  Polymomentum canonical theories, which are manifestly covariant
multi-parameter generalizations of the Hamiltonian formalism to field theory,
are considered as a possible basis of quantization. We arrive at a
multi-parameter hypercomplex generalization of quantum mechanics to field
theory in which the algebra of complex numbers and a time parameter are
replaced by the space-time Clifford algebra and space-time variables treated in
a manifestly covariant fashion. The corresponding covariant generalization of
the Schroedinger equation is shown to be consistent with several aspects of the
correspondence principle such as a relation to the De Donder-Weyl
Hamilton-Jacobi theory in the classical limit and the Ehrenfest theorem. A
relation of the corresponding wave function (over a finite dimensional
configuration space of field and space-time variables) to the Schroedinger wave
functional in quantum field theory is examined in the ultra-local
approximation.
