Temperature, periodicity and horizons
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We explain and explore the connections among the following propositions: (1) thermal equilibrium is characterized by the KMS condition, 〈At-iβB〉 = 〈BAt〉; (2) finite-temperature Green functions are periodic in imaginary time; (3) black holes are hot; and to an accelerating observer, empty space is hot. The KMS condition of quantum statistical mechanics is derived, with special attention to quantum field systems satisfying relativistic commutation relations and linear field equations. We display the analytic structure of the two-point function and show in what sense the KMS condition for such systems is a statement of periodicity. Then the application of these ideas to horizons in general-relativistic settings is reviewed. Other matters discussed include: the identification of the analytically continued two-point function with the Green function of an elliptic ("Euclideanized") operator; the analogous relation between a nonrelativistic propagator and a parabolic operator; the construction of thermal two-point functions as image sums; the (in)significance of time ordering; simplications of the KMS condition in the presence of discrete symmetries; the appearance of a "double" Fock space (artificially in general statistical mechanics, but naturally in space-times with horizons); and complications associated with the infrared behavior of the "particle" spectrum. © 1987.
author list (cited authors)
Fulling, S. A., & Ruijsenaars, S.