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

arXiv:1712.00943 · Progress of Theoretical and Experimental Physics

Secular terms in Dyson series to all orders of perturbation

Satoshi Iso, Hikaru Ohta, Takao Suyama

Published Jun 19, 2018• 6 citations
arXivHTMLPDF

Abstract

In classical and quantum systems, perturbation of an evolution equation is often invalidated by secular terms which diverge at late times. The diverging behavior of evolution can be remedied by various techniques of resummation such as the renormalization group or multi-scale analysis. In this paper, we prove that, in a generic quantum-mechanical system, secular terms can be systematically removed to all orders in the Dyson series by the method of improved (renormalized) perturbation. A recurrence relation to provide an explicit method to remove the secular terms is given. As a by-product, we give a simple method to obtain energy eigenvalues and decay rates to all orders of perturbation.

DOI
10.1093/ptep/pty079

Related papers

Renormalization-Group Resummation of a Divergent Series of the Perturbative Wave Functions of Quantum Systems

Teiji Kunihiro

The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x → ∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.

arXiv:hep-th/9801196•Progress of Theoretical Physics Supplement•Jan 1, 1998•17 citations

Theory of quantum resonance: A renormalization-group approach

Marco Frasca

The standard perturbation theory for time-dependent problems in quantum mechanics is reconsidered through renormalization-group methods. This approach justifies a posteriori the theory of quantum resonance given by the multiple-scale analysis of the perturbation series applied to the coupled equations of wave-function amplitudes. The resonance equations for the leading-order amplitude probabilties are then obtained up to second order and a general method for the algorithmic computation of higher-order terms of the perturbation series is given. The three-level model in a monochromatic wave with two resonant states is discussed in the light of the present results.

Physical Review A•Jul 1, 1998•17 citations

An integral-free representation of the Dyson series using divided differences

Amir Kalev, Itay Hen

The Dyson series is an infinite sum of multi-dimensional time-ordered integrals, which serves as a formal representation of the quantum time-evolution operator in the interaction-picture. Using the mathematical tool of divided differences, we introduce an alternative representation for the series that is entirely free from both time ordering and integrals. In this new formalism, the Dyson expansion is given as a sum of efficiently-computable divided differences of the exponential function, considerably simplifying the calculation of the Dyson expansion terms, while also allowing for time-dependent perturbation calculations to be performed directly in the Schrödinger-picture. We showcase the utility of this novel representation by studying a number of use cases. We also discuss several immediate applications.

arXiv:2010.09888•New Journal of Physics•Oct 1, 2021•8 citations

Aspects of Time-Dependent Perturbation Theory

Peter W. Langhoff, Saul T. Epstein, Martin Karplus

The Dirac variation-of-constants method has long provided a basis for perturbative solution of the time-dependent Schr\"odinger equation. In spite of its widespread utilization, certain aspects of the method have been discussed only superficially and remain somewhat obscure. The present review attempts to clarify some of these points, particularly those related to secular and normalization terms. Secular terms arise from an over-all time-dependent phase in the wave function, while normalization terms preserve the norm of the wave function. A proper treatment of the secular terms is essential in the presence of a physically significant level shift that can produce secular divergences in the time-dependent perturbation functions. The normalization terms are always important, although the formulation of a simple method for including them is of greatest utility in applications requiring higher-order perturbation theory (e.g., nonlinear optical phenomena), where difficulties have arisen in previous treatments. Although the Dirac perturbation technique includes the correct secular and normalization terms when properly executed, it is convenient to reinterpret the perturbation problem so that the secular and normalization terms can be factored from the wave function to all orders. It is shown that an appropriate over-all multiplicative, time-dependent normalization and phase factor can be obtained, and that it is simply the amplitude for finding the system in the unperturbed eigenstate at any time $t$. The regular part of the wave function remaining after this factorization provides a complete description of the physical properties of the system of interest and determines the over-all normalization and phase, as well. Most important, the regular function and its perturbation expansion satisfy equations which are more convenient for computational applications than are the customary Dirac equations, and, in contrast to the latter, they reduce directly to the familiar time-independent perturbation equations in the static limit. To illustrate the general development, the model problem of a linearly perturbed harmonic oscillator and the static, harmonic, and electromagnetic perturbations of arbitrary quantum-mechanical systems are treated explicitly. In the case of an adiabatically applied static perturbation, the familiar adiabatic theorem is recovered with the over-all phase factor giving the perturbed eigenvalue, while in the case of an harmonic perturbation, the overall phase factor obtained includes the system level shift appropriate for a quasiperiodic state. For an electromagnetic perturbation, compact expressions are obtained for various nonlinear optical susceptibilities in forms suitable for computations. Time-dependent Hartree-Fock approximations are treated explicity to demonstrate that difficulties can arise when normalization and secular terms are not extracted prior to application of the perturbation formalism. Connection is also made with other methods which can be employed to eliminate secular and normalization terms from the wave function; these include a projection procedure and multiple-time-scales perturbation theory. The elimination of secular divergences from the perturbation functions is shown to be important for the construction of a valid Fourier transform. Secular and normalization terms also arise in connection with variational principles for the time-dependent Schr\"odinger equation. By employing the Frenkel variational principle and an ansatz for the total wave function that explicitly isolates the secular and normalization terms, a computationally convenient variational functional is obtained. This form of the Frenkel principle provides a bound to the system level shift induced by an oscillatory perturbation and is equivalent to the Ritz variational principle in the static limit. Explicit expressions for the variational functional in the Hartree-Fock approximations are derived in forms suitable for computational applications to the interactions of radiation and matter.

