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See detailApproximate spectral functions in thermal field theory
Henning, P. A.; Polyatchenko, E.; Schilling, Tanja UL et al

in Physical Review. D, Particles and Fields (1996), 54(5239), 1-5

Causality requires that the ~anti!commutator of two interacting field operators vanishes for spacelike coordinate differences. This implies that the Fourier transform of the spectral function of this ... [more ▼]

Causality requires that the ~anti!commutator of two interacting field operators vanishes for spacelike coordinate differences. This implies that the Fourier transform of the spectral function of this quantum field should vanish in the spacelike domain. We find that this requirement imposes some constraints on the use of resummed propagators in high temperature gauge theory. [less ▲]

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