Boson realization of sp(4). I. The matrix formulation
Holstein and Primakoff derived long ago the boson realization of a su(2) Lie algebra for an arbitrary irreducible representation (irrep) of the SU(2) group. The corresponding result for su(1,1)approx. =sp(2) is also well known. This raises the question of whether it is possible to obtain in an explicit, analytic, and closed form, and for any integer d, the boson realization of a sp(2d) Lie algebra for an arbitrary irrep of the Sp(2d) group, which is a problem of considerable physical interest. The case d = 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp(4). The Dyson realization is well known, and the passage to bosons satisfying the appropriate Hermiticity conditions can be done by a similarity transformation through an operator K. What we want, though, is an explicit boson realization for sp(2d) similar to the one that exists for sp(2). In Sec. VI we show how we can get it for sp(4) if the operator K is known. Unfortunately while the matrix form of K/sup 2/ can be explicitly derived from definite recursion relations, the same cannot be said of K as it involves, in general, the solution of algebraic equations of high degree. Thus the conclusion, corroborated also by a classical analysis where K does not appear, is that an explicit, analytic, and closed boson realization of sp(4), and thus also of sp(2d), is only possible for particular irreps of the corresponding groups.
- Research Organization:
- Centro de Estudios Nucleares, Universidad Nacional Autonoma de Mexico, Apartado Postal 70-543, Mexico, D. F. 01000, Mexico
- OSTI ID:
- 5400920
- Journal Information:
- J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 26:9; ISSN JMAPA
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
653007 -- Nuclear Theory-- Nuclear Models-- (-1987)
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
73 NUCLEAR PHYSICS AND RADIATION PHYSICS
ANGULAR MOMENTUM
ANNIHILATION OPERATORS
BOSON EXPANSION
COMMUTATION RELATIONS
CREATION OPERATORS
FIELD THEORIES
HAMILTONIANS
IRREDUCIBLE REPRESENTATIONS
LIE GROUPS
MATHEMATICAL OPERATORS
MATRICES
MATRIX ELEMENTS
PARTICLE PROPERTIES
QUANTUM FIELD THEORY
QUANTUM OPERATORS
SP GROUPS
SPIN
SU GROUPS
SU-2 GROUPS
SYMMETRY GROUPS
TRANSFORMATIONS