  Conventional finite-temperature perturbation theory in which propagators have
poles at $k^{2}=m^{2}$ is shown to break down at the two-loop level for
self-interacting scalar fields. The breakdown is avoided by using free thermal
propagators that have poles at the same energy as the exact thermal propagator.
This quasiparticle energy ${\cal E}(\vec{k})$ is temperature-dependent,
complex, and gauge invariant. An operator theory containing two self-adjoint
scalar fields is presented in which all temperature dependence is incorporated
into the Hamiltonian. No thermal traces are required to compute thermal Green
functions. Choosing the spectrum of the unperturbed part of the Hamiltonian to
contain the exact quasiparticle energy ${\cal E}(\vec{k})$ produces a resummed
perturbation theory that has the correct poles and branch cuts. The location of
the poles and cuts is explained directly in terms of the spectrum of the
Hamiltonian.
