Two-state quantum systems interacting with their environments: A functional integral approach

Two-state quantum systems which interact with their environments are investigated with the aim of determining the effects of the environment on the system's behavior. A new functional integral formalism, analogous to the Feynman path integral formalism but based on integration over complex Poisson processes, is developed in order to describe two-state systems. The measure used in this formalism is well defined and the resulting functional integration scheme is rigorous, unlike most others presently in use. The evolution operators for the isolated two-state system and the interacting two-state system are expressed in terms of such functional integrals; the integrand is typically an operator constructed from the relevant Hamiltonian. Within this formalism the evolution operator for the interacting two-state system is related to that for the isolated two-state system. Here the integrand of the former contains an additional operator factor, called the influence operator, which completely describes the effects of the environment on the system. A method for determining the influence operator from the Hamiltonians is given. This provides a well-defined analog of the Feynman-Vernon influence functional method for the two- state system. An influence functional provides a similar relationship for the corresponding density operators. An explicit calculation of the influence functional is given for the spin Boson model.