Zeta function regularization of high-temperature expansions in field theory

Abstract

Exact high-temperature expansions of the thermodynamic potentials Ω B, F (β, M, μ) for the relativistic Bose and Fermi gases with temperature T = 1/ β, mass M and chemical potential μ are derived. (Calculations are done for arbitrary space-time dimension.) The calculations encounter various divergent series which are easily regulated using the Riemann ζ-fnction and related functions. The high- T expansions for the Fermi gas are given for the first time. Our results for the Bose gas agree exactly with the high- T series for Ω B obtained recently by Haber and Weldon (except for a one-term discrepancy arising from the branch points in this function). As will be reported more fully elsewhere, our method also solves the more general problem of calculating the high- T. series for the (one-loop) thermodynamic potentials Ω(β, M, μ, A 0) encountered in gauge theories, which depend on additional vacuum parameters (i.e. the diagonal elements of the constant euclidean gauge potential (A 0, 0) . Beyond this, innumerable mathematical identities of a particular type (useful in contexts other than temperature field theory) can easily be obtained by the ζ-function method.