Can space-time, in which phenomena occur, and the space of states in which phenomena are described by means of the Hermitian operator be related? I suspect it is because the hermetic operator is built on linear spaces that associate real states with vectors, but I'm not sure.
The standard Dirac bra-ket coordinate picture is $$ H=\iint\!\! dx dx'~|x\rangle \langle x|H|x'\rangle\langle x'| ~~~\leadsto , $$ so that, considering $\langle x|\psi\rangle =\psi(x)$ and $h(x,x')= \langle x|H|x'\rangle$, you readily have $$ |\phi\rangle =H|\psi \rangle ~~~\leftrightarrow ~~~\phi(x)= \int \!\! dx' ~~h(x,x') \psi (x'). $$
That is, you represent Hilbert-space states and operators through coordinate-space functions and convolutions.
Beyond this, there is a much subtler and disparate formulation which maps Hilbert-space operators to phase-space functions, through the Wigner map. This undergirds a qualitatively, distinctly different formulation of QM, equivalent to the Hilbert space you are studying, but I suspect this outranges your scope. In this formulation, the phase-space convolution law is very-very-very different, and is called the "star product".