Abstract
We review a proposal of how to complete non-relativistic Quantum Mechanics to a physically meaningful, mathematically precise and logically coherent theory. This proposal has been dubbed ETH-Approach to Quantum Mechanics, “E” standing for “Events,” “T” for “Trees,” and “H” for “Histories.” The ETH-Approach supplies the last one of three pillars Quantum Mechanics can be constructed upon in such a way that its foundations are solid and stable. Two of these pillars are well known. The third one has been proposed quite recently; it implies a general non-linear stochastic law for the time-evolution of states of individual physical systems.
Dedicated to the memory of Detlef Dürr.
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Notes
- 1.
We suspect, though, that our views of how to complete QM are likely to differ from what we think were his.
- 2.
We have added a remark on Bohmian mechanics on request of one of the editors.
- 3.
It really does not make much sense to present this approach to QM in a new way each time it has to be recalled, because people have chosen not to take notice of it.
- 4.
i.e., a von Neumann algebra.
- 5.
in the sense the late Rudolf Haag used this terminology; see [19].
- 6.
In more technical jargon, \(\mathfrak {P}(\omega _t)\) generates the center of the centralizer \(\mathcal {C}(\omega _t)\) of \(\omega _t\).
References
D. Dürr, S. Teufel, Bohmian Mechanics (Springer, Berlin and Heidelberg, 2009)
K. Kraus, States, Effects, and Operations, Lecture Notes in Physics, vol. 190. (Springer, Berlin, 1983)
V. Gorini, A. Kossakowski, E.C.G. Sudarshan, Completely positive semigroups of N-level systems. J. Math. Phys. 17(5), 821 (1976) G. Lindblad, On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)
G. Lüders, Über die Zustandsänderung durch den Messprozess. Ann. Phys. (Leipzig) 443(5–8), 322–328 (1950)
E.P. Wigner, Remarks on the mind-body question, in Symmetries and Reflections (Indiana University Press, Bloomington, 1967), pp. 171–184
J. Faupin, J. Fröhlich, B. Schubnel, On the probabilistic nature of quantum mechanics and the notion of closed systems. Ann. H. Poincaré 17, 689–731 (2016)
H.D. Everett III., Relative state formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957)
In: Interpretations of quantum mechanics, Wikipedia; see https://en.wikipedia.org/wiki/Interpretations{hyphen}of{hyphen}quantum{hyphen}mechanics
R.B. Griffiths, Consistent histories and the interpretation of quantum mechanics. J. Stat. Phys. 36(1), 219–272 (1984)
M. Gell-Mann, J.B. Hartle, Classical equations for quantum systems. Phys. Rev. D 47(8), 3345–3382 (1993)
G.C. Ghirardi, A. Rimini, T. Weber, Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 470–491 (1986)
G. ’tHooft, The Cellular Automaton Interpretation of Quantum Mechanics, Fundamental Theories of Physics, ed. by H. van Beijeren et al., vol. 185 (Springer, Cham, Heidelberg, New York, 2016)
J. Fröhlich, B. Schubnel, Quantum probability theory and the foundations of quantum mechanics, in The Message of Quantum Science, ed. by Ph. Blanchard, J. Fröhlich (Springer, Berlin, 2015)
Ph. Blanchard, J. Fröhlich, B. Schubnel, A Garden of forking paths - the quantum mechanics of histories of events. Nucl. Phys. B 912, 463–484 (2016)
J. Fröhlich, A brief review of the “ETH - approach to quantum mechanics,” in Frontiers in Analysis and Probability, ed. by N. Anantharaman, A. Nikeghbali, M. Rassias (Springer, Cham, 2020)
J. Fröhlich, A. Pizzo, The time-evolution of states in quantum mechanics according to the ETH-approach. Commun. Math. Phys. 389, 1673–1715 (2022)
J. Fröhlich, Relativistic quantum theory, in Do Wave Functions Jump? Perspectives of the Work of GianCarlo Ghirardi, ed. by V. Allori, A. Bassi, D. Dürr, N. Zanghi, Fundamental Theories of Physics (Springer, Cham, 2020). and paper in preparation
J. Fröhlich, Z. Gang, On the Evolution of States in a Quantum-Mechanical Model of Experiments. arXiv:2212.02599. Submitted for publication in Ann. H. Poincaré
R. Haag, Fundamental irreversibility and the concept of events. Commun. Math. Phys. 132, 245–251 (1990); See also: R. Haag, On quantum theory. Int. J. Quant. Inf. 17, 1950037 (2019)
A.M. Gleason, Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957); S. Maeda Probability measures on projections in von Neumann algebras. Rev. Math. Phys. 1, 235–290 (1989)
D. Buchholz, Collision theory for massless particles. Commun. Math. Phys. 52, 147–173 (1977); see also: D. Buchholz, J. Roberts, New light on infrared problems: sectors, statistics, symmetries and spectrum. Commun. Math. Phys. 330, 935–972 (2014)
G. Lüders, see [4]; J. Schwinger, The algebra of microscopic measurement. Proc. Natl. Acad. Sci. (USA) 45(10), 1542–1553 (1959); E.P. Wigner, in The Collected Works of Eugene Paul Wigner (Springer, Berlin, 1993)
Acknowledgements
One of us (J. F.) thanks his collaborators, in particular Baptiste Schubnel, in earlier work on related problems for the pleasure of cooperation, and Carlo Albert, Shelly Goldstein and Erhard Seiler for their very encouraging interest in our efforts.
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Fröhlich, J., Gang, Z., Pizzo, A. (2024). A Tentative Completion of Quantum Mechanics. In: Bassi, A., Goldstein, S., Tumulka, R., Zanghì, N. (eds) Physics and the Nature of Reality. Fundamental Theories of Physics, vol 215. Springer, Cham. https://doi.org/10.1007/978-3-031-45434-9_12
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