Nonequilibrium statistical quantum field theory for open systems

Recently, a number of authors have begun to study the evolution of quantum fields in the early Universe characterized by a time-dependent density matrix ${\rho }_S$(t). All of this work is predicated on the assumption that one’s ‘‘subsystem’’ of interest is in some sense ‘‘decoupled’’ from the rest of the Universe, so that ${\rho }_S$ satisfies a Liouville–von Neumann equation which implies, e.g., an isentropic evolution. Starting from ‘‘first principles,’’ i.e., the Schrödinger equation for the totality of ‘‘subsystem’’ plus surroundings (‘‘bath’’), it is shown here how such a picture can be derived as the limiting case of a more complete statistical description. Quite generally, one finds that ${\rho }_S$ and the ‘‘bath’’ density matrix ${\rho }_B$ satisfy coupled nonlinear generalizations of the Liouville–von Neumann equation and evidence a nonisentropic evolution. However, in a Vlasov-type approximation, ${\rho }_S$ and ${\rho }_B$ satisfy instead much simpler bilinear equations which imply an isentropic evolution. And finally, in the limit that the ‘‘back reaction’’ of ${\rho }_S$ on ${\rho }_B$ can be neglected in computing the evolution of ${\rho }_B$, one recovers a true Liouville–von Neumann equation for the evolution of ${\rho }_S$ in an external field.