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 de Physique Lettres

Path integrals and time-dependent mean-field theories

Jean-Paul Blaizot, Henri Orland

Published Jan 1, 1980• 20 citations
Publisher

Abstract

We present a path integral formulation of the many-body problem, using a continuous and overcomplete set of vectors in the Hilbert space. The saddle-point approximation for the calculation of the functional integral leads to classical equation of motion which can be quantized in a well prescribed way. The time dependent Hartree-Fock equations are an example of such classical equations. They are obtained by integrating over an overcomplete set of Slater determinant.

DOI
10.1051/jphyslet:0198000410305300

Related papers

Path integrals for the nuclear many-body problem

Jean-Paul Blaizot, Henri Orland

We present a general method for constructing path intergrals for the nuclear many-body problem. This method uses continuous and overcomplete sets of vectors in the Hilbert space. The state labels play the role of classical coordinates which are quantized as bosons. The equations of motion for the classical coordinates are obtained by calculating the functional integral in the saddle point approximation. In the particular case where the over-complete set considered is the set of all Slater determinants, the classical equations of motion are the time-dependent Hartree-Fock equations. The functional integral provides a way of requantizing these classical equations. This quantization involves boson degrees of freedom and is in some cases very similar to the method of boson expansion. It is shown that the functional integral formalism provides a unifying framework to describe various approaches to the nuclear many-body problem.NUCLEAR STRUCTURE Functional integrals on continuous over-complete sets. Time-dependent Hartree and Hartree-Fock theories. Boson representations for fermion systems.

Physical Review C•Oct 1, 1981•80 citations

Path integral approach for many-fermion systems and an extension of the time-dependent Hartree-Fock method

Kazunari Kaneko

The quantization of an extended time-dependent Hartree-Fock equation is performed by using the path integral formalism. The path integral form of time evolution operator between initial and final states is obtained with the use of the completeness relation of the fermion coherent states. It is shown that the extended time-dependent Hartree-Fock equation is naturally derived as a classical limit of the path integral.

Physical Review C•Aug 1, 1984•6 citations

Time-dependent mean-field theory and quantized bound states

Shimon Levit, John W. Negele, Zvi Paltiel

A theory is presented in which approximations to quantum observables are obtained by applying the stationary-phase approximation to an exact functional integral representation of the many-body evolution operator. The requirement that the leading correction to the stationary-phase approximation vanish leads to a time-dependent Hartree-Fock mean field. Application of the theory to the Fourier transform of the trace of the evolution operator and the study of its poles yields quantized bound states with large amplitude. The theory is shown to reproduce the familiar static Hartree-Fock and random-phase approximations in the appropriate limits and yields an excellent approximation to the entire spectrum of the exactly solvable Lipkin model.NUCLEAR STRUCTURE Derivation of time-dependent mean-field theory from functional integral representation. Quantum theory of large amplitude collective motion. Application to Lipkin model.

Physical Review C•Apr 1, 1980•80 citations

A Path Integral Formalism of Collective Motion

Hiroshi Kuratsuji

A Quantum mechanical formalism of collective motion is proposed by using the idea of path integral method. Time evolution of many· body system is described by the propagator which joins the many-body state vectors parametrized by complex collective parameters. A general form of path integral representation for the propagator is derived in the complex parameter space. In particular the case of the coherent state representation is discussed in detail. It is pointed out that the time-dependent variation principle and especially the time dependent Hartree-Fock are naturally obtained as a classical limit.

Progress of Theoretical Physics•Jan 1, 1981•16 citations

Path Integrals

Michael El-Batanouny

Presents functional integral methods of quantum many-body theory. Starting with Feynman’s path integral, it develops functional integrals of partition functions in imaginary time and extends these techniques to many-body systems. It expands the formulation in the coherent-state basis, and describes the application of the Hubbard–Stratononvich transformation and the saddle-point approximation.

Cambridge University Press eBooks•Mar 6, 2020•0 citations

Time-dependent mean-field approximation for nuclear dynamical problems

Shimon Levit

A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator $U({t}_{f},{t}_{i})$. In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of $U$ is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.

Physical Review C•Apr 1, 1980•64 citations
We present a path integral formulation of the many-body problem, using a continuous and overcomplete set of vectors in the Hilbert space. The saddle-point approximation for the calculation of the functional integral leads to classical equation of motion which can be quantized in a well prescribed way. The time dependent Hartree-Fock equations are an example of such classical equations. They are obtained by integrating over an overcomplete set of Slater determinant.
We present a general method for constructing path intergrals for the nuclear many-body problem. This method uses continuous and overcomplete sets of vectors in the Hilbert space. The state labels play the role of classical coordinates which are quantized as bosons. The equations of motion for the classical coordinates are obtained by calculating the functional integral in the saddle point approximation. In the particular case where the over-complete set considered is the set of all Slater determinants, the classical equations of motion are the time-dependent Hartree-Fock equations. The functional integral provides a way of requantizing these classical equations. This quantization involves boson degrees of freedom and is in some cases very similar to the method of boson expansion. It is shown that the functional integral formalism provides a unifying framework to describe various approaches to the nuclear many-body problem.NUCLEAR STRUCTURE Functional integrals on continuous over-complete sets. Time-dependent Hartree and Hartree-Fock theories. Boson representations for fermion systems.
A theory is presented in which approximations to quantum observables are obtained by applying the stationary-phase approximation to an exact functional integral representation of the many-body evolution operator. The requirement that the leading correction to the stationary-phase approximation vanish leads to a time-dependent Hartree-Fock mean field. Application of the theory to the Fourier transform of the trace of the evolution operator and the study of its poles yields quantized bound states with large amplitude. The theory is shown to reproduce the familiar static Hartree-Fock and random-phase approximations in the appropriate limits and yields an excellent approximation to the entire spectrum of the exactly solvable Lipkin model.NUCLEAR STRUCTURE Derivation of time-dependent mean-field theory from functional integral representation. Quantum theory of large amplitude collective motion. Application to Lipkin model.
A Quantum mechanical formalism of collective motion is proposed by using the idea of path integral method. Time evolution of many· body system is described by the propagator which joins the many-body state vectors parametrized by complex collective parameters. A general form of path integral representation for the propagator is derived in the complex parameter space. In particular the case of the coherent state representation is discussed in detail. It is pointed out that the time-dependent variation principle and especially the time dependent Hartree-Fock are naturally obtained as a classical limit.
A general method for a microscopic description of nuclear dynamical problems is discussed. The method is based on a functional integral representation for the many-body time evolution operator U(tf,ti)U({t}_{f},{t}_{i})U(tf​,ti​). In the stationary phase limit, a time-dependent mean-field approximation for the matrix elements of UUU is obtained. Using standard procedures this allows for extraction of quantum mechanical information about bound states, tunneling, or scattering phenomena in a many-nucleon system. The approximation represents a natural generalization of the time-dependent Hartree-Fock method.NUCLEAR STRUCTURE, NUCLEAR REACTIONS derived mean-field approximation for matrix elements of $\mathrm{exp}(\ensuremath{-}iHt)$ from exact functional integral; discussed relation to static HF, TDHF, and scattering calculations.