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Four-Point Spectral Functions and Ward Identities in Hot QED

We derive spectral representations for the different components of the 4-point function at finite temperature in the real time formalism in terms of five real spectral densities. We explicitly calculate all these functions in QED in the hard thermal loop approximation. The Ward identities obeyed by the 1-loop 3- and 4-point functions in real time and their spectral functions are derived. We compare our results with those derived previously in the imaginary-time formalism for retarded functions in hot QCD, and we discuss the generalization of our results to non-equilibrium situations.