  We prove in full generality the thermal operator representation for Matsubara
sums in a relativistic field theory of scalar and fermionic particles. It
states that the full result of performing the Matsubara sum associated to any
given Feynman graph, in the imaginary-time formalism of finite-temperature
field theory, can be directly obtained from its corresponding zero-temperature
energy integral, by means of a simple linear operator, which is independent of
the external Euclidean energies and whose form depends solely on the topology
of the graph.
