Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

A Projection Operator Method in the Heisenberg Picture

Kae Nemoto, Fumiaki Shibata

Published Nov 15, 1996• 4 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.65.3492

Related papers

A Systematic Projection Operator Formalism in Nonequilibrium Statistical Mechanics

Chikako Uchiyama, Fumiaki Shibata

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.

Journal of the Physical Society of Japan•Apr 15, 1996•5 citations

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

Tomoi Koide C

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.

arXiv:hep-th/0110224•Progress of Theoretical Physics•Mar 1, 2002•23 citations

Projection Operator Technique

Biman Bagchi

Overview Projection operator technique is simple in purpose. It allows us to separate the dynamical evolution of a quantity into two parts: known and unknown, the latter to be evaluated. However, this technique serves multiple purposes in TDSM. It is used to derive a reduced equation of motion for the relaxation of a dynamical variable. It is also used to derive expression for an experimentally observed TCF in terms of theoretically known TCFs. This technique was developed by Zwanzig, who was inspired by Kubo’s approach of “reduction in degrees of freedom” in an interacting many-body system. In the projection operator method, we divide the time evolution of a dynamical variable A ( t ) into two parts: (i) one through a relevant subspace that is closely related to A and varies on a similar time scale, and (ii) an orthogonal subspace, which could be fast and therefore can be averaged out to obtain a time-dependent friction kernel. The idea then is that the first contribution is known while the second may be described accurately by projecting it onto the relevant subspace [that is, (i)]. In the derivation of the reduced equation of motion, one starts with the Liouville equation and projects out irrelevant degrees of freedom to obtain, after specific algebraic manipulations, an equation of motion of the dynamical variable. The final relaxation equation has the form of the Langevin equation with microscopic expressions for the terms involved (frequency term, the friction, and the random force term). This final equation of motion is often termed relaxation equation or memory function equation. It finds wide use in the study of time-dependent statistical mechanics. In the present chapter we derive the memory function equation and discuss a few applications.

Chapman & Hall/CRC eBooks•Oct 2, 2023•0 citations

Projection Operator Techniques

Heinz‐Peter Breuer, Francesco Petruccione

Abstract This chapter is devoted to the foundations and to the basic mathematical structure of the non-Markovian quantum dynamics of open systems. It gives a survey of the Nakajima–Zwanzig projection operator methods with the help of which one derives so-called generalized master equations for the reduced system dynamics. In the non-Markovian regime, these master equations involve a retarded memory kernel, i.e. a time-convolution integral taken over the history of the reduced system. The chapter also describes an alternative method of particular relevance in many applications, which is based on a time-local quantum master equation, and which is known as the time-convolutionless projection operator method. This method serves as a starting point for a systematic expansion in terms of ordered cumulants and forms the basis for a non-Markovian stochastic unravelling of the reduced system dynamics.

•Jan 25, 2007•18 citations

Projection Operator Approach to Langevin Equations in 4 Theory

Tomoi Koide C, M. Maruyama, Fujio Takagi

We apply the projection operator method (POM) to ϕ4 theory and derive both quantum and semiclassical equations of motion for the soft modes. These equations have no time-convolution integral term, in sharp contrast with other well-known results obtained using the influence functional method (IFM) and the closed time path method (CTP). However, except for the fluctuation force field terms, these equations are similar to the corresponding equations obtained using IFM with the linear harmonic approximation, which was introduced to remove the time-convolution integral. The quantum equation of motion in POM can be regarded as a kind of quantum Langevin equation in which the fluctuation force field is given in terms of the operators of the hard modes. These operators are then replaced with c-numbers using a certain procedure to obtain a semiclassical Langevin equation. It is pointed out that there are significant differences between the fluctuation force fields introduced in this paper and those introduced in IFM. The arbitrariness of the definition of the fluctuation force field in IFM is also discussed.

arXiv:hep-ph/0102272•Progress of Theoretical Physics•May 1, 2002•4 citations

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

Tomoi Koide C, M. Maruyama

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.

arXiv:hep-th/0008182•Progress of Theoretical Physics•Sep 1, 2000•14 citations
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.
Overview Projection operator technique is simple in purpose. It allows us to separate the dynamical evolution of a quantity into two parts: known and unknown, the latter to be evaluated. However, this technique serves multiple purposes in TDSM. It is used to derive a reduced equation of motion for the relaxation of a dynamical variable. It is also used to derive expression for an experimentally observed TCF in terms of theoretically known TCFs. This technique was developed by Zwanzig, who was inspired by Kubo’s approach of “reduction in degrees of freedom” in an interacting many-body system. In the projection operator method, we divide the time evolution of a dynamical variable A ( t ) into two parts: (i) one through a relevant subspace that is closely related to A and varies on a similar time scale, and (ii) an orthogonal subspace, which could be fast and therefore can be averaged out to obtain a time-dependent friction kernel. The idea then is that the first contribution is known while the second may be described accurately by projecting it onto the relevant subspace [that is, (i)]. In the derivation of the reduced equation of motion, one starts with the Liouville equation and projects out irrelevant degrees of freedom to obtain, after specific algebraic manipulations, an equation of motion of the dynamical variable. The final relaxation equation has the form of the Langevin equation with microscopic expressions for the terms involved (frequency term, the friction, and the random force term). This final equation of motion is often termed relaxation equation or memory function equation. It finds wide use in the study of time-dependent statistical mechanics. In the present chapter we derive the memory function equation and discuss a few applications.
Abstract This chapter is devoted to the foundations and to the basic mathematical structure of the non-Markovian quantum dynamics of open systems. It gives a survey of the Nakajima–Zwanzig projection operator methods with the help of which one derives so-called generalized master equations for the reduced system dynamics. In the non-Markovian regime, these master equations involve a retarded memory kernel, i.e. a time-convolution integral taken over the history of the reduced system. The chapter also describes an alternative method of particular relevance in many applications, which is based on a time-local quantum master equation, and which is known as the time-convolutionless projection operator method. This method serves as a starting point for a systematic expansion in terms of ordered cumulants and forms the basis for a non-Markovian stochastic unravelling of the reduced system dynamics.
We apply the projection operator method (POM) to ϕ4 theory and derive both quantum and semiclassical equations of motion for the soft modes. These equations have no time-convolution integral term, in sharp contrast with other well-known results obtained using the influence functional method (IFM) and the closed time path method (CTP). However, except for the fluctuation force field terms, these equations are similar to the corresponding equations obtained using IFM with the linear harmonic approximation, which was introduced to remove the time-convolution integral. The quantum equation of motion in POM can be regarded as a kind of quantum Langevin equation in which the fluctuation force field is given in terms of the operators of the hard modes. These operators are then replaced with c-numbers using a certain procedure to obtain a semiclassical Langevin equation. It is pointed out that there are significant differences between the fluctuation force fields introduced in this paper and those introduced in IFM. The arbitrariness of the definition of the fluctuation force field in IFM is also discussed.
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.