  We study the conformally invariant quantum field theory in spaces of even
dimension D >= 4. The conformal transformations of current j_\mu and
energy-momentum tensor T_{\mu\nu} are examined. It is shown that the set of
conformal transformations of particular kind corresponds to the canonical
(unlike anomalous) dimensions l_j=D-1 and l_T=D of those fields. These
transformations cannot be derived by a smooth transiton from anomalous
dimensions. The structure of representations of the conformal group, which
correspond to these canonical dimensions, is analyzed, and new expressions for
the propagators < j_\mu j_\nu > and < T_{\mu\nu} T_{\rho\sigma}> are derived.
The latter expressions have integrable singularities. It is shown that both
propagators satisfy non-trivial Ward identities. The higher Green functions of
the fields j_\mu and T_{\mu\nu} are considered. The conformal QED and linear
conformal gravity are discussed. We obtain the expressions for invariant
propagators of electromagnetic and gravitational fields. The integrations over
internal photon and graviton lines are performed. The integrals are shown to be
conformally invariant and convergent, provided that the new expressions for the
propagators are used.
