© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Journal of Physics A Mathematical and General

Auxiliary fields for treating two-time Lagrangian quantum mechanics

G J Popadopoulos

Published Aug 1, 1985• 5 citations

Abstract

Functional integrals over complex and real auxiliary fields for propagators relating to two-time (memory) actions are constructed. The integrals over fields result from partial replacement of the path variable, via functional integration, in the path integral form of the propagator by a field variable in a way that allows the remaining path integration to be performed, thus leaving a functional integral. Two new cases of actions involving memory, a quadratic and a quartic are treated explicitly. A functional integral identity is also developed. Furthermore, the method of coupling an auxiliary dynamical field for generating memory actions is also employed and formulae are developed in the case of certain quadratic actions giving the propagator exactly. The method is flexible enough embracing all explicitly known cases of two-time quadratic actions and in addition enabling further cases to be treated. An example of this nature is given explicitly.

DOI: 10.1088/0305-4470/18/11/019

Related papers

Path integration of a general two-time quadratic action by prodistribution approach

Dinkar C. Khandekar, Sunil Datta, S. V. Lawande

The theory of prodistributions is applied to path integrate a general two-time quadratic action characterizing memory effects. The resulting Feynman propagator is exact and is in the form of an exponential integral over a single variable. A case where the integral is explicitly evaluated, for an arbitrary memory kernel, is presented to highlight the closed analytical structure of the propagator obtained. All special cases treated in the literature follow from this general expression.

Physical Review A•Jun 1, 1985•3 citations

Feynman propagator for time-dependent Lagrangians possessing an invariant quadratic in momentum

Asish Kumar Dhara, S. V. Lawande

A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.

Journal of Physics A Mathematical and General•Aug 21, 1984•42 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 and propagators for quadratic Lagrangians in three dimensions

Kai-Man Poon, Gerardo Muñoz

Functional methods are employed to derive the propagator for a three-dimensional system described by a Lagrangian at most quadratic in the velocity and position variables. Two physically relevant examples are presented to illustrate the concrete application of the general formula.

American Journal of Physics•Jun 1, 1999•7 citations

Time-dependent invariants and the Feynman propagator

Asish Kumar Dhara, S. V. Lawande

A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.

Physical Review A•Jul 1, 1984•45 citations

A Method to Evaluate Quantum Path Integrals

Babak Vakili

As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.

Theoretical Physics•Mar 22, 2017•0 citations

Functional integrals over complex and real auxiliary fields for propagators relating to two-time (memory) actions are constructed. The integrals over fields result from partial replacement of the path variable, via functional integration, in the path integral form of the propagator by a field variable in a way that allows the remaining path integration to be performed, thus leaving a functional integral. Two new cases of actions involving memory, a quadratic and a quartic are treated explicitly. A functional integral identity is also developed. Furthermore, the method of coupling an auxiliary dynamical field for generating memory actions is also employed and formulae are developed in the case of certain quadratic actions giving the propagator exactly. The method is flexible enough embracing all explicitly known cases of two-time quadratic actions and in addition enabling further cases to be treated. An example of this nature is given explicitly.

A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.

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.

A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.

As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.