Differential forms: from Clifford, through Cartan to Kahler
Limitations of the vector, tensor and Dirac calculi are illustrated to motivate the Kaehler calculus of integrands, which replaces all three of them and which we introduce in three steps. In a first step, we present the basics of the underlying Clifford algebra for that calculus, algebra valid for Euclidean and pseudo-Euclidean vector spaces of arbitrary dimension. The usual vector algebra is shown to be a corrupted form of Clifford algebra, corruption specific to dimension three and non-existing for other dimensions. The Clifford product is constituted by the sum of the exterior and interior products if at least one of the factors is a vector. Grossly speaking, these products play the role of the vector and scalar products of three dimensions, while generalizing them. It thus contains exterior algebra. As an intermediate step towards the Kaehler calculus, we briefly give the fundamentals of Cartans exterior calculus of scalar-valued differential forms, here viewed as ordinary scalar-valued integrands in multiple integrals. We also make a brief incursion into the exterior calculus of vector-valued differential forms, which is the moving frame version of differential geometry. We show the basics of the Kaehler calculus of differential forms. It is to the exterior calculus what Clifford algebra is to exterior algebra. Because of time and complexity constraints, we limit ourselves to scalar-valued differential forms, which is sufficient for relativistic quantum mechanics with electromagnetic coupling. In using this calculus, the problem with negative energy-solutions does not arise.