  Starting with a scalar field in a thermal bath and using the one loop quantum
correction potential, we rewrite the Klein-Gordon equation in its
thermodynamical representation and study the behavior of this scalar field due
to temperature variations in the equations of motion. We find the
generalization of a Gross-Pitaevskii like equation for a relativistic Bose gas
with finite temperature, the corresponding thermodynamic and viscosity
expressions, and an expression for the postulate of the first law of the
thermodynamics for this BECs. We also propose that the equations obtained might
help to explain at some level the phase transition of a Bose-Einstein
Condensate in terms of quantum field theory in a simple way.
