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I wonder if it's possible to discover another version of quantum theory that doesn't depend on complex numbers. We may discover a formulation of quantum mechanics using p-adic numbers, quaternions or a finite field etc. Also, physical states lives on a Hilbert space. What if we consider the infinite dimensional Hilbert space to be the tangent space of an infinite dimensional manifold at some point? Is it possible to make these generalized theories and if it's possible can it lead to new predictions or resolve some of the difficulties that are present?

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4 Answers 4

Various generalizations in the directions you mentioned have been tired, but none of them took off the ground. Ad hoc generalizations are usually incompatible with the existing knowledge.

Note that an improved theory must recover all well established phenomena in addition to resolving current difficulties. most new proposals fail on the first account.

It is like inventing a new arithmetic to replace the old one - you must recover the old results, which are known to be correct. This leaves very little freedom.

Progress in foundations of QM comes from detailed study of the difficulties, not from just trying one or the other generalization.

[Edit] Some important papers on p-adic quantum mechanics:

  • http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104178988
  • http://link.springer.com/article/10.1134%2FS1063778810020055 (a free copy is available from 144.206.159.178/FT/8304/650261/13044559.pdf)
  • http://arxiv.org/pdf/0904.4205

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    Thanks for answer,The reason I asked was that I have been reading about Archimedean property of R1 that made me think that it might be possible to replace Archimedean fields with non-Archimedean fields so that we can somehow control some of the infinities that arise in field theories –  ahmed Nov 20 '12 at 13:25
        
    @ahmed: for the p-adic case see my edit. –  Arnold Neumaier Nov 20 '12 at 16:14
        
    What does inventing a new arithmetic mean if you require it to recover the old results. And what does known to be correct mean anyway? - Do you mean with a Platonic philosophy in mind? –  NikolajK Nov 20 '12 at 21:40
        
    @NickKidman: If you want to do real physics you need to recover all old results, at least to the accuracy with which these agree with experiments. If you can't, it is just science play but no real science. –  Arnold Neumaier Nov 21 '12 at 9:11
        
    @ArnoldNeumaier: Mhm yes, but I was asking about the statement regarding arithmetic here, specifically –  NikolajK Nov 21 '12 at 13:40

    If you're pondering these questions, you might be interested in generalised probabilistic theories, e.g. this paper by Jon Barrett, but there have since been several more. Studying these theories is one attempt to understand why quantum mechanics has the mathematical structure it does.

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    There are attempts that try to start from the ground up. An example is string theory (M-theory), which although amiable, is still in its infancy and no confirmation of old results nor predictions of new measurable phenomena have been calculated.

    There are also attempts to formalize the math behind quantum mechanics and the space in which it is defined. For example, Clifford algebra can be applied to define a vector space of quantum mechanics, allowing you to define everything (operators, observables, wave functions, etc...) within that single vector space. See as a reference/introduction e.g. here and this Oersted Medal lecture on why Hestenes thinks all physics should be taught in terms of geometric algebra. There are other ways to incorporate (complex) Clifford algebra into the story, but none are as convincing (or readable/understandable) as Hestenes' form.

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    Do you mean "generalized quantum mechanics" or "another version" equivalent to existent ones?

    If you mean "generalized" then a number of generalizations of ordinary quantum mechanics are currently under active research: generalizations using Liouville space [1], generalizations using geometric calculus based in Clifford algebra and Fock parametrization [2], generalizations beyond the Hilbert space [3] (this is a recent work by myself),...

    In all the cases the goal is to solve difficulties with ordinary quantum mechanics.

    [1] The Liouville Space Extension of Quantum Mechanics

    [2] The Landscape of Theoretical Physics: A Global View; From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle

    [3] Positive Definite Phase Space Quantum Mechanics

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