Precanonical quantization and the Schrödinger wave functional revisited

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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 $\gamma^0\varkappa\rightarrow\delta(\mathbf{0})$, where the constant $\varkappa$ 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.