BORN RECIPROCITY IN TWO DIMENSIONS: QUAPLECTIC GROUP SYMMETRY, BRANCHING RULES AND STATE LABELLING

Abstract

We consider the continuous symmetry group underlying Born's reciprocity principle, namely the so-called quaplectic group, the semidirect product of time-space-energy-momentum coordinate transformations with the Weyl-Heisenberg group. In two dimensional Minkowski space this group is Q(1, 1) ≅ U(1, 1) ⋉ H(2), or in Euclidean space Q(2) ≅ U(2) ⋉ H(2). For the 'scalar' system in the sense of induced representations, unitary irreducible representations are carried on a Fock space equivalent to that used by Schwinger as a model of the SU(2) angular momentum algebra, or by Holman and Biedenharn as a model of SU(1,1). Using this construction we consider the branching rules and state labelling problem for the reduction of Q(2) and Q(1, 1) to the 'physical' Euclidean and Poincaré subalgebras, respectively. The results serve to illustrate the difficulties of any consideration of Born reciprocity as an extended symmetry principle of nature.