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Theoretical and mathematical physics

Fermionic Coherent States

Didier Robert, Monique Combescure

Published Jan 1, 2021• 1 citation
Publisher

Abstract

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.

DOI
10.1007/978-3-030-70845-0_16

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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.
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’.
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 methods that have been described in previous chapters can be used to study general quantum Bose systems. They are based in a direct way on the introduction of a generating function of symmetric wave functions of bosons. In the case of fermion systems, however, one faces the problem that fermion wave functions, or fermion correlation functions (or Green functions) are antisymmetric with respect to the exchange of a fermion pair. Thus, the construction of generating functions requires the introduction of an antisymmetric or Grassmann algebra of ‘classical functions’. It is then possible to generalize to Grassmann algebras the notions of derivatives and integrals, yielding quite parallel formalisms for bosons and fermions, in particular, to define a path integral for fermion systems, analogous to the holomorphic path integral for bosons. This chapter discusses differentiation and integration in Grassmann algebras, Gaussian integrals and perturbative expansion, partition function, and quantum Fermi gas.
It is well known from the work of Berezin [24] (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behaviour of a quantum system when the Planck constant ℏ\hbar ℏ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem. In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space Rn\mathbb R^nRn. So that in the Hamiltonian formalism, the phase space is Rn×Rn\mathbb R^n\times \mathbb R^nRn×Rn with its canonical symplectic form σ\sigma σ. The quantization problem has many solutions, so we choose a convient one, introduced by Weyl [259] (The Classical Groups, 1997) and Wigner [261] (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959). We study the symmetries of Weyl quantization, the operationnal calculus and applications to propagation of observables. We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given to the semi-classical limit after introducing an efficient modern tool: the semi-classical measures. We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
In this chapter we introduce anticommuting variables and derive the Grassmann integral representations for the partition function and the correlation functions. Grassmann integrals are a suitable algebraic tool for the rearrangement of fermionic perturbation series. We demonstrate this in the first section by considering a model with a quadratic perturbation which is explicitely solvable. In that case the perturbation series can be resumed completely. By doing this in two ways, first a direct computation without anticommuting variables and then a calculation using Grassmann integrals, we hope to be able to convince the reader of the utility of that formalism.