[weinzierl]

I often wonder how much the fact that quantum mechanics' original formulation was in terms of a wave function and differential equations has to do with the ubiquity and importance of these topics at that particular era.

For example, Werner Heisenberg's doctoral thesis[1] arose from a contract of his doctor father Arnold Sommerfeld from a company that dealt with the channelling of the Isar river through the city of Munich. Very practical problems involving differential equations - kind of the bread and butter of physicists and engineers at the time.

What if quantum mechanics was found today in a world where the bread and butter has shifted to computer science, linear algebra and discrete math? Would we still end up with waves and differential equations, or would another formulation arise more naturally?

EDIT: I think a beautiful (but imperfect) example to illustrate this dichotomy in the ways of thinking is how the Bell inequality can be approached with photons and polarization or as a game. Thinking about Alice and Bob or polarized light, which would you prefer?

[1] https://ntrs.nasa.gov/citations/19930093939

[eigenket]

It was formulated at least twice in different versions. First by Heisenberg with his matrix mechanics and shortly afterwards by Schrödinger in terms of wave-functions.

There was considerable disagreement between the factions of physicists who favoured the different versions which essentially ended when after some considerable theoretical effort (mostly by Dirac) it was shown that the two pictures are exactly equivalent.

Physicists still use whichever formulation is most suitable for whatever problem they're trying to solve, for example if you're analysing the something where you care about a bunch of bound states like the simple harmonic oscillator or the hydrogen atom then the matrix picture tends to be easier to work with.

You are right that wave mechanics was more popular than matrix mechanics because physicists were already very familiar with wave methods.

[ImHereToVote]

I wonder if there is a wave formulation for LLM's and transformers in general?

[rfonseca]

This paper [1] models some simple (r) NN as ODEs, and uses ODE tools to train and for inference. It’s a start.

[1] https://arxiv.org/abs/1806.07366

[mikk14]

I don't know if this is exactly what you are thinking about, but there are some physicists working to understand what happens in transformers: https://proceedings.neurips.cc/paper_files/paper/2023/file/b...

[api]

Is it really true that we don't really understand why transformers work so well?

I mean we obviously understand how they work at a pure mechanical level, and we have this analogy with lookup (keys, queries, values) and "attention," but do we really get it? Can someone explain to me why that design works so much better than lots of other things like RNNs?

Or did we just tinker a lot (a method known as "graduate student descent") guided by mathematical hunches and loose analogies with biological brains until we found something that kinda worked?

It wouldn't be the first time. AFAIK we got the idea of wings from birds and figured out how to fly with them before we had a really solid fluid mechanical understanding of how and why wings work the way they do. We just thought "hmm so birds fly, so lets try stuff that looks a bit like that..."

[ImHereToVote]

We really don't have a mathematical theory for large complexity. We are kinda in alchemy stage for this "science".

[eigenket]

You can probably write down a differential equation which models them but I doubt such a thing would be particularly interesting.

[ImHereToVote]

Perhaps neat to visualize.

[prof-dr-ir]

I would like to stress what is also mentioned in a sibling comment: quantum mechanics' original formulation was not in terms of a wave function and differential equations.

Heisenberg's matrix mechanics was the first, and formulated entirely in terms of linear algebra. See for example the introduction of basic linear algebra techniques in the famous Bohr-Jordan paper from 1925:

http://www.psiquadrat.de/downloads/bornjordan1925.pdf

[orra]

It's an interesting question but matrix mechanics is already a thing. https://en.wikipedia.org/wiki/Matrix_mechanics

For me it's a reminder that physics describes how quantum systems evolve, but it doesn't (in a sense) tell us what those systems are. Are particles waves? Matrices? Excitations of a field?

Each of these descriptions works, so I can't exclusively say any one of them is what particles really are. Bring a macroscopic being is philosophically frustrating.

[monktastic1]

In case it's not obvious from what others have said, doing QM on observables with discrete spectra looks very different from that on continuous spectra. Different mathematical tools are helpful in each case.

[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.

[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.

[techwiz137]

I think that without their contributions back then, we wouldn't have the computers of today.