Reviews of Modern Physics•Jul 1, 1972•619 citations

Real time evolution in quantum many-body systems with unitary perturbation theory

Andreas Hackl, Stefan K. Kehrein

We develop an analytical method for solving real time evolution problems of quantum many-body systems. Our approach is a direct generalization of the well-known canonical perturbation theory for classical systems. Similar to canonical perturbation theory, secular terms are avoided in a systematic expansion and one obtains stable long-time behavior. These general ideas are illustrated by applying them to the spin-boson model and studying its nonequilibrium spin dynamics.

arXiv:0709.2100•Physical Review B•Sep 17, 2008•66 citations

Perturbative dynamics of open quantum systems by renormalization group method

Shingo Kukita

We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. We compare this dynamics with the exact dynamics in an exactly solvable spin-boson model.

arXiv:1705.10522•Physical review. E•Oct 9, 2017•1 citation
In classical and quantum systems, perturbation of an evolution equation is often invalidated by secular terms which diverge at late times. The diverging behavior of evolution can be remedied by various techniques of resummation such as the renormalization group or multi-scale analysis. In this paper, we prove that, in a generic quantum-mechanical system, secular terms can be systematically removed to all orders in the Dyson series by the method of improved (renormalized) perturbation. A recurrence relation to provide an explicit method to remove the secular terms is given. As a by-product, we give a simple method to obtain energy eigenvalues and decay rates to all orders of perturbation.
The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x → ∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.
The standard perturbation theory for time-dependent problems in quantum mechanics is reconsidered through renormalization-group methods. This approach justifies a posteriori the theory of quantum resonance given by the multiple-scale analysis of the perturbation series applied to the coupled equations of wave-function amplitudes. The resonance equations for the leading-order amplitude probabilties are then obtained up to second order and a general method for the algorithmic computation of higher-order terms of the perturbation series is given. The three-level model in a monochromatic wave with two resonant states is discussed in the light of the present results.
The Dyson series is an infinite sum of multi-dimensional time-ordered integrals, which serves as a formal representation of the quantum time-evolution operator in the interaction-picture. Using the mathematical tool of divided differences, we introduce an alternative representation for the series that is entirely free from both time ordering and integrals. In this new formalism, the Dyson expansion is given as a sum of efficiently-computable divided differences of the exponential function, considerably simplifying the calculation of the Dyson expansion terms, while also allowing for time-dependent perturbation calculations to be performed directly in the Schrödinger-picture. We showcase the utility of this novel representation by studying a number of use cases. We also discuss several immediate applications.
The Dirac variation-of-constants method has long provided a basis for perturbative solution of the time-dependent Schr\"odinger equation. In spite of its widespread utilization, certain aspects of the method have been discussed only superficially and remain somewhat obscure. The present review attempts to clarify some of these points, particularly those related to secular and normalization terms. Secular terms arise from an over-all time-dependent phase in the wave function, while normalization terms preserve the norm of the wave function. A proper treatment of the secular terms is essential in the presence of a physically significant level shift that can produce secular divergences in the time-dependent perturbation functions. The normalization terms are always important, although the formulation of a simple method for including them is of greatest utility in applications requiring higher-order perturbation theory (e.g., nonlinear optical phenomena), where difficulties have arisen in previous treatments. Although the Dirac perturbation technique includes the correct secular and normalization terms when properly executed, it is convenient to reinterpret the perturbation problem so that the secular and normalization terms can be factored from the wave function to all orders. It is shown that an appropriate over-all multiplicative, time-dependent normalization and phase factor can be obtained, and that it is simply the amplitude for finding the system in the unperturbed eigenstate at any time ttt. The regular part of the wave function remaining after this factorization provides a complete description of the physical properties of the system of interest and determines the over-all normalization and phase, as well. Most important, the regular function and its perturbation expansion satisfy equations which are more convenient for computational applications than are the customary Dirac equations, and, in contrast to the latter, they reduce directly to the familiar time-independent perturbation equations in the static limit. To illustrate the general development, the model problem of a linearly perturbed harmonic oscillator and the static, harmonic, and electromagnetic perturbations of arbitrary quantum-mechanical systems are treated explicitly. In the case of an adiabatically applied static perturbation, the familiar adiabatic theorem is recovered with the over-all phase factor giving the perturbed eigenvalue, while in the case of an harmonic perturbation, the overall phase factor obtained includes the system level shift appropriate for a quasiperiodic state. For an electromagnetic perturbation, compact expressions are obtained for various nonlinear optical susceptibilities in forms suitable for computations. Time-dependent Hartree-Fock approximations are treated explicity to demonstrate that difficulties can arise when normalization and secular terms are not extracted prior to application of the perturbation formalism. Connection is also made with other methods which can be employed to eliminate secular and normalization terms from the wave function; these include a projection procedure and multiple-time-scales perturbation theory. The elimination of secular divergences from the perturbation functions is shown to be important for the construction of a valid Fourier transform. Secular and normalization terms also arise in connection with variational principles for the time-dependent Schr\"odinger equation. By employing the Frenkel variational principle and an ansatz for the total wave function that explicitly isolates the secular and normalization terms, a computationally convenient variational functional is obtained. This form of the Frenkel principle provides a bound to the system level shift induced by an oscillatory perturbation and is equivalent to the Ritz variational principle in the static limit. Explicit expressions for the variational functional in the Hartree-Fock approximations are derived in forms suitable for computational applications to the interactions of radiation and matter.
We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. We compare this dynamics with the exact dynamics in an exactly solvable spin-boson model.