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 Mathematical Physics

Path integrals on homogeneous manifolds

M. S. Marinov

Published May 1, 1995• 11 citations
Publisher

Abstract

Quantum kinematics are considered for two classes of systems: (i) the coordinate space is a homogeneous Riemannian manifold and the kinetic energy is the Laplace–Beltrami operator, (ii) the phase-space is a homogeneous Kählerian manifold where the algebra of observables is the universal enveloping algebra of the symmetry group. In both the cases, the path integrals are derived from more fundamental principles. The derivation enables a definite description of the classes of trajectories which constitute the domain of integration in the path integrals. The specification of the classes of functions is important for consistency of the approach. The standard action functional and the variational principle appear in the steepest descent method corresponding to the semiclassical approximation.

DOI
10.1063/1.531047

Related papers

Quantum mechanics on homogeneous spaces

H. D. Doebner, J. Tolar

A complete description of quantum kinematics on a homogeneous G−space M is presented using imprimitivity systems for G based on M. The kinematics on M is considered (if possible and consistent with this quantization) as kinematics on a G−orbit equivalent to M in some Euclidean space Rn. This method gives a physically justified and mathematically well−defined method of connecting the free Hamiltonian of a quantum system in Rn with an operator proportional to the Laplace−Beltrami operator on M (with the Riemannian structure inherited from Rn) which is proposed to be the free Hamiltonian on M.

Journal of Mathematical Physics•Apr 1, 1975•35 citations

Path Integral Quantization and Riemannian-Symplectic Manifolds

Sergei V. Shabanov, John R. Klauder

We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a

•Jan 1, 1998•6 citations

Quantum mechanics (QM): Path integrals in phase space

Jean Zinn‐Justin

Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.

•Apr 15, 2021•0 citations

Path Integrals In Quantum Mechanics: Generalizations

Jean Zinn‐Justin

Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.

Oxford University Press eBooks•Jun 6, 2002•0 citations

Quasiclassical path integral in coherent-state manifolds

E. A. Kochetov

Quantization of classical dynamical systems with a Poisson structure on homogeneous Kähler manifolds is considered. The quantization follows the method invented by Berezin and represents the unitary transition operator as a quasiclassical path integral in the coherent-state basis. In case the coherent-state manifold appears as a (degenerate) rank-one co-adjoint orbit of the symmetry group, an explicit representation of the transition amplitude in terms of classical data can be derived for large values of the highest weight, which corresponds to the quasiclassical approximation. This representation is further shown to perfectly agree, in contrast to some earlier approaches, with the known exact results and may provide non-trivial asymptotics of physical relevance.

Journal of Physics A Mathematical and General•May 15, 1998•37 citations

Quantum and classical mechanics on homogeneous Riemannian manifolds

Gérard G. Emch

Starting from the axioms of quantum mechanics as formalized by the systems of imprimitivity for homogeneous Riemannian manifolds, the classical theory is derived as a consequence, complete with its phase space realized as the space of pure classical states, a generalized version of the Wigner–Moyal correspondence rule, the Jordan and Lie algebra structures of functions on the cotangent bundle, given by point-wise multiplication and Poisson bracket, and the momentum map. A comparison is also given of the quantum and classical dynamics and equilibrium statistical mechanics of free particles on compact manifolds of constant negative curvature.

Journal of Mathematical Physics•Oct 1, 1982•20 citations
Quantum kinematics are considered for two classes of systems: (i) the coordinate space is a homogeneous Riemannian manifold and the kinetic energy is the Laplace–Beltrami operator, (ii) the phase-space is a homogeneous Kählerian manifold where the algebra of observables is the universal enveloping algebra of the symmetry group. In both the cases, the path integrals are derived from more fundamental principles. The derivation enables a definite description of the classes of trajectories which constitute the domain of integration in the path integrals. The specification of the classes of functions is important for consistency of the approach. The standard action functional and the variational principle appear in the steepest descent method corresponding to the semiclassical approximation.
A complete description of quantum kinematics on a homogeneous G−space M is presented using imprimitivity systems for G based on M. The kinematics on M is considered (if possible and consistent with this quantization) as kinematics on a G−orbit equivalent to M in some Euclidean space Rn. This method gives a physically justified and mathematically well−defined method of connecting the free Hamiltonian of a quantum system in Rn with an operator proportional to the Laplace−Beltrami operator on M (with the Riemannian structure inherited from Rn) which is proposed to be the free Hamiltonian on M.
We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a
Abstract In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to Hamiltonians that are more general functions of phase space variables. This results in integrals over paths in phase space involving the action expressed in terms of the classical Hamiltonian H(p,q). However, it is shown that, in the general case, the path integral is not completely defined, and this reflects the problem that the classical Hamiltonian does not specify completely the quantum Hamiltonian, due to the problem of ordering quantum operators in products. When the Hamiltonian is a quadratic function of the momentum variables, the integral over momenta is Gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is ambiguous, since a problem of operator ordering arises, and the ambiguity must be fixed. Hamiltonians that are general quadratic functions provide other important examples, which are analysed thoroughly. Such Hamiltonians appear in the quantization of the motion on Riemannian manifolds. There, the problem of ambiguities is even more severe. The problem is illustrated by the analysis of the quantization of the free motion on the sphere SN−1.
Chapter 2 constructed a path integral representation of the matrix elements of the statistical operator e-βH. This chapter extends the construction to hamiltonians which are general functions of phase space variables. This results in integrals over trajectories or paths in phase space. When the hamiltonian is quadratic in the momentum variables, the integral over momenta is gaussian and can be performed. In the separable example, the path integral of Chapter 2 is recovered. In the case of the charged particle in a magnetic field a more general form is found, which is somewhat ambiguous, reflecting the problem of order between quantum operators. Hamiltonians which are general quadratic functions of momentum variables provide other important examples, and these are analyzed thoroughly. Such hamiltonians arise in the quantization of the motion on Riemannian manifolds. The analysis is illustrated by the quantization of the free motion on the sphere SN-1.
Quantization of classical dynamical systems with a Poisson structure on homogeneous Kähler manifolds is considered. The quantization follows the method invented by Berezin and represents the unitary transition operator as a quasiclassical path integral in the coherent-state basis. In case the coherent-state manifold appears as a (degenerate) rank-one co-adjoint orbit of the symmetry group, an explicit representation of the transition amplitude in terms of classical data can be derived for large values of the highest weight, which corresponds to the quasiclassical approximation. This representation is further shown to perfectly agree, in contrast to some earlier approaches, with the known exact results and may provide non-trivial asymptotics of physical relevance.
Starting from the axioms of quantum mechanics as formalized by the systems of imprimitivity for homogeneous Riemannian manifolds, the classical theory is derived as a consequence, complete with its phase space realized as the space of pure classical states, a generalized version of the Wigner–Moyal correspondence rule, the Jordan and Lie algebra structures of functions on the cotangent bundle, given by point-wise multiplication and Poisson bracket, and the momentum map. A comparison is also given of the quantum and classical dynamics and equilibrium statistical mechanics of free particles on compact manifolds of constant negative curvature.