Precanonical quantization and the Schr\"odinger wave functional revisited

We address the issue of the relation between the canonical functional Schr"odinger representation in quantum field theory and the approach of precanonical field quantization proposed by the author, which requires neither a distinguished time variable nor infinite-dimensional spaces of field configurations. We argue that the standard functional derivative Schr"odinger equation can be derived from the precanonical Dirac-like covariant generalization of the Schr"odinger equation under the formal limiting transition ${\gamma }^0\mathrm{\varkappa }\to \delta (\mathbf{0})$, where the constant $\mathrm{\varkappa }$ naturally appears within the precanonical quantization as the inverse of a small “elementary volume” of space. We obtain a formal explicit expression of the Schr"odinger wave functional as a continuous product of the Dirac algebra valued precanonical wave functions, which are defined on the finite-dimensional covariant configuration space of the field variables and space-time variables.