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Quantum Probability Theory and the Foundations of Quantum Mechanics

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The Message of Quantum Science

Part of the book series: Lecture Notes in Physics ((LNP,volume 899))

Abstract

By and large, people are better at coining expressions than at filling them with interesting, concrete contents. Thus, it may not be very surprising that there are many professional probabilists who may have heard the expression but do not appear to be aware of the need to develop “quantum probability theory” into a thriving, rich, useful field featured at meetings and conferences on probability theory. Although our aim, in this essay, is not to contribute new results on quantum probability theory, we hope to be able to let the reader feel the enormous potential and richness of this field. What we intend to do, in the following, is to contribute some novel points of view to the “foundations of quantum mechanics”, using mathematical tools from “quantum probability theory” (such as the theory of operator algebras).

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Notes

  1. 1.

    This story is purely fictional, but quite plausible.

  2. 2.

    And that Heisenberg’s 1925 paper [46] cannot be understood.

  3. 3.

    “dévisser les problèmes” (in reference to A. Grothendieck).

  4. 4.

    “The one thing to say about art is that it is one thing. Art is art-as-art and everything else is everything else.” Ad Reinhardt, [63].

  5. 5.

    A result of the form of Eq. (7.21) was conjectured by J.F. in the 1990s. But the proof remained elusive.

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Acknowledgements

A rough first draft of this paper has been written during J.F.’s stay at the School of Mathematics of the Institute for Advanced Study (Princeton), 2012/2013. His stay has been supported by the ‘Fund for Math’ and the ‘Monell Foundation’. He is deeply grateful to Thomas C. Spencer for his most generous hospitality. He acknowledges useful discussions with Ph. Blanchard, P. Deift, S. Kochen and S. Lomonaco. He thanks D.Bernard for drawing his attention to [6] and W. Faris for correspondence. He is grateful to D. Buchholz, D. Dürr, S. Goldstein, J. Yngvason and N. Zanghi for numerous friendly and instructive discussions, encouragement and for the privilege to occasionally disagree in mutual respect and friendship.

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Authors and Affiliations

  1. Institut für Theoretische Physik, HIT K42.3, ETH Zürich, 8093, Zürich, Switzerland

    Jürg Fröhlich

  2. Departement Mathematik, ETH Zürich, 8092, Zürich, Switzerland

    Baptiste Schubnel

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  1. Jürg Fröhlich
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  2. Baptiste Schubnel
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Correspondence to Jürg Fröhlich .

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Editors and Affiliations

  1. Fakultät für Physik, Universität Bielefeld, Bielefeld, Germany

    Philippe Blanchard

  2. Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland

    Jürg Fröhlich

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Fröhlich, J., Schubnel, B. (2015). Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard, P., Fröhlich, J. (eds) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46422-9_7

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