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Their paper and this article both explain why this naive approach fails. Read at least one of them before assuming your simplistic and completely incorrect asse

Their paper and this article both explain why this naive approach fails. Read at least one of them before assuming your simplistic and completely incorrect assessment satisfies the requirements of the underlying physics. This idea and any such idea fails - that is why their work is worth understanding. Simply mapping one vector space to another of a different dimension doesn't satisfy other requirements that the physics requires. These authors aren't complete idiots.

kevinventullo 5y ago
The article takes a pretty simple view of these approaches. E.g. in the “too many dimensions” example, the problem is that they’re taking tensor products over R, when they should be taking tensor products over this matrix algebra. More generally, you can always obscure the language of complex numbers in real algebra terms (I.e. replace every instance of “C” with “R[x]/(x^2+1)”), so whatever they proved must make some assumptions about what exactly a “real-valued” formulation of quantum mechanics is.
- ChrisLomont 5y ago
  Look, I've a PhD in algebraic geometry, and a healthy dose of research level physics. I'm fully aware of the math side of such claims. I don't see this thread having any understanding of physics here, which is what this paper is about. When you write "when they should be" as if this naive (mis) understanding of the issues is somehow the correct formulation of the problem, it's not relevant. They address this in the paper. They take a standard set of rules for QM, replace a complex Hilbert space with a real Hilbert space, and ask can it reproduce QM. This includes any real valued structure underlying that space, including any of the field isomorphisms. They derive an experiment that can detect in our particular reality which case we live in. So far experiments rule out real valued QM, just like Bell's inequality allowed experimenters to rule out all possible hidden variable theories. So apparently there is more to the physics than this naive representation you claim is equivalent. So please stop with the "mathematically isomorphic as fields" argument. It's ruled out experimentally. This is much deeper, and appears fundamentally interwoven in reality, which is much cooler.
  - kevinventullo 5y ago
    Thank you for reading the paper. So, they’re asking if you can use real Hilbert spaces instead of complex. No cheating by using “real algebra”-valued Hilbert spaces. Okay, fine, but that’s assuming a whole lot about your model, so the question is fairly limited in scope, right? I dunno, to be honest I just don’t see what’s interesting about this pursuit at all, since complex numbers are far from the most exotic construction in theoretical physics.
    - ChrisLomont 5y ago
      > No cheating by using “real algebra”-valued Hilbert spaces No, read what I wrote, or read the paper. They showed that no "real algebra"-valued Hilbert space works. >I just don’t see what’s interesting about this pursuit at all Yes, because you don't care to see that your assumptions using naive math constructions are not applicable here. They didn't cheat and use circular arguments, they are not unaware of these naive math constructions, they didn't make assumptions that are not validated by physics. Doing something as trivial as you assume their work is would not get published in Nature. They far surpassed anything you've put up as some workaround or dumb loophole in their arguments. If you're not willing to follow an explanation, or to read and understand their paper, then please stop poo-pooing work you don't understand. Their work is actually a solid answer to a question that has been poked at by the best minds in physics since QM was founded over 100 years ago. It's a really deep and interesting result. If you want to see some of the work on this, and there is a lot, simply google "does quantum mechanincs require complex numbers," and you'll find a long history of trying to answer this question one way or another. And no, the answer is not simply replace C with R[z]/(z^2+1) and everything works.
      - kevinventullo 5y ago
        They showed that no "real algebra"-valued Hilbert space works. That is literally not true. The abstract clearly states that they only compare Real and Complex-valued Hilbert Spaces; they say nothing about general R-algebras. And of course they couldn’t prove what you claim because C itself is an R-algebra! I’m not claiming that they’re cheating or that what they did is trivial, just that 1) I don’t think they’ve answered the very broad question “Does quantum mechanics need imaginary numbers?” because as stated it’s sort of ill-posed and unfalsifiable. What they have answered is “Can this Hilbert space formulation of quantum mechanics with Postulates 1-4 work with real coefficients?” and the answer is no. The article itself points out non-Hilbert formulations that do in fact avoid complex numbers. Maybe this narrow question is interesting to physicists, I don’t know! 2) I don’t personally think it’s an interesting question, because I don’t think complex numbers are anywhere near the top of the list of unintuitive abstractions needed for modern physics or mathematics. Square root of -1 is a lot easier to swallow than even Cauchy Sequences or Dedekind cuts, much less I dunno, topoi of presheaves on the category of commutative von Neumann algebras of bounded operators (I wish I were making that up: https://aip.scitation.org/doi/abs/10.1063/1.4898185 https://aip.scitation.org/doi/abs/10.1063/1.4898185).

