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arXiv:hep-th/0008182 · Progress of Theoretical Physics

A New Expansion of the Heisenberg Equation of Motion with Projection Operator

Tomoi Koide C, M. Maruyama

Published Sep 1, 2000• 14 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1143/ptp.104.575

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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.
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.
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.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.
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.
The quantum theory of a damped harmonic oscillator is developed by considering an exactly solvable model, previously discussed by Unruh and Zurek [Phys. Rev. D 40, 1071 (1989)]. A one-dimensional harmonic oscillator is coupled to a scalar field. The field is assumed to be initially in thermal equilibrium. The Heisenberg equations of motion are solved without approximation, and the first and second moments, 〈q(t)〉, 〈p(t)〉, 〈q2{\mathit{q}}^{2}q2(t)〉, 〈p2{\mathit{p}}^{2}p2(t)〉, and 〈q(t)p(t)+p(t)q(t)〉 of the coordinate and momentum of the oscillator are calculated. The high-temperature, zero-temperature, free-particle, and weak-damping limits are discussed. A Wigner function of Gaussian form is constructed. The density-matrix equation derived by Louisell $[---Quantum Statistical Properties of Radiation (Wiley, New York, 1973)] is examined and shown to have a solution in agreement with our weak-damping limit.