Skip to main content
Physics

You are not logged in. Your edit will be placed in a queue until it is peer reviewed.

We welcome edits that make the post easier to understand and more valuable for readers. Because community members review edits, please try to make the post substantially better than how you found it, for example, by fixing grammar or adding additional resources and hyperlinks.

Required fields*

7
  • 1
    $\begingroup$ von Neumann? Density matrices? $\endgroup$ Commented Jan 31, 2023 at 20:37
  • $\begingroup$ @CosmasZachos No, my understanding is that what I am calling the "von Neumann formulation" goes through a generalized spectral theorem which lets us make sense of an eigendecomposition of unbounded operators (in terms of some integral over a projection-valued measure?). This is strikingly different than simply admitting the "eigenfunctions" of such operators which exist in a space conjugate to the nuclear space of our Hilbert space as in the rigged Hilbert space formalism. $\endgroup$ Commented Jan 31, 2023 at 20:56
  • 3
    $\begingroup$ The approach based on the rigged Hilbert space structure requires more mathematical hypotheses, and it is much more delicate to rigorously handle, than the von Neumann formulation. For that reason I strongly prefer the latter. Physically speaking, in all concrete situations, they are equivalent. I think that nobody uses the rigged H space for rigorous computations: an impossible heavy work to obtain formally evident results. $\endgroup$ Commented Jan 31, 2023 at 21:51
  • $\begingroup$ Personally, I sometimes pretend to use the rigged H space approch and finally I rigorously prove the found result with the vN formulation which is quite easy to use once guessed the formal result. I think this is the only safe way to exploit the rigged H. space formulation... $\endgroup$ Commented Jan 31, 2023 at 21:56
  • $\begingroup$ Thanks for your response Prof. Moretti. Is it correct to say that the rigged H space formulation is useful in the manner you describe because it effectively lets us make precise sense of all the crazy "momentum eigenstates" etc. that we take for granted in a "physics textbook"? @ValterMoretti $\endgroup$ Commented Jan 31, 2023 at 22:04