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:math-ph/0101026

Stationary Phase in Coherent State Path Integrals

Ray, Chris, Giulio Ruffini

Published Jan 23, 2001• 0 citations
arXivHTMLPDF

Abstract

In applying the stationary phase approximation to coherent state path integrals a difficulty occurs; there are no classical paths that satisfy the boundary conditions of the path integral. Others have gotten around this problem by reevaluating the action. In this work it is shown that it is not necessary to reevaluate the action because the stationary phase approximation is applicable so long as the path, about which the expansion is performed, satisfies the associated Lagrange's equations of motion. It is not necessary for this path to satisfy the boundary conditions in order to apply the stationary phase approximation.

DOI
10.48550/arxiv.math-ph/0101026

Related papers

On the stationary phase evaluation of path integrals in the coherent states representation

Yitzhak Weissman

The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.

Journal of Physics A Mathematical and General•Aug 21, 1983•37 citations

SU(2) coherent-state path integral

E. A. Kochetov

The SU(2) coherent-state path integral is used to represent the matrix element of a propagator in the SU(2) coherent-state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary-phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.

Journal of Mathematical Physics•Sep 1, 1995•85 citations

Path integrals and stationary-phase approximations

John R. Klauder

The general formalism for path integrals expressed in terms of an arbitrary continuous representation (generalized coherent states) is applied to give $c$-number formulations for the canonical algebra and for the spin algebra, and is used to derive meaningful stationary-phase approximations for these two cases. Some clarification of a recent discussion by Jevicki and Papanicolaou for the spin case is given.

Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 15, 1979•327 citations

Coherent-state path integral for a transition amplitude: A theory and applications

E. A. Kochetov, Vladimir Sergeevich Yarunin

The analytical methods of path integral (PI) calculations are presented for transition matrix elements in a basis of the generalized coherent states (CS'). Appropriate transformations in phase space are found and the stationary-phase method is developed with a particular emphasis on the PI boundary terms. Examples demonstrate the application of the general scheme for spin systems and the tests of the accuracy of the semiclassical approximation in nonlinear dynamics.

Physica Scripta•Jan 1, 1995•13 citations

Path Integrals in Quantum Mechanics

Pierre VAN BAAL

The integrals in the path integral are strongly oscillating and can only be defined by analytic continuation. It is important to note that the continuous expression is just a notation for the discrete version of the path integral, but formal manipulations will be much easier to perform in this continuous formulation. Furthermore, the integral is only defined through the analytic continuation in time. As long as the Hamiltonian is quadratic in the momenta, the stationary phase approximation for the momentum integral is exact. In practical situations one splits from the action the quadratic part in the coordinates and velocities and considers the rest as a perturbation.

CRC Press eBooks•Apr 19, 2016•2 citations

Time-sliced path integrals with stationary states

Chris M Field, Frank Willem Nijhoff

The path integral approach to the quantization of one degree-of-freedom Newtonian particles is considered within the discrete time-slicing approach, as in Feynman's original development. In the time-slicing approximation the quantum mechanical evolution will generally not have any stationary states. We look for conditions on the potential energy term such that the quantum mechanical evolution may possess stationary states without having to perform a continuum limit. When the stationary states are postulated to be solutions of a second-order ordinary differential equation (ODE) eigenvalue problem it is found that the potential is required to be a solution of a particular first-order ODE. Similarly, when the stationary states are postulated to be solutions of a second-order ordinary difference equation (O$Δ$E) eigenvalue problem the potential is required to be a solution of a particular first-order O$Δ$E. The classical limits (which are at times very nontrivial) are integrable maps.

arXiv:nlin/0604046•Journal of Physics A Mathematical and General•May 3, 2006•3 citations
In applying the stationary phase approximation to coherent state path integrals a difficulty occurs; there are no classical paths that satisfy the boundary conditions of the path integral. Others have gotten around this problem by reevaluating the action. In this work it is shown that it is not necessary to reevaluate the action because the stationary phase approximation is applicable so long as the path, about which the expansion is performed, satisfies the associated Lagrange's equations of motion. It is not necessary for this path to satisfy the boundary conditions in order to apply the stationary phase approximation.
The extremal paths that arise in the stationary phase evaluation of coherent state path integrals do not seem to have a simple physical interpretation, in contrast to the extremal paths that occur in the conventional path integrals. On the other hand, a recently derived semiclassical formula for the coherent state propagator involves a path that is determined in exactly the same manner as the extremal paths of the conventional path integrals. Since both the semiclassical and the stationary phase analyses yield asymptotic (h(cross) to 0) approximations, the stationary phase and the semiclassical expressions for the propagator should be identical. The author presents a simple and direct proof that, in spite of the apparent differences, this is indeed the case. The simplification in the semiclassical formula is due to the utilisation of an appropriate set of canonical variables to describe the classical dynamics. In order to illustrate the usefulness of the semiclassical formula they present an application to the problem of the degenerate parametric amplifier, which had been treated before by operator ordering and path integral methods. The semiclassical approach has a simple classical interpretation that is absent in the alternative treatments.
The SU(2) coherent-state path integral is used to represent the matrix element of a propagator in the SU(2) coherent-state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary-phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.
The general formalism for path integrals expressed in terms of an arbitrary continuous representation (generalized coherent states) is applied to give ccc-number formulations for the canonical algebra and for the spin algebra, and is used to derive meaningful stationary-phase approximations for these two cases. Some clarification of a recent discussion by Jevicki and Papanicolaou for the spin case is given.
The integrals in the path integral are strongly oscillating and can only be defined by analytic continuation. It is important to note that the continuous expression is just a notation for the discrete version of the path integral, but formal manipulations will be much easier to perform in this continuous formulation. Furthermore, the integral is only defined through the analytic continuation in time. As long as the Hamiltonian is quadratic in the momenta, the stationary phase approximation for the momentum integral is exact. In practical situations one splits from the action the quadratic part in the coordinates and velocities and considers the rest as a perturbation.
The path integral approach to the quantization of one degree-of-freedom Newtonian particles is considered within the discrete time-slicing approach, as in Feynman's original development. In the time-slicing approximation the quantum mechanical evolution will generally not have any stationary states. We look for conditions on the potential energy term such that the quantum mechanical evolution may possess stationary states without having to perform a continuum limit. When the stationary states are postulated to be solutions of a second-order ordinary differential equation (ODE) eigenvalue problem it is found that the potential is required to be a solution of a particular first-order ODE. Similarly, when the stationary states are postulated to be solutions of a second-order ordinary difference equation (OΔΔΔE) eigenvalue problem the potential is required to be a solution of a particular first-order OΔΔΔE. The classical limits (which are at times very nontrivial) are integrable maps.