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 bold 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.

  • 7
    $\begingroup$ I guess that the functional analysis role is to put all the arguments used in quantum mechanics on a rigorous mathematical setting. This is obviously very important, as without it, you may never know if you committed some sort of maths taboo based on your intuition. $\endgroup$ Nov 18, 2019 at 8:59
  • 3
    $\begingroup$ @Qmechanic How this post can be too broad as the expert Klaas Landsman states that there is no such real contribution? I wrote this post to give the opportunity to other experts which disagree with him to write what they consider as a real contribution in order to balance the debate. Are you Klaas Landsman? Could you please let this post open for few days? $\endgroup$ Nov 18, 2019 at 10:26
  • 1
    $\begingroup$ Functional analysis are used in many areas of quantum theory. Whether one wants to call them real contributions seems to be primarily opinion-based. Physicists are very good at inventing intuitive explanations for why their naive calculations fail. $\endgroup$
    – Qmechanic
    Nov 18, 2019 at 11:01
  • 2
    $\begingroup$ @lcv: his comment could itself be seen as contained in Landsman's view. I am looking for answers from people skeptic of this view. It is interesting (and perhaps revealer of something) that mathematicians on mathoverflow are much more loquacious on this question than physicists here on physics.stackexchange. $\endgroup$ Dec 11, 2019 at 21:03
  • 1
    $\begingroup$ @SebastienPalcoux Physicist's often come up with mathematics based on intuition to suit their needs even if they are not very well defined mathematically. The best example I can think of this is the Dirac Delta function which dirac made to fit a function that he needed. It took mathematicians almost 30 years to invent the theory of distributions to explain it. The point is that both works are very important as Dirac delta functions later found use in a lot of places. I think you will find this video useful youtube.com/watch?v=obCjODeoLVw. $\endgroup$ Dec 12, 2019 at 21:02