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International Journal of Quantum Chemistry

Generalized Dirac identities and explicit relations between the permutational symmetry and the spin operators for systems of identical particles

Jacob Katriel, Josef Paldus, Reuben Pauncz

Published Aug 1, 1985• 14 citations

Abstract

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.

DOI: 10.1002/qua.560280203

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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.

Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to nontrivial permutation symmetries (beyond the conventional paradigm of symmetric-antisymmetric states). The present work obtains the following. (a) Selection rules for additional nonseparable (dependent on spins and coordinates) k-body interactions: the Young diagrams, associated with the initial and final states of a transition, can differ by relocation of no more than k boxes between their rows. (b) Correlation rules in which eigenstate-averaged local correlations of k particles vanish if k exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry-in 1929, Dirac identified those with characters of the symmetric group-relating them to experimentally observable correlations of several particles. The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.

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

The general form of the operators commuting with the ground representation (appearing in many physical problems within the single-particle approximation) of the group is found. With the help of the modified group projector technique, this result is applied to a system of identical particles with spin-independent interaction, to derive the Dirac-Heisenberg Hamiltonian and its effective space for arbitrary orbital occupation numbers and arbitrary spin. This gives transparent insight into the physical contents of this Hamiltonian, showing that formal generalizations with spin greater than ½ involve non-trivial additional physical assumptions.