Alterman Summer School on Geometric Algebra and Kähler Calculus 2017

Monday, July 31, 2017 - 9:00am to Friday, August 4, 2017 - 5:00pm

The Alterman Summer School on Geometric Algebra and Kähler Calculus 2017 deals mainly with Clifford (geometric) algebra and the Kähler calculus based on (but not only) Clifford algebra of differential forms. Erich Kähler developed this calculus by joining Clifford's geometric algebra and Cartan´s exterior calculus. The Kähler calculus has immediate application to many branches of physics, especially quantum mechanics, whose foundations it affects.

The Alterman events are becoming a series of periodic conferences and summer schools sponsored by Mr. Eric Alterman. Thanks to his support, fees are unexpensive (150€ and 50€ for shared lodging). Students have a reduced fee (75€) and free shared lodging. The contents of the lectures and the unexpensive fees provide an unusual learning opportunity. This year is the second edition. The first edition (Alterman Conference 2016) was held in Brasov (Romania).

Faculty of Transportation Engineering, 20 Hr. Smirnenski Blvd.
1164 Sofia

Teachers: Rafał Abłamowicz, Danail Brezov, Zbigniew Oziewicz, José Vargas

July 31 (Monday): Basic Clifford Algebra and Exterior Calculus.
Welcome Ceremony
Keynote lecture: Mulling over Impredicative Systems (Prof. Roberto Poli).
The Clifford algebras of the Euclidean and hyperbolic planes.
Definition, fundamentals and perspective on Clifford algebra.
The Clifford algebra of space and spacetime. Reflections, rotations and Lorentz transformations.
Exterior algebra and calculus.

August 1 (Tuesday): Kähler algebra and Fundamentals of Kähler Calculus.
Keynote lecture: The Unity of Geometry (Prof. José Vargas)
Definitions of Kähler algebra and perspective on the Kähler calculus.
Idempotents, ideals, spinors, primitive idempotents, exterior systems, tensor product of Kähler and tangent Clifford algebras.
Basic Kähler calculus in Cartesian coordinates.
Particles and antiparticles, chirality, leptons, Cooper pairs, idempotents for particles.
The real-valued integrations of the complex calculus as an exercise in Kähler algebra.
The complex variable formalism as a minor extension of the Kähler calculus over the reals.

August 2 (Wednesday): Excursion.

August 3 (Thursday): Kähler calculus and Computer Calculations.
Exterior and interior differentiations.
Leibniz rules.
Laplacians, harmonic, strict harmonic and constant differentials.
Hodge dual and relation of interior differentiation to the co-derivative.
Green's identities and conservation laws.
The Kähler equation and its adjoint.
Clifford algebra with the CLIFFORD package for Maple in the computer lab.

August 4 (Friday): Quantum mechanics.
Lie operators in exterior and Kähler calculus.
Angular momentum.
Strict harmonic and spherical differential forms in 3-D.
The stationary Kähler-Dirac equation for each sign of “generalized charge” and indication of further specialization.
The Kähler-Dirac equation with electromagnetic coupling: emergence of standard charge even before electrons and positrons.
Electrons, positrons and the hydrogen atom.
Clifford algebra with the CLIFFORD package for Maple in the computer lab.