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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Feynman Functional Integrals for Systems of Indistinguishable Particles

Michael G. G. Laidlaw, C. DeWitt

Published Mar 15, 1971• 394 citations
Publisher

Abstract

The theory of path integration is extended to include systems whose configuration space is multiply connected, and it is seen that there are as many distinct propagators as there are scalar representations of the associated fundamental group. It is shown that the configuration space for a system of indistinguishable particles is multiply connected. There are only two propagators for this system, giving bosons and fermions, and showing that the Feynman formalism excludes parastatistics.

DOI
10.1103/physrevd.3.1375

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The theory of path integration is extended to include systems whose configuration space is multiply connected, and it is seen that there are as many distinct propagators as there are scalar representations of the associated fundamental group. It is shown that the configuration space for a system of indistinguishable particles is multiply connected. There are only two propagators for this system, giving bosons and fermions, and showing that the Feynman formalism excludes parastatistics.
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
For distinguishable particles it is well known that Brownian motion and a Feynman-Kac functional can be used to calculate the path integral (for imaginary times) for a general class of scalar potentials. In order to treat identical particles, we exploit the fact that this method separates the problem of the potential, dealt with by the Feynman-Kac functional, from the process which gives sample paths of a noninteracting system. For motion in one dimension, we emphasize that the permutation symmetry of the identical particles completely determines the domain of Brownian motion and the appropriate boundary conditions: absorption for fermions, reflection for bosons. Further analysis of the sample paths for motion in three dimensions allows us to decompose these paths into a superposition of one-dimensional sample paths. This reduction expresses the propagator (and consequently the energy and other thermodynamical quantities) in terms of well-behaved one-dimensional fermion and boson diffusion processes and the Feynman-Kac functional.
Quantum mechanics is often contrasted with its classical counterpart. This distinction is nearly “washed out” when quantum statistics for bosonic systems are formulated in terms of Feynman’s path integrals [1]. These provide a unique possibility of projecting subtle quantum mechanical properties into the language of classical statistics of closed trajectories in (d+1)-dimensional space. We note, however, that the corresponding projection and an intuition associated with it require that the statistics of path integrals be sign-positive (which is not the case for the generic fermionic system). Indeed, for the positive definite statistics, one can appeal to properties of “typical” configurations contributing to the answer and then estimate probabilities of atypical configurations to occur. If configurations contribute to the statistics with the sign or, in a more general case, with phase factors, then the notion of typical and relevant configurations is misleadingconfigurations with large modulus and opposite sign cancel each other to a large degree in the final answer, and only their combined effect has a physical meaning.
The motion of a nonrelativistic charged particle in a plane, multiply-connected region is studied within the framework of Feynman's path-integral approach to quantum mechanics. In particular, the authors study the simple case when the multiply-connected region is obtained by excluding a disc from the plane. If a nonzero magnetic flux is confined inside the disc, this is shown to include a change in the one-dimensional unitary representation of the fundamental group of the space which enters a proper definition of the path-integral. In this way, a simple explanation of the Aharonov-Bohm effect is shown to arise.
We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on R3N\mathbb{R}^{3N}R3N to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wave functions on the universal covering space of the configuration space.