A class of unitary irreducible representations of the Lie superalgebra osp(1|2n)

Using the equivalence of the defining relations of the orthosymplectic Lie superalgebra osp(1|2n) to the defining triple relations of n pairs of parabose operators b± i we construct a class of unitary irreducible (infinite-dimensional) lowest weight representations V (p) of osp(1|2n). We introduce an orthogonal basis of V (p) in terms of Gelfand-Zetlin patterns, where the subalgebra u(n) of osp(1|2n) plays a crucial role and we present explicit actions of the osp(1|2n) generators. Following some physical ideas we construct a class of infinite dimensional unitary irreducible representations of the Lie superalgebra (LS) osp(1|2n) [1] in an explicit form. In 1953 Green [2] introduced the so called parabose operators (PBOs) b±j (j = 1, 2,...) satisfying [{bξj, bηk}, bǫl] = (ǫ − ξ)δjlbηk + (ǫ − η)δklbξj, (1) j, k, l ∈ {1, 2,...} and η, ǫ, ξ ∈ {+,−} (to be interpreted as +1 and −1 in the algebraic expres-sions ǫ − ξ and ǫ − η) as a generalization of the ordinary Bose operators. The Fock space V (p) of n pairs of PBOs is a Hilbert space with a vacuum |0〉, defined by means of (j, k = 1, 2,..., n) 〈0|0 〉 = 1, b−j |0 〉 = 0, (b±j) † = b∓j, {b−j, b+k}|0 〉 = p δjk |0〉, (2) and by irreducibility under the action of the algebra spanned by the elements b+j, b j (j = 1,..., n), subject to (1). The parameter p is referred to as the order of the paraboson system. However the structure of the parabose Fock space is not known, also a proper basis has not been introduced. We solve these problems using the relation between n pairs of PBOs and the defining relations of the LS osp(1|2n), discovered by Ganchev and Palev [3]. The orthosymplectic superalgebra osp(1|2n) [1] consists of matrices of the form 0 a a1at1 b c −at d −bt , (3) where a and a1 are (1 × n)-matrices, b is any (n × n)-matrix, and c and d are symmetric (n × n)-matrices. The even elements have a = a1 = 0 and the odd elements are those wit