  We show that the one-loop effective action at finite temperature for a scalar
field with quartic interaction has the same renormalized expression as at zero
temperature if written in terms of a certain classical field $\phi_c$, and if
we trade free propagators at zero temperature for their finite-temperature
counterparts. The result follows if we write the partition function as an
integral over field eigenstates (boundary fields) of the density matrix element
in the functional Schr\"{o}dinger field-representation, and perform a
semiclassical expansion in two steps: first, we integrate around the
saddle-point for fixed boundary fields, which is the classical field $\phi_c$,
a functional of the boundary fields; then, we perform a saddle-point
integration over the boundary fields, whose correlations characterize the
thermal properties of the system. This procedure provides a
dimensionally-reduced effective theory for the thermal system. We calculate the
two-point correlation as an example.
