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Progress of Theoretical Physics

Quantum Field Theory of Fermions and Parafermions Constructed from Quantum Bose Fields. I: Simplification and Generalization

Andrés J. Kálnay

Published Dec 1, 1975• 18 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.54.1848

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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.
Special relativity implies that mass can be created from energy — it implies particle production. A relativistic quantum theory is of necessity a theory of many particles. We shall therefore first describe systems of many identical particles but in the simpler context of non-relativistic theory. As we shall soon see, many-particle systems can be described by field operators that create and destroy particles, with these operators acting in a vastly enlarged vector space that spans states with an arbitrary number of particles, including a state of no particles — the vacuum state. Both Bose and Fermi statistics are handled in this way, and the resulting formalism is compact, convenient, and powerful. For the Bose case, the functional integral representation of the quantum many-particle system is direct generalization of the coherent-state functional integral developed in the previous chapter. To describe the analogous formulation for the Fermi case, we must first develop the theory of anticommuting variables. The power and utility of field operator and functional integral methods will be illustrated by applying it to systems in thermal equilibrium, first to free Fermi and Bose gases, and then to a model of liquid helium which exhibits the phenomenon of spontaneous symmetry breaking, a phenomenon of great importance in condensed matter physics as well as in elementary particle theory.
In this article, we provide an alternative and general way to construct the result of the action of any particle-conserving bosonic or fermionic operator represented in second quantized form on a state vector, without resorting to the matrix representation of operators or even its elements. This approach is based on our proposal to compactly enumerate the configurations (i.e., determinants for fermions, permanents for bosons) that are the elements of the state vector. This extremely simplifies the calculation of the action of an operator on a state vector. The computations of statical properties and the evolution dynamics of a system become much more efficient, and applications to systems made of more particles become feasible. Explicit formulations are given for spin-polarized fermionic systems and spinless bosonic systems, as well as to general (two-component) fermionic systems, two-component bosonic systems, and mixtures thereof.
In a recent series of papers, we have analyzed a certain deformation of the canonical commutation relations producing an interesting functional structure which has been proved to have some connections with physics, and in particular with quasi-Hermitian quantum mechanics. Here we repeat a similar analysis starting with the canonical anticommutation relations. We will show that in this case, most of the assumptions needed in the former situation are automatically satisfied, making our construction rather friendly . We discuss some examples of our construction, again related to quasi-Hermitian quantum mechanics, and the bi-coherent states for the system. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.