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Complex fermion coherent states

Tomáš Tyc, Brett Hamilton, Barry C. Sanders, William D. Oliver

Published Jul 18, 2005• 0 citations
OpenAlex

Abstract

Whereas boson coherent states provide an elegant, intuitive and useful representation, we show that the desirable features of boson coherent states do not carry over very well to fermion fields unless one is prepared to use exotic approaches such as Grassmann fields. Specifically, we identify four appealing properties of boson coherent states (eigenstate of annihilation operator, displaced vacuum state, preservation of product states under linear coupling, and factorization of correlators) and show that fermion coherent states, and approximations to fermion coherent states, defined over the complex field, do not behave well for any of these four criteria.

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Whereas boson coherent states provide an elegant, intuitive and useful representation, we show that the desirable features of boson coherent states do not carry over very well to fermion fields unless one is prepared to use exotic approaches such as Grassmann fields. Specifically, we identify four appealing properties of boson coherent states (eigenstate of annihilation operator, displaced vacuum state, preservation of product states under linear coupling, and factorization of correlators) and show that fermion coherent states, and approximations to fermion coherent states, defined over the complex field, do not behave well for any of these four criteria.
Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.
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 mathematical methods that have been used to analyze the statistical properties of boson fields, and in particular the coherence of photons in quantum optics, have their counterparts for Fermi fields. The coherent states, the displacement operators, the P representation, and the other operator expansions all possess surprisingly close fermionic analogues. These methods for describing the statistical properties of fermions are based upon a practical calculus of anticommuting variables. They are used to calculate correlation functions and counting distributions for general systems of fermions.
This chapter is an introduction to some computation techniques for fermionic states. After defining Grassmann algebras it is possible to get a classical analogue for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin [23, 26], 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 formula 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 present a systematic analysis of coherent states of composite bosons consisting of two distinguishable particles. By defining an effective composite boson (coboson) annihilation operator, we derive its eigenstate and commutator. Depending on the elementary particles comprising the composite particles, we gauge the resemblance between this eigenstate and traditional coherent states through typical measures of nonclassicality, such as quadrature variances and Mandel's QQQ parameter. Furthermore, we show that the eigenstate of the coboson annihilation operator is useful in estimating the maximum eigenvalue of the coboson number operator.