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arXiv:hep-th/0110224 · Progress of Theoretical Physics

Derivation of Transport Equations Using the Time-Dependent Projection Operator Method

Tomoi Koide C

Published Mar 1, 2002• 23 citations
arXivHTMLPDF

Abstract

We develop a formalism to carry out coarse-grainings in quantum field theoretical systems by using a time-dependent projection operator in the Heisenberg picture. A systematic perturbative expansion with respect to the interaction part of the Hamiltonian is given, and a Langevin-type equation without a time-convolution integral term is obtained. This method is applied to a quantum field theoretical model, and coupled transport equations are derived.

DOI
10.1143/ptp.107.525

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We develop a formalism to carry out coarse-grainings in quantum field theoretical systems by using a time-dependent projection operator in the Heisenberg picture. A systematic perturbative expansion with respect to the interaction part of the Hamiltonian is given, and a Langevin-type equation without a time-convolution integral term is obtained. This method is applied to a quantum field theoretical model, and coupled transport equations are derived.
A projection operator method in Heisenberg picture is developed. Corresponding to the equations for the reduced density matrix, there are two types of Langevin-like equations; time convolution (TC) and time convolutionless (TCL) equations. When the whole system is composed of relevant system and irrelevant system, unnecessary information is eliminated by projection operator. Explicit expansion formulae are obtained for the both types of equations. Relaxation operators and fluctuating forces are expressed in terms of certain cumulants. The method is illustrated by simple examples.
A systematic damping theoretical formalism is presented to give basic equations of statistical physics. Our formalism naturally yields the so-called time convolution (TC) and time convolutionless (TCL) equations both for a density matrix and for dynamical variables. The method is explicitly used to obtain equations in the Heisenberg picture (equations of motion for dynamical variables). In contrast to the existing derivation of equations of motion, the present formalism gives a systematic method for deriving these equations without specifying the projection operator. Moreover, the formalism is valid even for time-dependent Hamiltonian systems. A simple example is given to illustrate our formalism.
We develop a systematic method of operator perturbation theory in the Heisenberg picture, which is the formal analog of time-independent stationary-state perturbation theory in the Schr\"odinger picture. However, we use the eigenoperators and operator eigenfrequencies to calculate the time evolution of the dynamical variables of interest. The spectrum of the unperturbed Hamiltonian is assumed to be discrete. The method is especially well-suited for treating systems where the unperturbed Hamiltonian represents noninteracting fermions and/or bosons, and therefore, we illustrate the method using the exactly solvable model of Jaynes and Cummings: a single two-level atom interacting with a single quantized field mode. The time evolution of these operator solutions is shown to be unitary order by order in powers of the interaction strength. We find that the time evolution of exact operator solutions is well approximated for significantly longer times by the time evolution of this method's operator solutions than by the time evolution of the same operator solutions calculated using the Dyson expansion. We also demonstrate that time-dependent operator solutions are very convenient for computing quantities such as correlation functions.
The time evolution of quantum many body systems in the relativistic quantum field theory is studied. We use Shibata-Hashitsume’s projection operator method which gives a systematic expansion of the Heisenberg equation of motion in terms of the interaction Hamiltonian. We calculate the time evolution of expectation values of creation and annihilation operators in a renormalizable model with unstable particles in a vacuum and in a medium. It is shown how to renormalize the ultraviolet divergences appearing in the equation of motion and calculate the medium effects on the time evolution. The mass and the wavefunction renormalizations are carried out in the lowest order. We find that the results coincide with the on-mass-shell values of the self-energy and its derivative which are calculated by the usual Feynman rule. We also discuss the initial divergence which remains at the initial time after mass renormalization. It is suggested that the projection operator method can be applied to the relativistic quantum field theory.
We derive a new expansion of the Heisenberg equation of motion based on the projection operator method proposed by Shibata, Hashitsume and Shingū. In their projection operator method, a certain restriction is imposed on the initial state. As a result, one cannot prepare arbitrary initial states, for example a coherent state, to calculate the time development of quantum systems. In this paper, we generalize the projection operator method by relaxing this restriction. We explain our method in the case of a Hamiltonian both with and without explicit time dependence. Furthermore, we apply it to an exactly solvable model called the damped harmonic oscillator model and confirm the validity of our method.
We systematically derive the quantum generalized nonlinear Langevin equation using Morozov's projection operator method. This approach extends the linear Mori-Zwanzig projection operator technique, allowing for the inclusion of nonlinear interactions among macroscopic modes. Additionally, we obtain the quantum generalized Fokker-Planck equation within the Heisenberg picture, which is consistent with Morozov's original formulation. These equations are fundamentally significant in nonequilibrium statistical physics, particularly in scenarios characterized by enhanced fluctuations, such as anomalous transport phenomena near critical points. The quantum nature of the derived generalized Langevin and Fokker-Planck equations is anticipated to provide a more detailed description than their classical equivalents. Specifically, the noise kernel in the quantum generalized Langevin equation is multiplicative, which broadens the applicability beyond Gaussian approximations. Given specific interactions, these equations are expected to be instrumental in investigating critical transport phenomena.