Introduction, Motivation

This is a technique to solve the Liouville-von-Neumann Equation,

$$\frac{d}{dt}\chi(t)=-i[H,\chi(t)],\leadsto \chi(t)= e^{-iHt} \chi(t=0) e^{iHt},$$ (151)

for the time-dependent density matrix $\rho(t)$ of system-bath Hamiltonians

$$H \equiv H_S+H_B+ H_{SB},$$ (152)

cf. Eq. (7.4). It is mainly useful for cases where the system Hamiltonian $H_S$ referees to a single (or a few) degrees of freedom, coupled via $H_{SB}$ to a bath $H_B$ of many degrees of freedom. The technique is based on double path integrals. The original reference is R. P. Feynman, F. L. Vernon, Ann. Phys. (N. Y.) 24, 118 (1963).

One of the applications of influence functional theories is the systematic derivation of a semiclassical dynamics (Fokker-Planck equations, ...) from an exact quantum-mechanical theory: