Infinite Irreducible Representations of the Lorentz Group
Abstract
It is shown that corresponding to every pair of complex numbers kappa , kappa ^{*} for which 2(kappa kappa ^{*}) is real and integral, there exists, in general, one irreducible representation germ{L}_{kappa,kappa*} of the Lorentz group. However, if 4kappa , 4kappa ^{*} are both real and integral there are two representations germ{D}_{kappa,kappa*}^{+} and germ{D}_{kappa,kappa*}^{} associated to the pair (kappa , kappa^{*}). All these representations are infinite except germ{D}_{kappa,kappa*}^{} which is finite if 2kappa , 2kappa ^{*} are both integral. For suitable values of (kappa , kappa ^{*}), germ{D}_{kappa,kappa*} or germ{D}_{kappa,kappa*}^{+} is unitary. germ{U} and germ{B} matrices similar to those given by Dirac (1936) and Fierz (1939) are introduced for these infinite representations. The extension of Dirac's expansor formalism to cover halfintegral spins is given. These new quantities, which are called expinors, bear the same relation to spinors as Dirac's expansors to tensors. It is shown that they can be used to describe the spin properties of a particle in accordance with the principles of quantum mechanics.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 May 1947
 DOI:
 10.1098/rspa.1947.0047
 Bibcode:
 1947RSPSA.189..372H