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Journal of physics. A, Proceedings of the Physical Society. General

When is the 'sum over classical paths' exact?

J. S. Dowker

Published Sep 1, 1970• 72 citations
Publisher

Abstract

The calculations of Schulman (1968) on the path integral formulation of the spherical top are generalized to the case of a free particle moving on the manifold of a simple Lie group. It is shown that the finite propagator takes on the same form as the short time one except for the phase factor exp (- 1 / 12 iRt) where R is the constant scalar curvature of the group manifold.

DOI
10.1088/0305-4470/3/5/001

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The calculations of Schulman (1968) on the path integral formulation of the spherical top are generalized to the case of a free particle moving on the manifold of a simple Lie group. It is shown that the finite propagator takes on the same form as the short time one except for the phase factor exp (- 1 / 12 iRt) where R is the constant scalar curvature of the group manifold.
A path integral for spinning particles is developed. It is a one-particle theory, equivalent to the usual quantum mechanics. Our method employs a classical model for spin which is quantized by path integration. The model, the spherical top, is a natural one from a group-theoretic point of view and has been used before in a similar context. The curvature and multiple connectedness of the top coordinate space [SO(3)\mathrm{SO}(3)SO(3)] provide some interesting features in the sum over paths. The Green's function which results from this procedure propagates all spins, and recovery of the usual Pauli spinors from this formalism is achieved by projection to a specific spin subspace.
The quantization of physical systems moving on group and symmetric spaces has been an area of active research over the past three decades. This book shows that it is possible to introduce a representation-independent propagator for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations. For a given set of kinematical variables this propagator is a single generalized function independent of any particular choice of fiducial vector and the irreducible representations of the Lie group generated by these kinematical variables, which nonetheless correctly propagates each element of a continuous representation based on the coherent states associated with these kinematical variables.Furthermore, the book shows that it is possible to construct regularized lattice phase space path integrals for a real, separable, connected and simply connected Lie group with irreducible, square-integrable representations, and although the configuration space is in general a multidimensional curved manifold, it is shown that the resulting lattice phase space path integral has the form of a lattice phase space path integral on a multidimensional flat manifold. Hence, a novel and extremely natural phase space path integral quantization is obtained for general physical systems whose kinematical variables are the generators of a connected and simply connected Lie group. This novel phase space path integral quantization is (a) exact, (b) more general than, and (c) free from the limitations of the previously considered path integral quantizations of free physical systems moving on group manifolds.To illustrate the general theory, a representation-independent propagator is explicitly constructed for SU(2) and the affine group
The simple physics of a free particle reveals important features of the path-integral formulation of relativistic quantum theories. The exact quantum-mechanical propagator is calculated here for a particle described by the simple relativistic action proportional to its proper time. This propagator is nonvanishing outside the light cone, implying that spacelike trajectories must be included in the path integral. The propagator matches the WKB approximation to the corresponding configuration-space path integral far from the light cone; outside the light cone that approximation consists of the contribution from a single spacelike geodesic. This propagator also has the unusual property that its short-time limit does not coincide with the WKB approximation, making the construction of a concrete skeletonized version of the path integral more complicated than in nonrelativistic theory.
The propagator relative to a particle constrained to move in a finite region of space is calculated in the framework of path integrals. This region of the three-dimensional space is delimited through a sector of opening angle α, and also through the action of two attractive harmonic potentials, one being central and located in the 0xy plan, and the other directed along the z axis, with respective pulsations ω and ω0. It is shown that for α=π/2 and π the propagator is the sum of propagators evaluated on classical paths. The important case of the edge (α=2π) is considered.