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Progress of Theoretical Physics

Coherent States of Fermi Operators and the Path Integral

Yoshio Ohnuki, Tarō Kashiwa

Published Aug 1, 1978• 193 citations
Publisher

Abstract

Coherent states of Fermi operators are explicitly constructed with the aid of Grassmann numbers, and their properties are discussed in detail. A rigorous form of the path integral for Fermi fields is formulated by means of the coherent ststes. A simple example is also discussed.
DOI
10.1143/ptp.60.548

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Operator ordering for Fermi systems is formulated generally with the aid of the Grassmann numbers. The ordering schemes are restricted to a special class called s-ordering in contrast with those for Bose systems. A relation, similar to that for Bose systems, is shown to exist between different values of the equal-time contractions in the Dyson–Wick expansions of the S matrix and different operator orderings in the interaction Hamiltonian. The S matrix is independent of the ordering scheme chosen. In the functional integral context, different orderings correspond to different discretization prescriptions for the definition of the functional integrals in terms of coherent states for Fermi operators.
The aim of this paper is to give a self-contained and unified presentation of a fermionic coherent state theory with the necessary mathematical details, discussing their definition, properties and some applications. After defining Grassmann algebras, it is possible to get a classical analog for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin (1966 The Method of Second Quantization (New York: Academic); 1987 Introduction to Superanalysis (Dordrecht: Reidel Publishing Company)), we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formulas for fermionic observables. In particular, there exists a Moyal product formula. As an application, we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators. We prove a Mehler formula for the propagator and Mehlig–Wilkinson-type formulas for the covariant and contravariant symbols of ‘metaplectic’ transformations for fermionic states. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
The authors examine several topical subjects, commencing with a general introduction to path integrals in quantum mechanics and the group theoretical backgrounds for path integrals. Applications of harmonic analysis, polar coordinate formulation, various techniques and path integrals on SU(2) and SU(1, 1) are discussed. Soluble examples presented include particle-flux system, a pulsed oscillator, magnetic monopole, the Coulomb problem in curved space and others.The second part deals with the SU(2) coherent states and their applications. Construction and generalization of the SU(2) coherent sta