Projective representations of SL(3,C) in the Z-operator formalism

An operator formalism, previously developed to discuss the Gel'fand-Naimark z-basis for the homogeneous Lorentz group, is now generalized to treat the projective representations of SL(3,C). We find the projective ``position'' operators Z1, Z2, Z3, as well as their canonical conjugate ``momenta'' II1, II2, II3 which form the building blocks of the generators of SL(3,C). The z-representation, the states in which the ``position'' operators Zi are diagonal, has very simple global transformation properties. This representation also leads to a Hilbert space endowed with an affine metric G, which relates the ``covariant'' and ``contravariant'' states. All unitary representations are unified by means of a single scalar product in which the matrix elements of G play the role of an intertwining operator.