Spectral-function sum rules and the pion electromagnetic mass difference at finite temperature

Spectral representations are derived for current correlation functions in relativistic many-body theory. Equal-time current commutation relations allow the derivation of spectral-function sum rules at finite temperature. It is shown that the spectral representation for the Schwinger term in the usual time-space current commutator has a finite temperature-dependent part. The mass relations between the vector and axial-vector meson masses derivable from Weinberg sum rules remain unaltered. These results are used to evaluate the temperature dependence of the pion electromagnetic mass difference $m_{{\pi }^{+}}-m_{{\pi }^0}$ in the soft-pion limit.