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.

  • 1
    $\begingroup$ 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, 2021 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, 2021 at 17:32
  • $\begingroup$ @CosmasZachos Yes. Thank you very much for clarifying. Can you also recommend bibliography? $\endgroup$
    – Dayzk
    Nov 16, 2021 at 0:27

1 Answer 1


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".

  • $\begingroup$ Thank you! it helped me to have things a little clearer :) $\endgroup$
    – Dayzk
    Nov 17, 2021 at 9:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.