The Foundations of Quantum Mechanics and the Evolution of the Cartan-Kähler Calculus

Abstract

In 1960In -1962. Kähler enriched É. Cartan's exterior calculus, making it suitable for quantum mechanics (QM) and not only classical physics. His "Kähler-Dirac" (KD) equation reproduces the fine structure of the hydrogen atom. Its positron solutions correspond to the same sign of the energy as electrons.The Cartan-Kähler view of some basic concepts of differential geometry is presented, as it explains why the components of Kähler's tensor-valued differential forms have three series of indices. We demonstrate the power of his calculus by developing for the electron's and positron's large components their standard Hamiltonian beyond the Pauli approximation, but without resort to Foldy-Wouthuysen transformations or ad hoc alternatives (positrons are not identified with small components in Kähler's work). The emergence of negative energies for positrons in the Dirac theory is interpreted from the perspective of the KD equation. Hamiltonians in closed form (i.e. exact through a finite number of terms) are obtained for both large and small components when the potential is time-independent.A new but as yet modest new interpretation of QM starts to emerge from that calculus' peculiarities, which are present even when the input differential form in the Kähler equation is scalar-valued. Examples are the presence of an extra spin term, the greater number of components of "wave functions" and the non-association of small components with antiparticles. Contact with geometry is made through a Kähler type equation pertaining to Clifford-valued differential forms.

Algebraic Quarks from the Tangent Bundle: Methodology

“…The claim is substantiated by the depth of results that Kähler himself obtained in using it in relativistic quantum mechanics [4], [6], by our derivation of the electromagnetic Hamiltonian without resort to Foldy-Wouthuysen transformations [8] and by what we are presently doing in connection with quark phenomenology. Of special interest in this regard is his treatment of total angular momentum [6], which we have adapted to our commutative algebra [2], [3].…”

“…As an example, consider Kähler's equation with electromagnetic coupling [4], [6]. Momenta and generalized momenta operators emerge while obtaining a classical Hamiltonian with electromagnetic coupling by the expedient process of considering the mass of the electron as the dominant energy, which is introduced in the exponent of a phase factor [8].…”

“…In section 27 of [6], Kähler obtains a second spin term which may cancel or equate the standard one, thus changing the perspective one may have of the gyromagnetic ratio. He actually emphasizes this difference at the end of his comprehensive 1962 paper [6] This ratio does not appear in the derivation of the Pauli term from the Kähler equation with electromagnetic coupling [8], and yet one obtains the right equation, and the spectrum of the hydrogen atom is not affected at all [6]. The hyperfine corrections to the gyromagnetic ratio should then be viewed as limitations of theory, rather than involving the unnecessary concept of gyromagnetic ratio.…”

Real Units Imaginary in Kaehler's Quantum Mechanics

“…If we were dealing with a vector space, the Lie operator ζ j ∂/∂x j acting on u would yield only the first of the two terms on the right hand side of (5).…”

Differential Geometry for Physicists and Mathematicians

“…The author also considers the extension of the Euclidean frame bundle to an affine frame bundle by the action of the linear group on the Euclidean fibers. He briefly presents: the Hodge dual of a differential r-form using Clifford algebra, the Kähler calculus of scalar-valued differential forms referring to [1] where computations in relativistic quantum mechanics with Kähler calculus are given. The Lie algebra of the Euclidean group is found.…”