The Z 2 × Z 2 -graded Lie superalgebra p s o ( 2 m + 1 | 2 n ) and new parastatistics representations

Source abstract

Abstract When the relative commutation relations between a set of m parafermions and n parabosons are of ‘relative parafermion type’, the underlying algebraic structure is the classical orthosymplectic Lie superalgebra (LSA) o s p ( 2 m + 1 | 2 n ) . The relative commutation relations can also be chosen differently, of ‘relative paraboson type’. In this second case, the underlying algebraic structure is no longer an ordinary LSA, but a Z 2 × Z 2 -graded LSA, denoted here by p s o ( 2 m + 1 | 2 n ) . The identification of this new algebraic structure was performed by Tolstoy, amongst others. In the present paper, we investigate the subalgebra structure of p s o ( 2 m + 1 | 2 n ) . This allows us to study the parastatistics Fock spaces for this new set of m + n para-operators, as they correspond to lowest weight representations of p s o ( 2 m + 1 | 2 n ) . Our main result is the construction of these Fock spaces, with a complete labeling of the basis vectors and an explicit action of the para-operators on these basis vectors.