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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 ubiq

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 https://ntrs.nasa.gov/citations/19930093939

techwiz137 3y ago
I think that without their contributions back then, we wouldn't have the computers of today.
orra 3y ago
It's an interesting question but matrix mechanics is already a thing. https://en.wikipedia.org/wiki/Matrix_mechanics 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.
eigenket 3y ago
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 3y ago
  I wonder if there is a wave formulation for LLM's and transformers in general?
  - eigenket 3y ago
    You can probably write down a differential equation which models them but I doubt such a thing would be particularly interesting.
    - ImHereToVote 3y ago
      Perhaps neat to visualize.
  - mikk14 3y ago
    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/b2b3e1d9840eba17ad9bbf073e009afe-Paper-Conference.pdf https://proceedings.neurips.cc/paper_files/paper/2023/file/b...
    - api 3y ago
      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..."
      - 3y ago
lisper 3y ago
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.
- weinzierl 3y ago
  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.
- eigenket 3y ago
  This page has lecture notes by Aaronson on exactly that topic https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html
- prof-dr-ir 3y ago
  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 https://news.ycombinator.com/item?id=38255476
  - nabla9 3y ago
    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.
prof-dr-ir 3y ago
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 http://www.psiquadrat.de/downloads/bornjordan1925.pdf
monktastic1 3y ago
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.

techwiz137 3y ago

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

orra 3y ago

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

eigenket 3y ago

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 3y ago
I wonder if there is a wave formulation for LLM's and transformers in general?
- eigenket 3y ago
  You can probably write down a differential equation which models them but I doubt such a thing would be particularly interesting.
  - ImHereToVote 3y ago
    Perhaps neat to visualize.
- mikk14 3y ago
  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/b2b3e1d9840eba17ad9bbf073e009afe-Paper-Conference.pdf https://proceedings.neurips.cc/paper_files/paper/2023/file/b...
  - api 3y ago
    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..."
    - 3y ago

lisper 3y ago

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.

weinzierl 3y ago
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.
eigenket 3y ago
This page has lecture notes by Aaronson on exactly that topic https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html
prof-dr-ir 3y ago
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 https://news.ycombinator.com/item?id=38255476
- nabla9 3y ago
  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.

prof-dr-ir 3y ago

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

monktastic1 3y ago

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.