Non-perturbative constraints on perturbation theory at finite temperature

It has long been understood that the inclusion of temperature in the perturbative treatment of quantum field theories leads to complications that are not present at zero temperature. In these proceedings we report on the non-perturbative obstructions that arise, and how these lead to deviations in the predictions of lattice scalar correlation functions in massive ${\phi }^4$ theory. Using the known non-perturbative spectral constraints satisfied by finite-temperature correlation functions we outline why the presence of distinct particle-like excitations could provide a resolution to these issues.