On irreducible representations of the ultrahyperbolic BMS group Original Research Article

The ordinary Bondi–Metzner–Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space–times. As such, B is the best candidate for the universal symmetry group of General Relativity. However, in studying quantum gravity, space–times with signatures other than the usual Lorentzian one, and complex space–times, are frequently considered. Generalisations of B appropriate to these other signatures have been defined earlier. Here, the generalisation B(2,2) appropriate to the ultrahyperbolic signature (+,+,−,−) is described in detail, and the irreducible unitary representations (IRs) of B(2,2) are analysed. It is proved that all induced IRs of B(2,2) arise from IRs of compact “little groups”. These little groups, which are closed subgroups of K=SO(2)×SO(2), are classified here in detail, with particular attention paid to those of infinite order.