Thermal Green Functions in Coordinate Space for Massless Particles of any Spin

Abstract

The thermal Wightman functions for free, massless particles of spin 0, 1/2, 1, 3/2, and 2 are computed directly in coordinate space by solving the appropriate differential equation and imposing the Kubo-Martin-Schwinger condition. The solutions are valid for real, imaginary, or complex time. The Wightman functions for spin 1 gauge bosons and for spin 2 gravitons are directly related to the fundamental functions for spin 0. The Wightman functions for spin 3/2 gravitinos is directly related to that for spin 1/2 fermions. Calculations for spin 1, 3/2, and 2 are done in covariant gauges. In the deep space-like region the Wightman functions for bosons fall like $T/r$ whereas those for the fermions fall exponentially. In the deep time-like region all the Wightman functions fall exponentially.