kevinventullo 5y ago

The article takes a pretty simple view of these approaches. E.g. in the “too many dimensions” example, the problem is that they’re taking tensor products over R, when they should be taking tensor products over this matrix algebra. More generally, you can always obscure the language of complex numbers in real algebra terms (I.e. replace every instance of “C” with “R[x]/(x^2+1)”), so whatever they proved must make some assumptions about what exactly a “real-valued” formulation of quantum mechanics is.

ChrisLomont 5y ago
Look, I've a PhD in algebraic geometry, and a healthy dose of research level physics. I'm fully aware of the math side of such claims. I don't see this thread having any understanding of physics here, which is what this paper is about. When you write "when they should be" as if this naive (mis) understanding of the issues is somehow the correct formulation of the problem, it's not relevant. They address this in the paper. They take a standard set of rules for QM, replace a complex Hilbert space with a real Hilbert space, and ask can it reproduce QM. This includes any real valued structure underlying that space, including any of the field isomorphisms. They derive an experiment that can detect in our particular reality which case we live in. So far experiments rule out real valued QM, just like Bell's inequality allowed experimenters to rule out all possible hidden variable theories. So apparently there is more to the physics than this naive representation you claim is equivalent. So please stop with the "mathematically isomorphic as fields" argument. It's ruled out experimentally. This is much deeper, and appears fundamentally interwoven in reality, which is much cooler.
- kevinventullo 5y ago
  Thank you for reading the paper. So, they’re asking if you can use real Hilbert spaces instead of complex. No cheating by using “real algebra”-valued Hilbert spaces. Okay, fine, but that’s assuming a whole lot about your model, so the question is fairly limited in scope, right? I dunno, to be honest I just don’t see what’s interesting about this pursuit at all, since complex numbers are far from the most exotic construction in theoretical physics.
  - ChrisLomont 5y ago
    > No cheating by using “real algebra”-valued Hilbert spaces No, read what I wrote, or read the paper. They showed that no "real algebra"-valued Hilbert space works. >I just don’t see what’s interesting about this pursuit at all Yes, because you don't care to see that your assumptions using naive math constructions are not applicable here. They didn't cheat and use circular arguments, they are not unaware of these naive math constructions, they didn't make assumptions that are not validated by physics. Doing something as trivial as you assume their work is would not get published in Nature. They far surpassed anything you've put up as some workaround or dumb loophole in their arguments. If you're not willing to follow an explanation, or to read and understand their paper, then please stop poo-pooing work you don't understand. Their work is actually a solid answer to a question that has been poked at by the best minds in physics since QM was founded over 100 years ago. It's a really deep and interesting result. If you want to see some of the work on this, and there is a lot, simply google "does quantum mechanincs require complex numbers," and you'll find a long history of trying to answer this question one way or another. And no, the answer is not simply replace C with R[z]/(z^2+1) and everything works.
    - kevinventullo 5y ago
      They showed that no "real algebra"-valued Hilbert space works. That is literally not true. The abstract clearly states that they only compare Real and Complex-valued Hilbert Spaces; they say nothing about general R-algebras. And of course they couldn’t prove what you claim because C itself is an R-algebra! I’m not claiming that they’re cheating or that what they did is trivial, just that 1) I don’t think they’ve answered the very broad question “Does quantum mechanics need imaginary numbers?” because as stated it’s sort of ill-posed and unfalsifiable. What they have answered is “Can this Hilbert space formulation of quantum mechanics with Postulates 1-4 work with real coefficients?” and the answer is no. The article itself points out non-Hilbert formulations that do in fact avoid complex numbers. Maybe this narrow question is interesting to physicists, I don’t know! 2) I don’t personally think it’s an interesting question, because I don’t think complex numbers are anywhere near the top of the list of unintuitive abstractions needed for modern physics or mathematics. Square root of -1 is a lot easier to swallow than even Cauchy Sequences or Dedekind cuts, much less I dunno, topoi of presheaves on the category of commutative von Neumann algebras of bounded operators (I wish I were making that up: https://aip.scitation.org/doi/abs/10.1063/1.4898185 https://aip.scitation.org/doi/abs/10.1063/1.4898185).