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International Symposium on Quantum Interaction

QI 2013: Quantum Interaction pp 6–21Cite as

Quantum Mechanics: Harbinger of a Non-commutative Probability Theory?

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8369))

Abstract

In this paper we discuss the relevance of the algebraic approach to quantum phenomena first introduced by von Neumann before he confessed to Birkoff that he no longer believed in Hilbert space. This approach is more general and allows us to see the structure of quantum processes in terms of non-commutative probability theory, a non-Boolean structure of the implicate order which contains Boolean sub-structures which accommodates the explicate classical world. We move away from mechanical ‘waves’ and ‘particles’ and take as basic what Bohm called a structure process. This enables us to learn new lessons that can have a wider application in the way we think of structures in language and thought itself.

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Notes

  1. 1.

    Unfortunately I have to make it clear that the spirit of the view Bohm and I were developing together had little in common with the proponents of the subject now called “Bohmian mechanics”.

  2. 2.

    Of course position and momentum have different dimensions so we choose \(x\leftrightarrow \widehat{X}\) and \(p\leftrightarrow \epsilon \widehat{P}\). Note that we are not appealing to anything quantum mechanical at this stage. It is only in quantum mechanics that we write \(\epsilon =1/\hbar \).

  3. 3.

    All type I von Neumann algebras have matrix representations.

  4. 4.

    This is the structure used in the Huygens construction and hence the Feynman path integral method [31].

  5. 5.

    See Bartlet [36] for a discussion of this point.

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Author information

Authors and Affiliations

  1. TPRU, Birkbeck, University of London, Malet Street, London, WC1E 7HX, UK

    B. J. Hiley

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  1. B. J. Hiley
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Editors and Affiliations

  1. ETH Zurich, Zurich, Switzerland

    Harald Atmanspacher

  2. School of Management, University of Leicester, Leicester, United Kingdom

    Emmanuel Haven

  3. Queensland University of Technology, Brisbane, Queensland, Australia

    Kirsty Kitto

  4. Department of Physics and Astronomy Centre for Interdisciplinary Science, University of Leicester, Leicester, United Kingdom

    Derek Raine

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Hiley, B.J. (2014). Quantum Mechanics: Harbinger of a Non-commutative Probability Theory?. In: Atmanspacher, H., Haven, E., Kitto, K., Raine, D. (eds) Quantum Interaction. QI 2013. Lecture Notes in Computer Science(), vol 8369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54943-4_2

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  • DOI: https://doi.org/10.1007/978-3-642-54943-4_2

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