  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.
