On the precanonical structure of the Schrödinger wave functional

We show that the Schrödinger wave functional may be obtained as the product integral of precanonical wave functions on the space of field and space-time variables. The functional derivative Schrödinger equation underlying the canonical field quantization is derived from the partial derivative covariant analogue of the Schrödinger equation, which appears in the precanonical field quantization based on the De Donder-Weyl generalization of the Hamiltonian formalism for field theory. The representations of precanonical quantum operators typically contain an ultraviolet parameter varkappa of the dimension of the inverse spatial volume. The transition from the precanonical description of quantum fields in terms of Clifford-valued wave functions and partial derivative operators to the standard functional Schrödinger representation obtained from canonical quantization is accomplished if frac{1}{varkappa}→ 0 and frac{1}{varkappa}γ₀ is mapped to the infinitesimal spatial volume element dmathbf{x}. Thus the standard QFT obtained via canonical quantization corresponds to the quantum theory of fields derived via precanonical quantization in the limiting case of an infinitesimal value of the parameter frac{1}{varkappa}.