Generalized Wigner functions in curved spaces: A new approach

It is well known that, given a quantum field in Minkowski space, one can define Wigner functions $f_W^N$($x_1$,$p_1$,...,$x_N$,$p_N$) which (a) are convenient to analyze since, unlike the field itself, they are c-number quantities and (b) can be interpreted in a limited sense as ‘‘quantum distribution functions.’’ Recently, Winter and Calzetta, Habib and Hu have shown one way in which these flat-space Wigner functions can be generalized to a curved-space setting, deriving thereby approximate kinetic equations which make sense ‘‘quasilocally’’ for ‘‘short-wavelength modes.’’ This paper suggests a completely orthogonal approach for defining curved-space Wigner functions which generalizes instead an object such as the Fourier-transformed $f_W^1$(k,p), which is effectively a two-point function viewed in terms of the ‘‘natural’’ creation and annihilation operators $a^{\mathrm{°}}$(p-(1/2k) and a(p+(1/2k). The approach suggested here lacks the precise phase-space interpretation implicit in the approach of Winter or Calzetta, Habib, and Hu, but it is useful in that (a) it is geared to handle any ‘‘natural’’ mode decomposition, so that (b) it can facilitate exact calculations at least in certain limits, such as for a source-free linear field in a static spacetime.