On Stochastic Quantization of Dissipative Systems at Finite Temperature

Abstract

Quantization of dissipative systems has been discussed, in particular, in connec­ tion with non-equilibrium quantum phenomena. To describe a dissipative system, one may naturally consider it as a part of a whole system. From a physical point of view, macroscopic dissipation results from the microscopic interaction between the system in question and its environment. Realizing this idea explicitly, Caldeira and Leggett (C-L) developed a model to describe dissipation. 1) They artificially introduced harmonic oscillators as the environment; which couples with the system in question linearly. By integrating out the environmental degrees of freedom in the C-L model, one can in principle derive an effective action for quantum effects with dissipation. In this short note, we derive a Langevin equation to describe a finite temperature dissipative system in the context of the stochastic quantization method (SQM).2) Since SQM can be formulated without an action principle (it is formulated in terms of a Langevin equation which includes a phenomenological equation of motion as the drift force), we naively expect that the artificial introduction of the environmental degrees of freedom is not necessary for the description of dissipative systems. In order to investigate whether this naive expectation is true or not, we derive a Langevin equation for finite temperature dissipative systems with higher derivative friction terms from the effective action which is derived by the C-L model approach at finite temperature. Then we show that the Langevin equation takes a form which includes a drift force slightly modified from the phenomenological equation of motion. In terms of the Langevin equation, SQM can describe the system, which interacts with its environment, by a phenomenological frictional coefficient. We also discuss the causal structure of the two-point function and show the consistency of our result with the fluctuation-dissipation theorem in higher derivative friction cases. Let us consider the system with dissipation for which the phenomenological equation of motion is given by