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

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 and on PhysicsOverflow.

• I agree with the entire quote, not just the bolded part. – Nik Weaver Dec 11 '19 at 12:12
• I do not agree closing. There is a well-established notion of what real physics is (e. g. "something for which physics Nobel prize can be awarded" or "predicting or explaining experimentally observable phenomena") as well as some common notion of what a real contribution of functional analysis would be. At least, "realness" defines a rather obvious order in both aspects, and hence this question is as good as asking for the strongest known version of a theorem. I would very much like to see answers of the form "a (non-trivial) theorem A informed prediction of phenomenon B", if possible. – Kostya_I Dec 12 '19 at 12:09
• ... most of answers so far are rather weak in this sense, they are advanced versions of "look, you don't need maths IRL but it's helps you to learn to organize your thoughts" cocktail party Maths apology. Closing the question that has chances to get real answers is a pity. – Kostya_I Dec 12 '19 at 12:15
• @Kostya_I Theorems don't make predictions - models do. The hard part about building mathematical models is choosing the right definitions, e.g. Einstein metrics on manifolds or Lie group representations on Hilbert spaces - this is what you need to do calculations. If those calculations consistently agree with experiment then the model is accepted - nobody outside of mathematics could care less whether they are properly justified by theorems. Clarifying and validating definitions (with theorems!) is the primary purpose of mathematical work - that's a boast, not an apology. – Paul Siegel Dec 13 '19 at 0:54
• @PaulSiegel, examples of theorems that do make predictions are abundant. Wigner's theorem on classification of representations of Poincaré group was proven with the purpose of classifying particles one can expect to discover. The behavior of Fermi-Pasta-Ulam chain was a complete mystery before KAM theorem came along. Spin-statistics theorem... Index theorem... Also, many theorems just underscore and organize general features of calculations, so I do not agree with your countraposing the two. – Kostya_I Dec 13 '19 at 7:41

I'm can't claim to have studied the relevant history in a lot of detail, but count me a skeptic of Landsman's claim. Let's take this little paper and the companion that it cites as a test case, which I hope we can all agree is "real physics". The authors are clearly well versed in the calculus of variations and the representation theory of Lie groups. Both of these subjects are heavily intertwined with functional analysis - functional analysis is even foundational for the former. Are we to believe that these physicists were entirely ignorant of the subject? Or is the argument that functional analysis only influenced them indirectly through its contact with those mathematical applications?

I think Landsman's argument makes an error common among pure mathematicians about how mathematics is actually applied to the sciences. We tend to think about theorems, because those are the main objects of study in our work, but for consumers of mathematics it is the definitions that are important. The role of theorems is to validate the correctness and importance of definitions, and sometimes provide tools for manipulating them. The definitions of functional analysis - (un)bounded linear operators, Hilbert spaces, states, and so on - appear all over the place in quantum mechanics. And many of the big open problems in theoretical physics call primarily for definitions rather than theorems: Is there a measure space on which path integrals make sense? What is the correct notion of Dirac operator on the loop space of a manifold? Is there a gauge theory which includes both gravity and the standard model? And so on.

• I agree with the importance of definitions. For a recent example see this article arxiv.org/abs/1911.07895 by Binder and Rychkov where they establish some theorems but perhaps more importantly give definitions of QFT models with $N$ component valued fields even if $N$ is not a positive integer. BTW the big open problem "Is there a measure spece on which path integrals make sense" has a solution: for Euclidean path integrals on flat space, the measure space is the space of temperate distributions $\mathcal{S}'(\mathbb{R}^d)$. – Abdelmalek Abdesselam Dec 11 '19 at 18:10
• +1 for the first two sentences of the second paragraph alone. – Noah Schweber Dec 11 '19 at 19:18
• The fact that lots of physicists are confused by the notion of the spectrum in functional analysis stems from the fact that most of them took at least one quantum mechanics course in which the spectrum of a linear operator on Hilbert space was defined (probably correctly, at least in context). That physics professors and textbook authors view this formalism as a foundational topic in the field - and thus most physicists are even aware of its existence - is in my mind evidence of the influence of functional analysis. – Paul Siegel Dec 12 '19 at 12:35
• @FrancoisZiegler For better or for worse, it’s exactly the other way around. Physicists used (and use) these concepts well before they have been stress-tested from a mathematics point of view. The math has almost always, to my knowledge, come later. As I said, it’s somewhat impressive that this hasn’t led to more problems. – Aaron Bergman Dec 12 '19 at 13:00
• @TimothyChow Actually, I was referring specifically to functional analysis. In other areas, particularly geometry, I would say that the subtle issues brought to the fore by mathematicians have proved to be very important — understanding global effects, for example. For whatever reason, to my knowledge, this just hasn’t been the case for analysis. I don’t know why. Back when I did this for a living, I would worry from time to time that QFT has not been made rigorous, and maybe that means there is something important we’re missing there. – Aaron Bergman Dec 13 '19 at 1:27

