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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.

© 1972 The American Institute of Physics

Received 06 July 1971
Published online 31 October 2003