Conformal algebra on Fock space and conjugate pairs of operators

Using the moment construction, we represent the generators of the conformal algebra as bilinear products of creation and annihiliation operators on the Fock space of the massless real scalar field in four dimensions. A complete set of one-particle eigenstates of the dilatation generator is given. Next, a complete set of one-particle eigenstates of the conformal generator is constructed in two distinct ways, once directly and once through an expansion in terms of dilatation eigenstates. The second approach uses an analytic continuation of the dilatation eigenvalue away from the real axis; the validity of the method is illustrated by the consistency with the first approach. Drawing upon this technique, we finally ponder the idea of building conjugates to the four components of the momentum operator by suitably modifying the action of the conformal generators on dilatation eigenstates. The construction of eigenstates of these new operators proceeds as for the conformal generator itself.