The foundations of quantum mechanics and the evolution of the Cartan-Kähler calculus
type: Item
instance of: scholarly article
title: The foundations of quantum mechanics and the evolution of the Cartan-Kähler calculus
zbMATH Open document ID: 1165.81009
published in: Foundations of Physics
publication date: 2009-04-21T00:00:00Z
review text: In the years following the development of quantum mechanics Dirac asked the question: what is the first-order differential operator whose square is the d'Alembertian operator? His answer gave rise to the famous Dirac equation describing electrons as well as positrons. Shortly after, van der Waerden asked the question: what are the (nonunitary) representations of the Lorentz group that describe Dirac particles? His answer now plays a leading role in our understanding of Dirac's theory. Other people see the origin in the theory of Clifford algebra. Yet another approach was suggested by E. Kähler in 1961 when he wrote an article (in German) titled \textit{Die Dirac-Gleichung}. Kähler's sophisticated calculus of differential forms generalized the exterior calculus of E.Cartan. But hardly anybody paid attention to the new approach. There is yet a subtle difference between the Dirac and the Kähler equation in that Kähler replaces Dirac differential forms by those that have 16 independent complex components though the physical consequences remain the same. Vargas recalls all the details in his article, claiming that a new interpretation of QM may emerge from the Cartan-Kähler calculus which may be even closer to Einstein's view of QM. But, in my view, these new methods have not triggered any breakthrough and have until now not provided new answers to old questions. They certainly require further effort to convince physicists that it is worthwhile to explore the consequences.
reviewed by: Gert Roepstorff
Mathematics Subject Classification ID: 81P05
Mathematics Subject Classification ID: 81Q70
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 81R25
zbMATH DE Number: 5545054
zbMATH Keywords: foundations of quantum mechanics
zbMATH Keywords: Dirac equation
zbMATH Keywords: Cartan-Kähler calculus
zbMATH Keywords: Clifford-valued differential forms
zbMATH Keywords: spinors
zbMATH Keywords: Killing symmetries
zbMATH Keywords: minimal electromagnetic coupling
author: José G. Vargas
MaRDI profile type: Publication
full work available at URL: s10701-008-9223-3
OpenAlex ID: W2029605785
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DOI: 10.1007/S10701-008-9223-3
description: scientific article; zbMATH DE number 5545054