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A Brief Review of the “ETH-Approach to Quantum Mechanics”

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Frontiers in Analysis and Probability

Abstract

To begin with, some of the conundrums concerning Quantum Mechanics and its interpretation(s) are recalled. Subsequently, a sketch of the “ETH-Approach to Quantum Mechanics” is presented. This approach yields a logically coherent quantum theory of “events” featured by isolated physical systems and of direct or projective measurements of physical quantities, without the need to invoke “observers.” It enables one to determine the stochastic time evolution of states of physical systems. We also briefly comment on the quantum theory of indirect or weak measurements, which is much easier to understand and more highly developed than the theory of direct (projective) measurements. A relativistic form of the ETH-Approach will be presented in a separate paper.

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Notes

  1. 1.

    I think it is more appropriate to speak of the “foundations of QM,” rather than “interpretations of QM.” We have to understand what QM tells us about Nature, what it means - once this is accomplished, the correct interpretation of the theory will come almost automatically.

  2. 2.

    In classical theories, these operators generate an abelian (C ∗-) algebra, and time evolution is given by a ∗-automorphism group of this algebra generated by a vector field on its spectrum; while, in QM, the algebra generated by operators representing physical quantities (and events) is non-commutative, and time evolution is given by a ∗-automorphism group of such an algebra only if the system is isolated.

  3. 3.

    It is advocated by certain groups of people that the problem arising from this fact can be remedied by invoking the phenomenon of “decoherence” and appealing to the “consistency” of histories of events [1]. But I find the arguments supporting this point of view unconvincing.

  4. 4.

    This is the case unless perfect “decoherence” holds.

  5. 5.

    The role of space-time in a relativistic version of the “ETH-Approach” is discussed in [16].

  6. 6.

    In local relativistic quantum theories with massless particles, the algebra \(\mathcal {N}\) tends to be a von Neumann algebra of type III; see [15].

  7. 7.

    I sometimes fear that unrealistically simple examples advanced with the intention to clarify aspects of the foundations of QM have had the opposite effect: They have contributed to clouding our views.

  8. 8.

    and the algebras \(\mathcal {E}_{\geq t}, t \in \mathbb {R}\), are von Neumann algebras of type III.

  9. 9.

    a rather unfortunate name!

  10. 10.

    The set \(\mathfrak {X}_{S}\) can also be defined in terms of a certain “flag manifold” associated with the Hilbert space \(\mathcal {H}\).

  11. 11.

    This digression can be omitted at first reading, and the reader is invited to proceed to point V., below.

  12. 12.

    One speaks of a “weak measurement” of A.

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Authors and Affiliations

  1. Department of Physics, ETH Zurich, 8093, Zurich, Switzerland

    Jürg Fröhlich

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  1. IRMA, University of Strasbourg, Strasbourg, France

    Nalini Anantharaman

  2. Institute of Mathematics, University of Zürich, Zurich, Switzerland

    Ashkan Nikeghbali

  3. Institute of Mathematics, University of Zürich, Zurich, Switzerland

    Michael Th. Rassias

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Fröhlich, J. (2020). A Brief Review of the “ETH-Approach to Quantum Mechanics”. In: Anantharaman, N., Nikeghbali, A., Rassias, M.T. (eds) Frontiers in Analysis and Probability. Springer, Cham. https://doi.org/10.1007/978-3-030-56409-4_2

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