I am currently reading the quantum mechanics text by Ballentine and, over and over, arguments are made (e.g. Chapter 4.6 on constraining the wavevectors of free particles to be real) which rely on rigged Hilbert space as the "correct" context for a quantum theory. Is this generally accepted as true? I ask this because I have this vague notion that there exists another capitulation/mathematization of quantum theory which used operator theory/spectral theory/follows von Neumann, and it seems to me that this is a completely different approach than the rigged Hilbert space approach. Thus my question, overall, is whether there is consensus from mathematical physicists as to which is "most correct", or are they perhaps equivalent?
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1$\begingroup$ von Neumann? Density matrices? $\endgroup$– Cosmas ZachosCommented Jan 31, 2023 at 20:37
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$\begingroup$ @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. $\endgroup$– EE18Commented Jan 31, 2023 at 20:56
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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. $\endgroup$– Valter MorettiCommented Jan 31, 2023 at 21:51
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$\begingroup$ 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... $\endgroup$– Valter MorettiCommented Jan 31, 2023 at 21:56
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$\begingroup$ 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 $\endgroup$– EE18Commented Jan 31, 2023 at 22:04
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1 Answer
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I do not think there is a unique answer. It mostly depends on personal taste.
However, I think that almost all mathematical physicists agree on the fact that the QM 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. Which, on the other hand, can be immediately applied to more general quantum theories than QM, whereas this extension is not similarly easy for the rigged Hilbert space formulation.
Personally speaking, for that reason I strongly prefer the von Neumann framework (though I am conscious that it may be dangerous if used uncritically). In all my career, I never met a colleague who really used the rigged Hilbert space for rigorous computations.
However a distinction is necessary between Dirac's improper eigenvectors formalism in QM and the theory of rigged Hilbert spaces to formalize QM.
What is evident is the formal powerfulness of the non-rigorous Dirac formalism. That is much more powerful, for practical manipulations, than the rigorous and solid approach by von Neumann.
The rigged Hilbert space formulation, essentially due to Gelfand, is a fruitful attempt to demonstrate that Dirac's manipulations can be formulated into a rigorous setting.
However physicists are not interested in that rigorous re-formulation of something already evident, on the one hand.
On the other hand, mathematical physicists do not need it, because von Neumann formalism is already pretty enough.
For that reason the rigorous rigged Hilbert space formulation is a sort of strange animal to admire in the zoo of mathematics applied to physics. Its existence is a proof of the fact that when the physical ideas are really good, then they can be formulated in a rigorous mathematical framework (the converse fact is tragically false).
The safe view on the issue, in my honest opinion, is to use Dirac’s formalism to guess the physical result and, if one also needs a rigorous proof of that, to pass to the von Neumann formalism to consolidate the guessed result.
All that, as a byproduct, leads to an illustration of the difference between mathematical physics and theoretical physics. The former proceeds in terms of theorems, so that the mathematical rigour is strictly necessary; the latter mainly looks at the physical plausibility of achieved theoretical results, without paying much attention to mathematical rigour. Both use mathematics, but for the former it is a guide, for the latter it is a slave.
answered Jan 31, 2023 at 22:20
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2$\begingroup$ This answer is very enlightening to me! As a mathematician using some of these ideas from physics in application to number theory, naturally I don't mind a little more overhead... and, for me, various rigged-Hilbert-Space ideas (e.g., $L^2$ Sobolev spaces) in combination with the Stone-vonNeumann spectral theory of unbounded operators is the most explanatory for me. I've long suspected that physicists took "physical phenomena" as entirely adequate substitutes for "proof", and I can understand that! Using things outside their testable range is more risky. :) Thanks again for your explanation! $\endgroup$ Commented Feb 2, 2023 at 17:48
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$\begingroup$ Yes. The difference between mathematical physics and theoretical physics is that the former proceeds in terms of theorems, so that the mathematical rigour is a guide, the latter mainly looks at the physical plausibility of achieved results, without paying much attention to mathematical rigour. $\endgroup$ Commented Feb 2, 2023 at 18:04