  The imaginary-time formulation is investigated in the field theory. The Dirac
and photon's Lagrangians are treated in the imaginary-time and space and are
invariant under scale transformations, which are motivated by the
Tolman-Ehrenfest relation and Wilson's approach in the renormalization group.
With the functional approaches, the propagators of leptons and photons can be
derived both for the real- and the imaginary-time. The theory is free of UV
divergence in the integration range of the imaginary-time, $(0,\beta\,]$, which
is required by the KMS condition. The results of one-loop radiative corrections
are proved to be consistent with those in QED. The renormalization group
equations with respect to vacuum's temperature give no Landau pole and show
identical renormalization coefficients as those from the $\overline{MS}$ scheme
of the renormalization in the limit of zero temperature.
