The Keldysh path integral can be thought of as a reformulation of the quantum optical master equation, which describes the markovian time evolution of the density operator of an open quantum system in the Schrödinger-picture.
The Keldysh-formalism allows one to calculate Green's functions. But fundamentally the Green's functions should be defined in the Heisenberg-picture, because they contain operators evaluated at different times.
But the formulation of the Heisenberg-picture for open quantum systems requires the introduction of noise terms into the equations of motion. Without the noise terms, the product rule does not hold for the time derivative.
My question is, that does anyone know any good references for linking these Heisenberg picture calculations based on quantum stochastic calculus to the Green's functions in the Keldysh-formalism?
If the enviromental variables are explicitly taken into account then the original system + the environment can be regarded as a larger closed system. One can then use the unitary time evolution operator to switch between the Schrödinger and the Heisenberg picture. My question was about linking these different formalisms at the level of the smaller system, where the enviromental degrees of freedom are not present. For example, the Keldysh path integral can be derived from the markovian master equation directly, without going back to the system + environment formulation.