High-temperature expansion of the one-loop effective action induced by scalar and Dirac particles

The complete nonperturbative expressions for the high-temperature expansion of the one-loop effective action induced by the charged scalar and the charged Dirac particles both at zero and finite temperatures are derived with account for possible nontrivial boundary conditions. The background electromagnetic field is assumed to be stationary and such that the corresponding Klein-Gordon operator or the Dirac Hamiltonian are self-adjoint. The contributions of particles and antiparticles are obtained separately. The explicit expressions for the $C$-symmetric and the non $C$-symmetric vacuum energies of the Dirac fermions are derived. The leading corrections to the high-temperature expansion due to the nontrivial boundary conditions are explicitly found. The corrections to the logarithmic divergence of the effective action that come from the boundary conditions are derived. The high-temperature expansion of the naive one-loop effective action induced by charged fermions turns out to be divergent in the limit of a zero fermion mass.