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> 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

> 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

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).