# Can the Hermitian operator be related to state space to describe physical phenomena?

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.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Nov 15 '21 at 15:36
• You mean, how do you represent hermitian operators in Hilbert space in terms of coordinate space functions, or, even, phase-space functions serving as convolution kernels? Nov 15 '21 at 17:32
• @CosmasZachos Yes. Thank you very much for clarifying. Can you also recommend bibliography? Nov 16 '21 at 0:27

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 $$|\phi\rangle =H|\psi \rangle ~~~\leftrightarrow ~~~\phi(x)= \int \!\! dx' ~~h(x,x') \psi (x').$$