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International Journal of Modern Physics A

FERMIONIC QUANTUM MECHANICS AND SUPERFIELDS

Robert Marnelius

Published Jan 20, 1990• 15 citations
Publisher

Abstract

The explicit forms of consistent eigenstate representations for finite dimensional fermionic quantum theories are considered in detail. In particular are the possible Grassmann characters of the eigenstates determined. A straightforward Schrödinger representation is shown to exist if they are even or odd. For an odd number of real eigenvalues, the eigenstates cannot be even or odd. Still a consistent Schrödinger picture is shown to exist provided the basic canonical operators are antilinearly represented. Since the wave functions within the Schrödinger picture are superfields, the class of superfields which also are first quantized wave functions is determined.

DOI
10.1142/s0217751x90000143

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The explicit forms of consistent eigenstate representations for finite dimensional fermionic quantum theories are considered in detail. In particular are the possible Grassmann characters of the eigenstates determined. A straightforward Schrödinger representation is shown to exist if they are even or odd. For an odd number of real eigenvalues, the eigenstates cannot be even or odd. Still a consistent Schrödinger picture is shown to exist provided the basic canonical operators are antilinearly represented. Since the wave functions within the Schrödinger picture are superfields, the class of superfields which also are first quantized wave functions is determined.
The standard quantum states of nnn complex Grassmann variables with a free-particle Lagrangian transform as a spinor of SO(2n). However, the same `free-fermion' model has a non-linearly realized SU(n∣1)SU(n|1)SU(n∣1) symmetry; it can be viewed as the mechanics of a `particle' on the Grassmann-odd coset space SU(n∣1)/U(n)SU(n|1)/U(n)SU(n∣1)/U(n). We implement a quantization of this model for which the states with non-zero norm transform as a representation of SU(n∣1)SU(n|1)SU(n∣1), the representation depending on the U(1) charge of the wave-function. For n=2n=2n=2 the wave-function can be interpreted as a BRST superfield.
The problem that usually arises in the study of a quantum system is the integration of the eigenvalue equation of the Hamiltonian, namely the search for eigenstates and the corresponding eigenvalues of energy. The eigenstates, in the representation of the position, are described by eigenfunctions with the same position variable. The relations work modulo a normalization constant. The number of nodes of a function allows to determine its parity: an odd number of nodes characterizes as an antisymmetric function, while a even number of nodes is a symmetric one. The state of a system can be characterized as bosonic or fermionic depending on whether the wave function is even or odd, respectively. The two operators are nilpotent, when applied more than once to an eigenstate, it generates a null ket, regardless of the starting state.
Properties and implications of the SO(2N+1) regular representation of a fermion system, which is a representation on the SO(2N+1) group proposed recently by us, are studied. By using the independent variables of the SO(2N+1) group, a representation of fermion annihilation-creation and pair operators is obtained which is an extension of the Dyson representation to cover both paired and unpaired modes. The structures of the SO(2N+1) spinor space and the SO(2N+1) wave function are determined. An integral and a differential Schrödinger equation on the SO(2N+1) spinor space are derived which are applicable to both even and odd fermion systems. The structures of the symmetry projected SO(2N+1) wave function and Schrödinger operator are determined. It is shown that odd fermion systems can be treated with an analogous way as even fermion systems by using a special SO(2N+1) canonical transformation which is very similar to the Bogoliubov transformation but corresponds to an improper rotation in the O(2N) group.
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’.
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.
In previous papers a Bose representation of fermions and parafermions was constructed in such a way that (1) the only fundamental quantum fields were Bose fields and (2) no property of the Fermi or para-Fermi systems (as e.g., the Pauli principle for fermions) be lost. The use of that construction and also the proofs of the theorems were made difficult by the complexity of many of the mathematical expressions we used previously. In the present paper we simplify much the above-mentioned construction. Moreover no results (concerning the Bose representation) proved in the previous papers are lost in the present reformulation, but all results are more general: In the previous papers we made the construction of the Bose representation of fermions for the case of pure Fock representations of Quantum Field Theory. The present work is consistent with Fock as well as with non-Fock representations of the Fermi anticommutation relations.