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AIP conference proceedings

Permutational symmetry of many-particle spin-functions: The dirac identities

Jacob Katriel

Published Jan 1, 2000• 0 citations
Publisher

Abstract

Explicit expressions are derived for the action of all conjugacy classsums of the symmetric group within the space spanned by n identical spin-1/2 particles, in terms of the total spin operator. For identical particles with elementary spin σ all conjugacy class-sums are presented in terms of the lowest 2σ single cycle conjugacy class-sums.

DOI
10.1063/1.1337716

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Explicit expressions are derived for the action of all conjugacy classsums of the symmetric group within the space spanned by n identical spin-1/2 particles, in terms of the total spin operator. For identical particles with elementary spin σ all conjugacy class-sums are presented in terms of the lowest 2σ single cycle conjugacy class-sums.
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.
Abstract The irreps ( S U ( 2 ) , H , U ) of SU(2) of dimension (2S + 1)N, i.e. operators acting on the space H = H N = C ( 2 S + 1 ) N of N identical particles with spin S, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case S = 1/2, Dirac [1] discovered that there is another rep given by ( S ( N ) , H , V ) where S ( N ) is the permutation group, Thus, the standard “linear” Hamiltonian, or Heisenberg interaction Hamiltonian H 0 = ∑ 1 ≤ i ≤ N S → i ⋅ S → j , where σ → i = 2 S → i is the vector of Pauli matrices, can be interpreted as the sum of the “Exchange Operators” Pij between particles i and j. Schrödinger [3] generalized to higher spin numbers S the Exchange Operator
It is shown that the method of Dirac is an effective way of teaching three-particle symmetry classifications to students in the first-year quantum mechanics class, who wish to understand the symmetry properties associated with elementary particles before or without learning group theory. According to the method spelled out for the N-particle case by Dirac in Secs. 55 and 56 of his classic book Principles of Quantum Mechanics, explicit calculations are carried out for three particles. It is shown that this method leads to the three-particle wave functions and their symmetric combinations as given by Feynman et al. In the following paper, the general principles derived here are applied to the quark model of hadrons.