Innerer und äusserer Differentialkalkül

Kahler, Erich

doi:10.1515/9783112550526

Cited by 13 publications

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“…a conservation law of the form d(j (1) + j (2) ) = 0, follows. The j (1) and j (2) are spacetime currents that come with corresponding densities, ± 1 2 ( ± u, ± u).…”

Section: Kähler Version Of Relativistic Quantum Mechanicsmentioning

confidence: 99%

“…a conservation law of the form d(j (1) + j (2) ) = 0, follows. The j (1) and j (2) are spacetime currents that come with corresponding densities, ± 1 2 ( ± u, ± u). The sign definiteness of the charge densities, which are of identical form except for sign, speaks of the fact that, in Kähler's quantum mechanics, the wave function is about amplitude of charge density, not of probability density.…”

Section: Kähler Version Of Relativistic Quantum Mechanicsmentioning

confidence: 99%

“…The argument leading to this equation from "first principles" is given in section 33 of [1]. We proceed to illustrate how the argument goes.…”

Section: Kähler Version Of Relativistic Quantum Mechanicsmentioning

confidence: 99%

“…In 1960-1962, Kähler produced his calculus (KC) [1], [2] and [3]. In the last decade of his long life, he returned to this topic [7], not to the topic of Kähler metrics and Kähler manifolds [8], for which he is best known.…”

Section: Introductionmentioning

confidence: 99%

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The Exterior Calculus and Relativistic Quantum Mechanics

Vargas¹

2019

Preprint

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In 1960-62, E. Kähler developed what looks as a generalization of the exterior calculus, which he based on Clifford rather than exterior algebra [1], [2] and [3]. The role of the exterior derivative, du, was taken by the more comprehensive derivative ∂u (≡ dx µ ∨ d µ u), where "∨" stands for Clifford product. The d µ u represents a set of quantities to which he referred as covariant derivative, and for which he gave a long, ad hoc expression. We provide the geometric foundation for this derivative, based on Cartan's treatment of the structure of a Riemannian differentiable manifold without resort to the concept of the so called affine connections.Buried at advanced points in his presentations [1], [3] is the implied statement that ∂u = du + * −1 d u * , the sign at the front of the coderivative term is a matter of whether we include the unit imaginary or not in the definition of Hodge dual, * . We extract and put together the pieces of theory that go into his derivation of that statement, which seems to have gone unnoticed in spite of its relevance for a quick understanding of what his "Kähler calculus".Kähler produced a most transparent, compelling and clear formulation of relativistic quantum mechanics (RQM) as a virtual concomitant of his calculus. We shall enumerate several of its notable features, which he failed to emphasize. The exterior calculus in Kähler format thus reveals itself as the computational tool for RQM, making the Dirac calculus unnecessary and its difficulties spurious.

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“…a conservation law of the form d(j (1) + j (2) ) = 0, follows. The j (1) and j (2) are spacetime currents that come with corresponding densities, ± 1 2 ( ± u, ± u).…”

Section: Kähler Version Of Relativistic Quantum Mechanicsmentioning

confidence: 99%

Section: Kähler Version Of Relativistic Quantum Mechanicsmentioning

confidence: 99%

“…The argument leading to this equation from "first principles" is given in section 33 of [1]. We proceed to illustrate how the argument goes.…”

Section: Kähler Version Of Relativistic Quantum Mechanicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Exterior Calculus and Relativistic Quantum Mechanics

Vargas¹

2019

Preprint

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“…We obtain Kähler's expression for angular momentum starting which the action of a partial derivative on a differential form. He obtained it in the general framework of what Cartan called [2] infinitesimal transformations, which he cleverly transformed into partial derivatives [3]. But this is a distraction that obscures the humble nature of the Lie derivative of a differential form.…”

Section: Introductionmentioning

confidence: 99%

Real Units Imaginary in Kaehler's Quantum Mechanics

Vargas

2012

Preprint

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Inspired by a similar, more general treatment by Kähler, we obtain the spin operator by pulling to the Cartesian coordinate system the azimuthal partial derivative of differential forms. At this point, no unit imaginary enters the picture, regardless of whether those forms are over the real or the complex field. Hence, the operator is to be viewed as a real operator. Also a view of Lie differentiation as a pullback emerges, thus avoiding conceps such as flows of vector fields for its definition.Enter Quantum Mechanics based on the Kähler calculus. Independently of the unit imaginary in the phase factor, the proper values of the spin part of angular momentum emerge as imaginary because of the idempotent defining the ideal associated with cylindrical symmetry. Thus the unit imaginary has to be introduced by hand as a factor in the angular momentum operator -and as a result also in its orbital part-for it to have real proper values. This is a concept of real operator opposite to that of the previous paragraph. Kähler stops short of stating the antithesis in this pair of concepts, both of them implicit in his work.A solution to this antithesis lies in viewing units imaginary in those idempotents as being the real quantities of square −1 in rotation operators of real tangent Clifford algebra. In so doing, one expands the calculus, and launches in principle a geometrization of quantum mechanics, whether by design or not. * Dedicated to Professor A. Rueda, in recognition for the impact he had on my professional development.

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Bemerkungen zum Inneren Differentialkalkül

Weier

1967

Mathematische Nachrichten

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Der Innere Differentialkalkul ist in [I] und [2] von E. KAHLER begriindet worden. Bezeichnet V den Produktoperator und S den Differentialoperator dieses Kalkuls, so lassen sich V und B in einfachen Fallen durch ( V ) = ( A ) + (*I> 6 = rot + div kennzeichnen. Die Gleichung ( V ) = ( A ) + ( a ) ist naherhin so gemeint. Sei V ein n-dimensionaler Vektorraum mit Skalarprodukt und (el, . . . , en) eine orthonormale Basis von V , ferner t ein schiefsymmetrischer Tensor iiber V . Dann ist ej V t = e j A t + e j . t , (en, A * * + A e,~) V t = (el., A * * A e+ V ( e ,~~ V t ) , im ubrigen ist das V-Produkt linear. Zur Definition des inneren Produktes 2 , . . . e2 . t : sind s = (61l' ' "") und t = (t I b ) heliebige Tensoren iiber I' mit u 5 b, so ist ' . ' P a (8 * .,?&a = ( l / u ! ) s " ' t L b l . . . / " " A , . . . , I & a 'Ersichtlich sind v und 6 inhomogene Operatoren in dem Sinne, daS homogene Tensoren in inhomogene iiberfiihrt werden. Die Grunde fur die KAHLERschen Definitionen treten deutlich hervor, wenn man diese Definitionen abwandelt und nach den Eigenschaften der abgewandelten Begriffe fragt. Weiterhin wird eine konkrete Modifizierung diskutiert :Schon an dieser Stelle sei bemerkt, daD weder der KAHLERSChe Kalkul noch der durch v -und -6 bestimmte Kalkul scharfe Dualitiitseigenschafteii be-

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Innerer und äusserer Differentialkalkül

Cited by 13 publications

References 0 publications

The Exterior Calculus and Relativistic Quantum Mechanics

The Exterior Calculus and Relativistic Quantum Mechanics

Real Units Imaginary in Kaehler's Quantum Mechanics

Bemerkungen zum Inneren Differentialkalkül

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