Operator Treatment of the Gel'fand-Naimark Basis for SL (2, C)

Abstract

An operator form is developed to treat the Gel'fand-Naimark z basis for the homogeneous Lorentz group. It is shown that the operator Z with eigenvalues z is a definite operator-valued function of the generators of SL (2, C). A unified formulation of the unitary representations of the Lorentz group is obtained in a Hilbert space endowed with an affine metric operator G whose functional dependence on the generators is derived explicitly. The Dirac bra-ket formalism is extended by making a distinction between covariant and contravariant state vectors. The matrix elements of G are shown to coincide with the intertwining operator of Gel'fand and co-workers. The principal series, the supplementary series, and the two kinds of integer point representations are unified by means of a single scalar product involving the metric operator.