Royal Society Publishing

Infinite Irreducible Representations of the Lorentz Group

Harish-Chandra

Abstract

It is shown that corresponding to every pair of complex numbers $\kappa $, $\kappa ^{\ast}$ for which 2($\kappa -\kappa ^{\ast}$) is real and integral, there exists, in general, one irreducible representation $\germ{L}_{\kappa,\kappa ^{\ast}}$ of the Lorentz group. However, if 4$\kappa $, 4$\kappa ^{\ast}$ are both real and integral there are two representations $\germ{D}_{\kappa,\kappa ^{\ast}}^{+}$ and $\germ{D}_{\kappa,\kappa ^{\ast}}^{-}$ associated to the pair ($\kappa $, $\kappa ^{\ast}$). All these representations are infinite except $\germ{D}_{\kappa,\kappa ^{\ast}}^{-}$ which is finite if 2$\kappa $, 2$\kappa ^{\ast}$ are both integral. For suitable values of ($\kappa $, $\kappa ^{\ast}$), $\germ{D}_{\kappa,\kappa ^{\ast}}$ or $\germ{D}_{\kappa,\kappa ^{\ast}}^{+}$ 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 half-integral 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.

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