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Canadian Journal of Physics

Quantization on the circle

M. Merad

Published May 1, 2006• 0 citations
Publisher

Abstract

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

DOI
10.1139/p06-069

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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
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
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.
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 .