor and other indices as well as dependence on spatial variables where necessary. The results are not altered by their inclusion. The eΦ−βH factor is used to shift the bra states in the trace by −ıβ in time so that all times involved lie on a path, C, which must run in the complex time plane so that it ends −ıβ below its starting point. ∗E-mail: [email protected] Another limitation on C that is often imposed comes from studying the behaviour of the Green functions. Inserting complete sets of energy eigenstates into the thermal Wightman function, say Γa1...aN , where the N fields are in a fixed order, we have Γa1...aN (τ1, ..., τN) = Tr{eΦ−βHφa1(τa1)φa2(τa2)...φaN (τaN )}/Tr{eΦ−βH}. = (−1)Φp ∑ m1,m2,...,mN [exp{−ıEm1(b− τa1 + τaN )}.