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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.

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    $\begingroup$ 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. $\endgroup$
    – Community Bot
    Nov 15 '21 at 15:36
  • $\begingroup$ 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? $\endgroup$ Nov 15 '21 at 17:32
  • $\begingroup$ @CosmasZachos Yes. Thank you very much for clarifying. Can you also recommend bibliography? $\endgroup$
    – Dayzk
    Nov 16 '21 at 0:27
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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".

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  • $\begingroup$ Thank you! it helped me to have things a little clearer :) $\endgroup$
    – Dayzk
    Nov 17 '21 at 9:36

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