Quantum Brownian Motion: The Functional Integral Approach

H. Grabert, P. Schramm und G.-L. Ingold

Phys. Rep. 168, 115–207 (1988) DOI: 10.1016/0370-1573(88)90023-3

The quantum mechanical dynamics of a particle coupled to a heat bath is treated by functional integral methods and a generalization of the Feynman-Vernon influence functional is derived. The extended theory describes the time evolution of nonfactorizing initial states and of equilibrium correlation functions. The theory is illuminated through exactly solvable models.


  • Introduction
Part I. General theory
  • Microscopic model and preparation of the initial state
    • The model Hamiltonian
    • Initial states and preparation function
  • Functional integral representation of the density matrix and elimination of the environment
    • Euclidean functional integral
    • Real time functional integral
    • Integration over the environmental coordinates and influence functional
    • Reduced dynamics and propagating function
  • Minimal action paths and damping kernel
    • The potential renormalization
    • Minimal action paths
    • Formulation of the theory in terms of the damping kernel
Part II. Damped harmonic oscillator
  • Time evolution of a damped harmonic oscillator
    • Extremal imaginary time path and reduced equilibrium density matrix
    • Extremal real time paths and minimal effective action
  • Equilibrium correlation functions and response function
    • Linear response of the coordinate to an applied force
    • Coordinate autocorrelation function
    • The fluctuation dissipation theorem
    • Linear response of the momentum to an applied force
    • Further correlation functions
    • Variances
    • Propagating function
    • Effect of initial correlations
  • Ohmic dissipation
  • Relaxation of nonequilibrium initial states
    • Approach to equilibrium
    • Relaxation of expectation values
    • Relaxation of factorizing initial states
    • Coherent and squeezed states
Part III. Free Brownian motion
  • Time evolution of a damped free particle
    • The displacement correlation function
    • The propagating function
  • Ohmic dissipation
  • Frequency-dependent damping
    • Spectral density and damping coefficient
    • The antisymmetrized displacement correlation function
    • The symmetrized displacement correlation function
  • Relaxation of nonequilibrium inital states
    • Time evolution of a Gaussian density matrix
    • Asymptotic spreading of the state
    • Long time behaviour for arbitrary initial states at finite temperatures
    • Long time behaviour for arbitrary initial states at zero temperature
  • Elimination of an environmental oscillator
  • Determination of the auxiliary function Ψ(t,t')
  • Microscopic origin of the inhomogeneity in the equation of motion for <q>t