The foundations of quantum mechanics and the evolution of the Cartan-Kähler calculus
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.
scientific article; zbMATH DE number 5545054
scientific article