Quantum Langevin equation

The macroscopic description of a quantum particle with passive dissipation and moving in an arbitrary external potential is formulated in terms of the generalized Langevin equation. The coupling with the heat bath corresponds to two terms: a mean force characterized by a memory function μ(t) and an operator-valued random force. Explicit expressions are given for the correlation and commutator of the random force. The random force is never Markovian. It is shown that μ̃(z), the Fourier transform of the memory function, must be a positive real function, analytic in the upper half-plane and with Re[μ̃(ω+i$0^{+}$)] a positive distribution on the real axis. This form is then derived for the independent-oscillator model of a heat bath. It is shown that the most general quantum Langevin equation can be realized by this simple model. A critical comparison is made with a number of other models that have appeared in the literature.