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  Any real contribution of functional analysis to quantum theory as a branch of physics?

+ 2 like - 0 dislike

In the last paragraph of this last paper of Klaas Landsman, you can read:  

Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of Hilbert, von Neumann established the link between quantum theory and functional analysis that has lasted. Moreover, partly through von Neumann's own contributions (which are on a par with those of Bohr, Einstein, and Schrodinger), the precision that functional analysis has brought to quantum theory has greatly benefited the foundational debate. However, it is simultaneously a loser's history: starting with Dirac and continuing with Feynman, until the present day physicists have managed to bring quantum theory forward in utter (and, in my view, arrogant) disregard for the relevant mathematical literature. As such, functional analysis has so far failed to make any real contribution to quantum theory as a branch of physics (as opposed to mathematics), and in this respect its role seems to have been limited to something like classical music or other parts of human culture that adorn life but do not change the economy or save the planet. On the other hand, like General Relativity, perhaps the intellectual development reviewed in this paper is one of those human achievements that make the planet worth saving.

To balance this interesting debate, if there actually exists real reasons to disagree with above bolded sentence of Klaas Landsman, let me ask the following:  

What are the real contributions of functional analysis to quantum theory as a branch of physics?  

Here "real" should be understood in the sense underlying the above paragraph.  

This question was asked on physics.stackexchange on 2019-11-18.

asked Dec 9, 2019 in Theoretical Physics by Sébastien Palcoux (110 points) [ revision history ]
edited Dec 10, 2019 by Sébastien Palcoux

Functional analysis had a lot of impact on the mathematical physics side of quantum theory, but this is not what Landsman would call ''quantum theory as a branch of physics''.  Physicists usually have a weaker standard of rigor, hence can escape the need for functional analysis.

An exception may be the causal approach to quantum field theory, which provides a mathematically clean approach to perturbative renormalization free of infinities. But even that is not at the forefront of research in physics.

Please add the link to the Physics SE question.

@ArnoldNeumaier : Done!

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