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arXiv:quant-ph/9607019

Coherent State Path Integrals for Systems with Constraints

John R. Klauder

Published Jul 22, 1996• 4 citations
arXivHTMLPDF

Abstract

We outline the principal results of a recent examination of the quantization of systems with first- and second-class constraints from the point of view of coherent-state phase-space path integration. Two examples serve to illustrate the procedures.
DOI
10.48550/arxiv.quant-ph/9607019

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Systems with constraints pose problems when they are quantized. Moreover, the Dirac procedure of quantization prior to reduction is preferred. The projection operator method of quantization, which can be most conveniently described by coherent state path integrals, enables one to directly impose a regularized form of the quantum constraints. This procedure also overcomes conventional difficulties with normalization and second class constraints that invalidate conventional Dirac quantization procedures.
Coherent states can be used for diverse applications in quantum physics including the construction of coherent state path integrals. Most definitions make use of a lattice regularization; however, recent definitions employ a continuous-time regularization that may involve a Wiener measure concentrated on continuous phase space paths. The introduction of constraints is both natural and economical in coherent state path integrals involving only the dynamical and Lagrange multiplier variables. A preliminary indication of how these procedures may possibly be applied to quantum gravity is briefly discussed.
The definition and interpretation of canonical, phase space path integrals has evolved over many years to achieve a form that now admits a correct and rigorous formulation, which is also covariant under canonical coordinate transformations. Such formulations involve coherent state representations, which, in their modern version, were originally introduced as an alternative tool to construct phase space path integrals. Moreover, coherent state representations lead to physical interpretations that are more natural than those afforded by more traditional representations. Suitable continuous time regularization procedures lead to a covariant phase space path integral formulation that greatly clarifies the vague phrase that canonical quantization requires Cartesian coordinates.
Canonical quantization entails using Cartesian coordinates, and Cartesian coordinates exist only in flat spaces. This situation can either be questioned or accepted. In this paper we offer a brief and introductory overview of how a flat phase space metric can be incorporated into a covariant, coordinate-free quantization procedure involving a continuous-time (Wiener measure) regularization of traditional phase space path integrals. Additionally we show how such procedures can be extended to incorporate systems with constraints and illustrate that extension for special systems.
This book is a self-contained and concise introduction to the techniques and applications of path integral quantization and functional techniques, aimed at students and practitioners. The first half of the text focuses on quantum mechanics, including a review of the action formulation of classical mechanics and quantum mechanics in the Dirac operator and state formalism, and further examination of the path integral. The second part examines relativistic field theories, reviewing special relativity, as well as derivation of the path integral representation of the vacuum transition element for quantized scalar, spinor, and vector fields from the coherent state representation of the respective field theories
We define phase space path integrals for systems moving on a Riemannian manifold and subject to a generalized potential by a skeletonization procedure which is manifestly covariant under point transformations. We achieve this goal by introducing a natural analog S(x″,t″‖x′,p′,t′) of the Hamilton principal function with phase space initial data. One class of such functions is based on the parallel transport of momentum, a second class is obtained by a modification of the first class, and a third class is based on the geodesic deviation transport of momentum. The third class of principal functions is geometrically privileged. We skeletonize the canonical action integral by replacing it by a manifestly covariant chain of phase space principal functions. Different functions lead to the same functional as we infinitely refine the skeletonization along a smooth path. Our phase space path integral is always taken with the natural Liouville measure, but the integration over momentum variables brings down a nontrivial measure to the remaining configuration space path integral. Because nondifferentiable rather than smooth paths dominate the path integral, different phase space principal functions generate different configuration space path measures. Such measures lead to quantum propagators which satisfy Schrödinger’s equations with all possible scalar curvature terms ∼ℏ2R. The geometrically privileged phase space principal function (based on the geodesic deviation transport) leads to the Schrödinger equation without any curvature term.