Schr\"odinger Functional of a Quantum Scalar Field in Static Space-Times from Precanonical Quantization

The functional Schr\"odinger representation of a scalar field on an $n$-dimensional static space-time background is argued to be a singular limiting case of the hypercomplex quantum theory of the same system obtained by the precanonical quantization based on the space-time symmetric De Donder-Weyl Hamiltonian theory. The functional Schr\"odinger representation emerges from the precanonical quantization when the ultraviolet parameter $\varkappa$ introduced by precanonical quantization is replaced by $\underline{\gamma}{}_0\delta^\mathrm{inv}(\mathbf{0})$, where $\underline{\gamma}{}_0$ is the time-like tangent space Dirac matrix and $\delta^\mathrm{inv}(\mathbf{0})$ is an invariant spatial $(n-1)$-dimensional Dirac's delta function whose regularized value at $\mathbf{x}=\mathbf{0}$ is identified with the cutoff of the volume of the momentum space. In this limiting case, the Schr\"odinger wave functional is expressed as the trace of the product integral of Clifford-algebra-valued precanonical wave functions restricted to a certain field configuration and the canonical functional derivative Schr\"odinger equation is derived from the manifestly covariant Dirac-like precanonical Schr\"odinger equation which is independent of a choice of a codimension-one foliation.