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Canadian Journal of Physics

Unitary-group canonical states and matrix elements

M.F. Soto, Ronald Mirman

Published Aug 1, 1989• 0 citations
Publisher

Abstract

States of unitary groups are realized as multinomials in boson operators, symmetrized to give symmetric-group basis states. From these, matrix elements of the group generators are calculated using the procedures discussed here. A table of basis states and matrix elements of unitary-group representations, and the values of the invariants so generated, is given for SU(1) through SU(4), for all symmetric groups from S(1) through S(3).

DOI
10.1139/p89-135

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States of unitary groups are realized as multinomials in boson operators, symmetrized to give symmetric-group basis states. From these, matrix elements of the group generators are calculated using the procedures discussed here. A table of basis states and matrix elements of unitary-group representations, and the values of the invariants so generated, is given for SU(1) through SU(4), for all symmetric groups from S(1) through S(3).
A detailed exposition of explicit formulas used in the evaluation of raising-lowering forms of two-body-operator matrix elements is presented. The methods are based on the use of the unitary group distinct row table and graphical representation of the many-particle basis. All matrix elements are expressible in terms of scalar and simple matrix factors. In order to facilitate the derivations of simple, computationally efficient forms for the matrix factors, we utilize a recently developed calculus based on elementary graphs. The methods are applicable to systems of particles involving spins greater than (1/2 and reduce to previously known results for the case of spin (1/2.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.