This reminds me the following anecdote. K. Friedrichs once met Heisenberg on a conference. He thanked Heisenberg for creation of quantum mechanics which benefited mathematics so much, and added:

"But mathematicians gave much in return."

Heisenberg: "What?"

Friedrichs: "For example, von Neumann explained the difference between symmetric and self-adjoint operators".

Heisenberg: "And what is the difference?"

Reference: J. Horwath, ed. A panorama of Hungarian mathematics, in the twentieth century, I, Springer 2006, Page 227; he refers to P. Lax, Func. Anal., John Willey, 2002.

• This answer also demonstrates that there is a difference between asking what mathematicians have contributed and what functional analysis has contributed. (Unless one insists on defining "functional analysis" as something created solely by mathematicians.) – Timothy Chow Dec 11 '19 at 15:52
• Heisenberg couldn’t even be bothered to use the sharp bound (ℏ/2) in his own inequality. – Francois Ziegler Dec 12 '19 at 19:13
• I would be very interested to know if there are actually situations where a physicist not knowing the difference between symmetric and self-adjoint could go astray. – Kostya_I Dec 13 '19 at 11:20
• @Kostya_I: I agree, this would be interesting. But difficult to find. Good physicists usually have very strong intuition which guards them from mistakes, and as the same time look down at mathematical niceties. – Alexandre Eremenko Dec 13 '19 at 14:42

As jjcale mentions in a comment, the index of a Fredholm operator is very important in physics. One way to define the Chern number of a topological insulator is in terms of the index of a Fredholm operator, as explained in [1].

There is also the concept of an index of a pair of projections. This is seen a lot recently in physics papers, for example in [2]. That paper uses as a reference on the index of certain pairs of projections a paper in the Journal of Functional Analysis [3].

For physics published this year, see [4]. That paper discusses trace class operators and cites a text in functional analysis.

[1] Bellissard, Jean, Andreas van Elst, and Hermann Schulz‐Baldes. "The noncommutative geometry of the quantum Hall effect." Journal of Mathematical Physics 35.10 (1994): 5373-5451.

[2] Akagi, Yutaka, Hosho Katsura, and Tohru Koma. "A New Numerical Method for Topological Insulators with Strong Disorder." Journal of the Physical Society of Japan 86.12 (2017): 123710.

[3] Avron, J., Ruth Seiler, and Barry Simon. "The index of a pair of projections." Journal of Functional Analysis 120.1 (1994): 220-237.

[4] Zhi Li and Roger S. K. Mong, "A Local formula for the Z_2 invariant of topological insulators" Phys. Rev. B 100, 205101 – Published 4 November 2019.

• I think this is the best answer yet. – Nik Weaver Dec 13 '19 at 2:42

It is probably dangerous to answer without reading Landsman’s whole paper (and the question seems likely to be closed as “opinion-based”), but I’ll record my first reaction as much like lcv’s (a), namely, it sounds a little bit strawman-ish to separate the two and then pit one (FA) against the other (QM).

Footnotes (by Born) on pp. 583, 585 of the famous 1926 Dreimännerarbeit (see commented translation pp. 351, 352) immediately identified Hilbert’s work on linear operators as the correct framework for QM, and are a literal blueprint for its extension to unbounded operators by von Neumann. If this, and Weyl’s group reformulation of $$[P,Q]=I$$, are not contributions of functional analysis, then I don’t know what could be!

