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On quantum jumps and attractors of the Maxwell–Schrödinger equations

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Abstract

Our goal is the discussion of the problem of mathematical interpretation of basic postulates (or “principles”) of Quantum Mechanics: transitions to quantum stationary orbits, the wave-particle duality, and the probabilistic interpretation, in the context of semiclassical self-consistent Maxwell–Schrödinger equations. We discuss possible dynamical interpretation of these postulates relying on a new general mathematical conjecture on global attractors of G-invariant nonlinear Hamiltonian partial differential equations with a Lie symmetry group G. This conjecture is inspired by the results on global attractors of nonlinear Hamiltonian PDEs obtained by the author together with his collaborators since 1990 for a list of model equations with three basic symmetry groups: the trivial group, the group of translations, and the unitary group \(\mathbf {U}(1)\). We sketch these results.

Résumé

Notre objectif est de discuter du problème de l’interprétation mathématique des postulats (ou "principes") de base de la mécanique quantique: les transitions aux orbites stationnaires quantiques, la dualité onde-particule et l’interprétation probabiliste, dans le contexte des équations semi-classiques autoconsistantes de Maxwell–Schrödinger. Nous discutons de l’interprétation dynamique possible de ces postulats en nous appuyant sur une nouvelle conjecture mathématique générale sur les attracteurs globaux d’équations aux dérivées partielles hamiltoniennes non linéaires G-invariantes avec un groupe de symétrie de Lie G. Cette conjecture est inspirée des résultats sur les attracteurs globaux des EDP non linéaires hamiltoniennes obtenus par l’auteur et ses collaborateurs depuis 1990 pour une liste d’équations modèles avec trois groupes de symétrie de base: le groupe trivial, le groupe des translations, et le groupe unitaire U(1). Nous esquissons ces résultats.

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Acknowledgements

The author thanks Andrew Comech, Sergey Kuksin, Alexander Shnirelman and Herbert Spohn for numerous fruitful discussions.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, 1090, Wien, Austria

    Alexander I. Komech

  2. Dobrushin Laboratory, IITP of Russian Academy of Sciences, Moscow, Russia

    Alexander I. Komech

  3. Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

    Alexander I. Komech

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To Sasha Shnirelman on the occasion of his 75th birthday.

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Komech, A.I. On quantum jumps and attractors of the Maxwell–Schrödinger equations. Ann. Math. Québec 46, 139–159 (2022). https://doi.org/10.1007/s40316-021-00179-1

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