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Journal of Physics A Mathematical and General

Path integral representations for a system constrained on a manifold

Yoshio Ohnuki

Published Jan 9, 2004• 1 citation
Publisher

Abstract

A systematic study is made of path integral representations for a particle constrained to move on a manifold diffeomorphic to S D by applying the irreducible representations of the Dirac algebra on it. Especially, we derive two types of path integral representation for this system, one of which is of a new form with a simple and compact expression, and the other is a rigorous version of the Faddeev–Senjanovic formula. It is also shown that the parameter α [0, 1) specifying the irreducible representation for D = 1 has a close connection with the Aharonov–Bohm gauge potential produced by the magnetic flux .

DOI
10.1088/0305-4470/37/4/021

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A systematic study is made of path integral representations for a particle constrained to move on a manifold diffeomorphic to S D by applying the irreducible representations of the Dirac algebra on it. Especially, we derive two types of path integral representation for this system, one of which is of a new form with a simple and compact expression, and the other is a rigorous version of the Faddeev–Senjanovic formula. It is also shown that the parameter α [0, 1) specifying the irreducible representation for D = 1 has a close connection with the Aharonov–Bohm gauge potential produced by the magnetic flux .
Abstract In this chapter, we generalize the previous path integral expression to a wider class of physical systems. To this end, we first discuss a change of variables in the path integral and the relationship to the operator formalism, and build up the polar coordinate path integral formula. Next we determine those cases where the p and q are confined within some restricted region, known as constrained systems, by applying the Faddeev-Senjanovic path integral formula to a sphere. Further we discuss quantum mechanics on a ring and the Aharonov-Bohm effect as the simplest examples of multiply connected space characterized by unshrinkable closed loops in configuration space. Also we make some comments on quantum mechanics for a more general manifold. In the next section the path integral for spin is de- rived with a newly introduced coherent state that also gives an alternative expression for the harmonic oscillator. Then, finally, cases where the WKB approximation happens to be exact are analysed.
We present, via the path-integral approach, the quantum study of a particle without spin constrained to move on a circle and subjected to the action of an external field (V, A). In the first stage, we follow the Faddeev–Senjanovic constraints technique that is essentially based on the Dirac algorithm; and in the second stage, we use the path-integral coherent state relative to the circle, compatible with the topological properties. In the two cases, the free particle and the problem of the magnetic field are shown as an illustrative calculation.PACS Nos.: 03.65–w, 03.65Ca, 03.65.Db
Exact Green’s function related to a Dirac particle submitted to the combination of Aharonov–Bohm and Coulomb potentials in [Formula: see text]) coordinate space is analytically calculated via path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger model. The energy spectrum as well as the corresponding normalized wave functions are extracted following this approach. The interesting properties of the spinors are thus deduced after symmetrization. According to the symmetric form for Green’s function, it is shown that the non-relativistic limit of the Dirac particle is undertaken with much ease.
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.
The concept of constrained Hamiltonian dynamical systems and their quantization, introduced by Dirac, constitutes one of the most fundamental contributions to the development of the canonical formalism in recent times. Dirac’s concepts were later incorporated into the path-integral formulation of quantum mechanics by Faddeev, which played an important role in understanding some subtle aspects in the quantization of non-Abelian gauge theory. An example is given from nonrelativistic quantum mechanics which illustrates the formalism through simple analogy.