Consider also that, as far as could be determined at this question, the first two times our very phrase “linear algebra” (I don’t mean the thing) appears in the literature are (b,c)

1. In Hermann Weyl’s 1919 book on general relativity; the phrase didn’t catch on then.

2. In Hermann Weyl’s 1928 book on quantum mechanics: then it caught on.

So QM and FA both played a role in establishing the other as a field of study.

a. Which I would qualify to: FA (some say linear algebra) “is essentially the same thing” as those parts of QM we understand.

b. The phrase also appears, just once fleetingly, in Courant and Hilbert’s 1924 book, of which H. Hameka writes in his nice informed account, p. 11: “by a fortunate coincidence, linear algebra was the subject of the first chapter in the newly published book Methods of Mathematical Physics by Courant and Hilbert.” Born and Jordan cited it in 1925 (translation p. 279); like Landsman, I think the idea that physicists needed no one’s help came largely from Dirac’s failure to cite almost anyone.

c. Correction: I have since found the expression defined in Hellinger-Toeplitz (1910, p. 292): “diejenigen Partien der Algebra, die man etwa unter dem Sammelnamen einer linearen Algebra vereinigen könnte: bilineare Formen (Rangverhältnisse), Trägheitsgesetz der quadratischen Formen, Formenscharen (Elementarteilertheorie von Weierstraß, Kronecker, Frobenius usw.).” There remains that it only caught on after 1928. (The Dreimännerarbeit cites Hellinger along with Hilbert; Hellinger and Toeplitz were Born’s classmates, and after 1904 all 3 and Courant reunited in Göttingen as the “group from Breslau”.)

It seems to me that the fissures between (sub-)disciplines are somewhat more complex than the simple functional analysis vs. quantum theory dichotomy that Landsman emphasizes.

The study of foundational questions in physics is pursued by a comparatively small group of researchers. After all, physics has to deal with observations of the real world, and the real world is messy, dirty, and usually much more complicated than those few idealizations we are able to treat with anything approaching mathematical rigor. That is not to say that it's not important to maintain and develop the contact with the foundations; but it lies in the nature of the subject that most questions asked in physics are not foundational ones.

I am not an expert in these foundational questions, but from what I have seen, physicists who are can't be accused of not paying attention to the relevant mathematical literature. And I can't help sensing an undertone in what superficially sounds like "quantum theorists aren't paying attention to functional analysis" of "mainstream physicists aren't paying attention to their colleagues working on the foundations of quantum theory."

While it may well be valid, I think this sentiment disregards the many ways functional analysis has permeated physicists' frame of mind in the course of almost a century. A physicist's intuition about quantum theory is a linear algebra intuition. It is communicated to students from the very beginning. Homage is paid to Hilbert by even calling things "Hilbert spaces" that aren't. Though it isn't usually required by the curriculum, a good advisor will urge physics students to take courses in functional analysis if there is an opportunity. Hearing the "aaah!" from a student first realizing that the solutions of a Schr\"odinger equation can be organized into a vector space is memorable. There can be no question that learning this language, even at a fairly crude level, has been instrumental in better understanding quantum systems, even in very applied settings.

On the other hand, important new developments in functional analysis do find their way into the attention of physicists - and this goes beyond the focus on foundational questions. An example are free random variables, connected to elementary particle theory here, which also led to several further explorations in the aftermath. Granted, this didn't end up in a wholesale revolution, but it's another one of those puzzle pieces that give us an additional way to think about and understand elementary particle physics.

The Birman-Schwinger principle bounds the number of eigenvalues of a Schrödinger operator below certain level (in terms of an integral operator involving the potential and the resolvent of the Laplacian).

This has been used in many "real physics" articles, e. g. "Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter" by Lieb and Thirring.

On might argue that Birman-Schwinger principle does not constitute a very deep application of functional analysis, which is probably true, but nevertheless even its general rigorous formulation (and, of course, proof) requires some FA.