Representations of the full inhomogeneous lorentz group

Abstract

The unitary representations of the restricted inhomogeneous Lorentz group derived by Shaw is extended to the full group. It is found that a single irreducible representation admits a Wigner time inversion and by adjoining two irreducible representations with both signs of the energy provides a representation of the full group where both the time inversion and space inversion are represented by antiunitary transformations. The Case-Jehle two-component wave equation for a spin-1/2 massive particle is rederived and generalized to arbitrary spin. By considering a direct sum (m, 0) ⊗D(s,0)(Λ) ⊗ (m, 0) ⊗D(0,s)(Λ) with the same sign of the energy, the representation space admits a Wigner time inversion and a unitary space inversion. By invoking Foldy’s condition of strong covariance, it is found necessary to include both signs of the energy resulting in a unitary representation of the full group and a transformation of charge conjugation. The relation of the present work with others is discussed.