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Journal of Physics A Mathematical and General

Extension of the Dirac identity

Daniel Klein

Published Oct 1, 1980• 11 citations
Publisher

Abstract

A constructive proof is presented to show that, for any permutation P on spin space, P+P -1 can be expressed as a linear combination of spin-operator products Pi s m .s n with all the particle indices m, n in a product distinct.
DOI
10.1088/0305-4470/13/10/011

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Abstract The well known one‐to‐one correspondence between the eigenstates of the total spin for a system of spin‐½ particles and irreducible representations of the symmetric group with up to two rows in the Young shape is the basis of interesting formal developments in quantum chemistry and in the theory of magnetism. As an explicit manifestation of this correspondence the class operators of the symmetric group are demonstrated to be expressible in terms of the total spin operator. This correspondence does not hold for higher elementary spins. The extension to arbitrary spin is investigated using Schrödinger's generalization of the Dirac identity, which expresses the transpositions in terms of two‐particle spin operators. It is shown that additional operators, which for σ = ½ reduce to the total spin operator, are needed for a complete classification. Some aspects of the formalism are developed in detail for σ = 1. In this case a classification identical with that provided by the irreducible representations of the symmetric group is obtained in terms of the eigenstates of two commuting operators, one of which is the total spin operator.