Finite- and infinite-dimensional representations of the orthosymplectic superalgebra OSP(3,2)

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Abstract
The shift operator technique is used to give a complete analysis of all finite- and infinite-dimensional irreducible representations of the orthosymplectic superalgebra osp(3,2). For all cases, the star or grade star conditions for the algebra are investigated. Only two finite-dimensional representations are grade star representations, if the representation space is required to be a graded Hilbert space. When the even part is so(3)⊕sp(2,R)≊su(2)⊕su(1,1), an infinite class of infinite-dimensional star representations is found. One of them can be realized in terms of two-valued functions of a complex variable. This representation reduces to the sum of two metaplectic representations of sp(2). We show that it is precisely this ‘‘metaplectic representation for osp(3,2)’’ which gives the spin-energy eigenstates for the one-dimensional harmonic oscillator with spin 1/2 states.