New S-function series and application of group theory in supersymmetries
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis is devoted to the study of S-function series and the application of group theory in two physical problems: the decomposition of the basic spin irreps of the extended Poincare supersymmetry and the construction of a colour superalgebra containing generalized quasispin subalgebra, as a receptacle for the dynamical algebra U(M/N) of nuclei supersymmetry. The sixteen classical S-function series which have long been recognized as important in obtaining the branching rules and Kronecker product rules for the classical Lie groups, are classified into six families of three types. Families of new series of type I and III are generated. Techniques are developed to identify the S-function contents of the new series. More than forty new series with well defined generating functions and standard S-function expansions are listed. Possible applications of the new series are mentioned. S-function techniques are used to obtain branching rules for the basic spin irreps of the special orthogonal group SO₂k under the restriction SO₂k SOD_₂ x K where D is the space-time dimension of the extended D-dimensional Poincare supersymmetry and K is the appropriate automorphism group which is D-dependent. General results for D ≤ 10 capable of extending to decompositions of irreps giving rise to helicities greater than two are given together with a general method for D > 10. A number of explicit decompositions are tabulated. Formulas for calculating spin plethysms of SOn (for n < 10) to any order are n given. Several new branching rules for subgroups of SO₂k are developed. Dynamical supersymmetries in nuclei and the harmonic oscillator (boson, fermion) realizations of classical Lie algebras and superalgebras are reviewed. A non-compact Z₂⊕Z₂ graded colour superalgebra SpO(2M1/2N/O) constructed out of the supercreation-annihilation operators is identified as a receptacle for the dynamical superalgebra U(M/N) of nuclei supersymmetry. Various subalgebras of the big algebra are discussed. The existence of a generalised quasispin algebra is demonstrated and its applications discussed.