Quantum noise and quantum Langevin equations

The quantum Langevin equation of Ford, Kac, and Mazur is rederived and shown to be equivalent to an adjoint equation. This latter can be handled by means of van Kampen's cumulant expansion to yield derivations of the quaiclassical Langevin equation, stochastic electrodynamics, quantum optical, and quantum Brownian motion master equations (under appropriate conditions). The result of Benguria and Kac—that the quantum Langevin equation yields the Boltzmann distribution over energy levels in thermodynamic equilibrium—is also verified.