The Hamiltonian formulation of mechanics can be extended to field theory without distinguishing the time variable. Classical fields are sections of finite-dimensional bundles over the spacetime. The analogue of Poisson brackets is defined on differential forms. Its geometric prequantization implies a prequantum wave function valued in inhomogeneous differential forms. Its normalization introduces the metric structure. A generalization of quantum formalism with Clifford-algebra-valued operators and wave functions, and a Dirac-K\"ahler-like generalization of the Schr\"odinger equation, emerges as a result. As a simple example we discuss the case of quantum scalar field theory and the relation of our constructions to the standard quantum field theory. We also outline an application of the formalism to the problem of quantization of gravity and the description of quantum space-time.