Wigner functions and density matrices in curved spaces as computational tools

Abstract

This paper contrasts two alternative approaches to statistical quantum field theory in curved spacetimes, namely (1) a canonical Hamiltonian approach, in which the basic object is a density matrix ϱ characterizing the noncovariant, but globally defined, modes of the field; and (2) a Wigner function approach, in which the basic object is a Wigner function f defined quasilocally from the Hadamard, or correlation, function G 1( x 1, x 2). The key object is to isolate on the conceptual biases underlying each of these approaches and then to assess their utility and limitations in effecting concrete calculations. The following questions are therefore addressed and largely answered. What sorts of spacetimes (e.g., de Sitter or Friedmann-Robertson-Walker) are comparatively easy to consider? What sorts of objects (e.g., average fields or renormalized stress energies) are easy to compute approximately? What, if anything, can be computed exactly? What approximations are intrinsic to each approach or convenient as computational tools? What sorts of “field entropies” are natural to define?