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Communications in Theoretical Physics

Closed Time-Path Green Functions by Stochastic Quantization

Chen Ren

Published Jun 1, 1990• 0 citations
Publisher

Abstract

The Langevin equation and the Gaussian average of the stochastic quantization of Parisi and Wu are continued on the closed time-path, at the same time, the new non-Gaussian source is added into it. After assuming some properties of new source, the closed time-path Green functions are computed by using the adapted Langevin equations. While they are transformed into the physical representation, some important characters of closed time-path Green function, i.e. the normalization and causality, are obtained also.

DOI
10.1088/0253-6102/13/4/441

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The Langevin equation and the Gaussian average of the stochastic quantization of Parisi and Wu are continued on the closed time-path, at the same time, the new non-Gaussian source is added into it. After assuming some properties of new source, the closed time-path Green functions are computed by using the adapted Langevin equations. While they are transformed into the physical representation, some important characters of closed time-path Green function, i.e. the normalization and causality, are obtained also.
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The closed time path Green's function (CTPGF) formalism is applied to the critical dynamics. The related results for the CTPGF approach are briefly reviewed. Three different forms of CTPGF's are defined, transformations from one to another form and other useful computation rules are given. The path integral presentation of the generating functional for CTPGF's is used to derive the Ward-Takahashi identities under both linear and nonlinear transformations of field variables. The generalized Langevin equations for the order parameters and conserved variables are derived from the vertex functional on the closed time path. The proper form of the equations for the conserved variables, including automatically the mode coupling terms, is determined according to the Ward-Takahashi identities and the linear response theory. All existing dynamic models are reobtained by assuming the corresponding symmetry properties for the system. The effective action for the order parameters is deduced by averaging over the random external field. The Lagrangian formulation of the statistical field theory is obtained if the random field one-loop approximation and the second-order approximation of order-parameter fluctuations on different time branches are both taken. The various possibilities of improving the current theory of critical dynamics within the framework of CTPGF's are discussed. The problem of renormalization for the finite-temperature field theory is considered. The whole theoretical framework is also applicable to systems near the stationary states far from equilibrium, whenever there exists an analog of the potential function (free energy).
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A theoretical scheme using closed time-path Green's functions is proposed to describe the quantum statistical properties of the order parameter in terms of a generating functional. The dynamic evolution, is generated by a driving source, while the statistical correlation by a fluctuation source. The statistical causality is shown to hold explicitly and to give rise to a number of important consequences. The problem of determining the quantum statistical properties for the order parameter is reduced to finding a solution of the functional equation for it.
The master equation for a linear open quantum system in a general environment is derived using a stochastic approach. This is an alternative derivation to that of Hu, Paz and Zhang, which was based on the direct computation of path integrals, or to that of Halliwell and Yu, based on the evolution of the Wigner function for a linear closed quantum system. We first show by using the influence functinal formalism that the reduced Wigner function for the open system coincides with a distribution function resulting from averaging both over the initial conditions and the stochastic source of a formal Langevin equation. The master equation for the reduced Wigner function can then be deduced as a Fokker-Planck equation obtained from the formal Langevin equation.