Representations of the orthosymplectic Lie superalgebra osp(1|4) and paraboson coherent states

Soon after parastatistics has been introduced [1], it was discovered that it has a deep algebraic structure. It turned out that any n pairs of parafermion operators generate the simple Lie algebra so(2n + 1) [2, 3], and n pairs of paraboson creation and annihilation operators b 1 , . . . , bn generate a Lie superalgebra [4], isomorphic to one of the basic classical Lie superalgebras in the classification of Kac [5], namely to the orthosymplectic Lie superalgebra osp(1|2n) [6]. Actually, the paraboson operators were introduced earlier by Wigner [7] in a search of the most general commutation relations between the position operator q̂ and the momentum operator p̂ of a onedimensional oscillator, so that the Heisenberg equations are compatible with Hamilton’s equations. The operators p̂, q̂ turned out to generate osp(1|2) and Wigner was the first who found a class of (infinite-dimensional) representations of a Lie superalgebra [8]. Later on the results of Wigner gave rise to more general quantum systems (see [9] for references in this respect and for a general introduction to parastatistics), and in particular to Wigner quantum systems introduced by Palev [8, 10]. However only quite recently, the paraboson Fock spaces for the Lie superalgebra osp(1|2n) were constructed [11]. These are lowest weight representations V (p) characterized by a positive parameter p, called the order of the statistics. In [11], an explicit basis for the representation spaces V (p) is introduced with the matrix elements of these representations. Because of the computational difficulties in the construction of the paraboson Fock spaces the paraboson coherent states (eigenstates of paraboson operators) were constructed only for one pair of paraboson operators [12]. For this single-mode case, the results are quite complete: in terms of the paraboson representations [13], the coherent states and an expression for the resolution of the identity operator was given [14]. Note that, again only for this single-mode case, an alternative approach using Macfarlane’s construction [15] of Green’s ansatz has more recently been used [16, 17] to construct paraboson coherent states (though this time not in irreducible representations). In the present paper we use the results of [11] for the n = 2 case in order to obtain “coherent state” representations of two pairs of paraboson operators b 1 , b 2 . Coherent states play vital roles [18–21] in many contexts such as quantum optics, semiclassical quantization of systems with spin degrees of freedom, construction of quantum mechanical path integrals, the geometric quantization of coadjoint orbits, and so on. One specific motivation of our study lies in the realm of