Talk:Dirac–von Neumann axioms

Errors in this page

This article contains no precise reference to where this list of axioms actually appears. I don't think this is just a citation issue -- I think the axioms contain errors as they are stated. In the cited references, both Dirac and Von Neumann explicitly contradict the axioms as they are stated.

I have checked all the references in the article, and none states these axioms anywhere. The other references, Strocchi and Takhtajan, state axioms different and inequivalent to these, in the language of C* algebras (an inequivalent extension of the Hilbert space construction).

As such, I believe this page should be deleted. There are two different rigorous formalisms which do what this article attempts: C* algebras, based on Von Neumann's approach, and rigged Hilbert spaces, based on Dirac's. Those have their own pages already.

Here are some issues with the current presentation. The list is not exhaustive.

It is stated that the dimension of the Hilbert space should be countably infinite. That's not correct, because finite-dimensional Hilbert spaces are clearly admissible (e.g. a single spin-1/2 particle, whose description is two dimensional). So this statement should probably be replaced with the statement that the dimension of the space is countable.

But Dirac explicitly rejects that proposal in his textbook, by introducing the Dirac delta distribution (`delta function'). To see that the delta function implies an uncountable dimension of the Hilbert space, simply note that it enumerates points on the real interval. This is the definition of uncountability.

The very idea that there can be a Dirac--Von Neumann set of axioms seems to be flawed from the outset, given that the two authors explicitly define contradictory approaches. To see this, see Von Neumann's introductory remarks in his book, which state that Dirac's approach involves `mathematical fictions' (e.g. delta functions).

In fact, the disagreement between the two, and the different approaches to axiomatisation that followed, constitutes a large field of enquiry in the philosophy of physics. See for example [1] which is entirely devoted to this question. The existence of the current page seems to ignore and contradict that century's worth of work.

It is true that some references, such as those on this page not written by Dirac or Von Neumann, do attribute sets of axioms to the combination of Dirac and Von Neumann. This would appear to be something of a misattribution. Both the references here in fact refer to C* algebras, deriving from Von Neumann and contradicting Dirac. In any case, they do not list the axioms stated above. 151.38.184.129 (talk) 12:48, 21 December 2024 (UTC)