[lisper]

Scott Aaronson has an insightful essay on this in chapter 9 of his excellent book, "Quantum Computing Since Democritus". The TL;DR is that quantum mechanics can be derived as a generalization of probability theory using the 2-norm instead of the 1-norm. In Aaronson's words:

"Quantum mechanics is what you would inevitably come up with if you started with probability theory, and then said, let's try to generalize it so that the numbers we used to call 'probabilities' can be negative numbers. As such, the theory could have been invented by mathematicians in the nineteenth century without any input from experiment. It wasn't, but it could have been."

Well worth a read. In fact, I'd say it's worth buying the book just for this one chapter.

[prof-dr-ir]

To counter your enthusiasm I must say that I rather disliked his reasoning. My problem is essentially that what Aaronson's calls "the theory" is a somewhat bastardized version of quantum mechanics that might suffice for quantum computing but, in my opinion, not for physics.

I discussed the difference earlier: https://news.ycombinator.com/item?id=38255476

[eigenket]

I think this is a very common view, and a somewhat mistaken one.

Quantum mechanics is used in very different ways by people in quantum information, condensed matter, many body physics, quantum field theory, nuclear physics and may other (sub)fields.

Of course it will be difficult for a quantum information theorist if they try to apply what they know directly to a hydrogen atom, but (speaking from experience) it will also be quite difficult for someone trained in what you call Hermitian quantum mechanics to directly apply what they know to quantum field theory, or quantum information or any other subfield that uses different language.

I strongly disagree with your summary "if you know Hermitian quantum mechanics then unitary quantum mechanics is conceptually straightforward. If you know unitary quantum mechanics then you will have a lot of new concepts and mathematics to learn before you understand hermitian quantum mechanics".

I challenge anyone trained in Hermitian quantum mechanics to make progress on (for example) proving or disproving the generalised quantum Stein's lemma, or any of the unsolved problems here https://arxiv.org/abs/2002.03233 using those methods.

[prof-dr-ir]

With conceptually straightforward I meant that the concepts are easy to pick up.

For example, the paper you cite is entirely understandable for anyone with some training in hermitian QM. In contrast, good luck trying to understand elementary concepts like the spectrum of the hydrogen atom or interference of matter waves from unitary QM.

Of course the field of quantum info has progressed enormously and has its own interesting challenges, for which hermitian QM is all but useless.

[eigenket]

I think maybe we have some difference in how we're talking about things.

Concepts like the spectrum of the hydrogen atom or interference phenomena aren't particularly difficult to understand conceptually: the Hamiltonian has some eigenvectors and eigenvalues, you use the Dirac equation and work them out. The "matter waves" interfere essentially in the same way that waves on the surface of a pond do.

The things that you're calling conceptual understanding I guess must be different to this: maybe something like detailed calculations of the structure of the spectrum?

[prof-dr-ir]

In my definition, someone cannot claim to know QM if they do not know (a) Heisenberg's uncertainty principle, (b) how to compute interference patterns or (c) how to compute the spectrum of hydrogen.

Aaronson claims that QM "can be derived" (quote from the original comment) are followed by an introduction to some aspects of QM that leads to strictly none of these things. That is why I am unhappy with it, and I still do not see why I am "somewhat mistaken" (quote from you).

In fact, I can go even further and say that (from a quick glance at least) in his whole book positions and momenta make no appearance, let alone the correspondence principle. (I do not even see hermitian operators!) Without them I just do not see any reasonable "derivation" of (or, more properly, argumentation for) what I call QM.

[eigenket]

Ah, I see. You've changed from talking about concepts and conceptual understanding to talking about computations.

I completely agree with you that from the perspective of a quantum information theorist computing the spectrum of the hydrogen atom is a rather complicated thing. I disagree wholeheartedly that this is part of the essence of quantum mechanics.

The hydrogen atom is one system, understanding conceptually that its behavior is governed by a self-adjoint operator and its spectrum is very relevant to the whole of quantum physics. Understanding exactly the details of the calculation I think are not. Especially because if you do the calculation within quantum mechanics without quantum field theory you will obtain a somewhat incorrect result anyway (you will miss interesting phenomena like Lamb shift).

Similarly interference is an interesting phenomenon that one needs to understand to understand quantum mechanics, but understanding the specific calculation of how interference makes nice patterns in some example setup isn't particularly enlightening.

I agree that the Heisenberg uncertainty principle is important, but it certainly be derived from Aaronson's point of view (e.g. the standard Robertson-Schrödinger inequality is easily obtained).

As an aside, I also think that self-adjoint operators and the correspondence principle are a fairly terrible way to think about observables in quantum mechanics. An obvious fact is that every measurement of (for example) the position of a particle has some unavoidable experimental error (our apparatus only has finite resolution) so the thing we actually measure in reality is some fuzzy observable which can not be represented as a self-adjoint operator. A POVM is a much more natural candidate (as a physicist it is natural to assume that the thing you get by adding some classical noise to your measurement is still a measurement).

[nabla9]

To counter your unenthusiasm, think about it from a mathematician's perspective. Mathematics of QM does not live in some separate corner created to do physics. The need for QM created short-lived confusion, now it's all embedded into a much larger coherent mathematical structure.

For a pure mathematician, quantum mechanics is a lovely introduction to Hilbert Spaces.

[abdullahkhalids]

That's only kind of true. Standard continuous-variable quantum mechanics has all sorts of consistency problems, and the only way to get reasonable predictions out is to paper over infinities and pretend they don't exist.

I know there are what are called C*-algebras, which help solve some of these issues, but I don't know anything about them. I do know that the Hilbert Space approach is not sufficient.

[ernesto95]

True, but from a mathematician's point of view, the theory quickly becomes complicated (and in some sense limited) if you really want to do things rigorously when working with continuous systems, something that does not happen with finite dimensional systems as the parent comment probably alludes to.

[eigenket]

This page has lecture notes by Aaronson on exactly that topic

https://www.scottaaronson.com/democritus/lec9.html

[weinzierl]

I was thinking about Aaronson when I added the CHSH game to the comment, because one of his essays (which I unfortunately could not find anymore) made that click for me.