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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Description of Spin and Statistics in Nonrelativistic Quantum Theories Based on Local Currents

J. Grodnik, D. H. Sharp

Published Mar 15, 1970• 20 citations
Publisher

Abstract

We continue the investigation of the preceding paper into the irreducible representations of local non-relativistic current algebras. Here, we concentrate on the new features which arise in classifying the representations of the current algebra with regard to particle statistics, and in including internal variables such as spin.
DOI
10.1103/physrevd.1.1546

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In this paper we show how to specify the particle content of a nonrelativistic quantum theory of NNN identical spinless particles in terms of observables like the particle number density and the flux density of particles. Our approach to this problem is through a study of the irreducible representations of the local, equal-time current algebra. It is shown how these representations define a functional representation of the current algebra, and that the Hamiltonian can be written in terms of the currents in a nonsingular fashion in any irreducible representation.
Abstract In the previous sections we treated the irreducible representation space associated with the vacuum state. In this chapter we treat other representation spaces necessary to describe particle systems with the vacuum as a background. Such representation spaces are characterized as transportable localized excitations on the vacuum background. If we assume Haag’s duality of the vacuum representation, such a localized excitation is described by a localized endomorphism of the observables and its equivalence classes correspond to different representations called sectors(eigensubspaces of the superselection rules in each of which unrestricted superposition principle holds). From the discussion about composition and conjugation of localized endomorphisms, we define the statistics (of a sector) which represents the behaviour of more than two equivalent localized excitations under permutation, and we derive the connection of spin and statistics. We also introduce the lattice of local field algebras as operators intertwining different sectors, and the gauge group acting on them. Then the local observable algebras are the gauge invariant part of the local field algebras and sectors are inequivalent representations of the gauge group. The local field algebras can be chosen to satisfy the standard commutation or anticommutation relations between space-like separated regions and we can derive the connection of spin and statistics of sectors. In this way we obtain a deep understanding of the reason why Fermi fields in spatially separated regions anticommute.
In this work we consider non-relativistic quantum mechanics, obtained from a classical configuration space 𝒬 of indistinguishable particles. Following an approach proposed in [8], wave functions are regarded as elements of suitable projective modules over C ( 𝒬 ) . We take furthermore into account the G -Theory point of view (cf. [HPRS,S]) where the role of group action is particularly emphasized. As an example illustrating the method, the case of two particles is worked out in detail. Previous works (cf. [BR1,BR2]) aiming at a proof of a spin-statistics theorem for non-relativistic quantum mechanics are re-considered from the point of view of our approach, enabling us to clarify several points.
In this chapter we study the application of geometric algebra to both nonrelativistic and relativistic quantum mechanics. We concentrate on the quantum theory of spin-1/2 particles, whose dynamics is described by the Pauli and Dirac equations. For interactions where spin and relativity are not important the dynamics reduces to that of the Schrödinger equation. There are many good textbooks describing this topic and we will make no attempt to cover it here. We assume, furthermore, that most readers have a basic understanding of quantum mechanics, and are familiar with the concepts of states and operators. Both the Pauli and Dirac matrices arise naturally as representations of the geometric algebras of space and spacetime. It is no surprise, then, that much of quantum theory finds a natural expression within geometric algebra. To achieve this, however, one must reconsider the standard interpretation of the quantum spin operators. Like much discussion of the interpretation of quantum theory, certain issues raised here are controversial. There is no question about the validity of our algebraic approach, however, and little doubt about its advantages. Whether the algebraic simplifications obtained here are indicative of a deeper structure embedded in quantum mechanics is an open question. In this chapter we only consider the quantum theory of single particles in background fields. Multiparticle systems are considered in the following chapter.