  This paper investigates a new formalism to describe real time evolution of
quantum systems at finite temperature. A time correlation function among
subsystems will be derived which allows for a probabilistic interpretation. Our
derivation is non-perturbative and fully quantized. Various numerical methods
used to compute the needed path integrals in complex time were tested and their
effectiveness was compared. For checking the formalism we used the harmonic
oscillator where the numerical results could be compared with exact solutions.
Interesting results were also obtained for a system that presents tunneling. A
ring of coupled oscillators was treated in order to try to check
selfconsistency in the thermodynamic limit. The short time distribution seems
to propagate causally in the relativistic case. Our formalism can be extended
easily to field theories where it remains to be seen if relevant models will be
computable.
