Related papers: Complex Geometry and Dirac Equation
I apply the algebraic framework developed in arXiv:1101.4542 to study geometry of elliptic spaces in 1, 2, and 3 dimensions. The background material on projectivised Clifford algebras and their application to Cayley-Klein geometries is…
A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of…
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors…
This paper studies unitary representations with Dirac cohomology for complex groups, in particular relations to unipotent representations
The separability of the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington-Finkelstein-type coordinates is shown. To this end, the Kerr geometry is described in the Newman-Penrose formalism by…
The two-dimensional Dirac equation has been widely used in graphene physics, the surface of topological insulators, and especially quantum scarring. Although a numerical approach to tackling an arbitrary confining problem was proposed…
We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kaehler manifolds.
Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such…
We develop a systematic method to derive the Majorana representation of the Dirac equation in (1+3)-dimensions. We compare with similar approach in (2+2)-dimensions . We argue that our formalism can be useful to have a better understanding…
The aim of this paper is to give an alternative proof of Kac's theorem for weighted projective lines (\cite{W}) over the complex field. The geometric realization of complex Lie algebras arising from derived categories (\cite{XXZ}) is…
There were many attempts to geometrize electromagnetic field and find out new interpretation for quantum mechanics formalism. The distinctive feature of this work is that it combines geometrization of electromagnetic field and…
We develop a real-analytic framework, called perplex analysis, in which the complex, split-complex, and dual numbers arise as members of a single four-parameter family of two-dimensional commutative real algebras. Within this unified…
An alternative, pedagogically simpler derivation of the allowed physical wave fronts of a propagating electromagnetic signal is presented using geometric algebra. Maxwell's equations can be expressed in a single multivector equation using…
The Dirac equation is considered with the recently proposed generalized gravitational interaction (Kepler or Coulomb), which includes post-Newtonian (relativistic) and quantum corrections to the classical potential. The general idea in…
We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call \emph{Dirac maps}, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex…
Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to…
A deformed differential calculus is developed based on an associative star-product. In two dimensions the Hamiltonian vector fields model the algebra of pseudo-differential operator, as used in the theory of integrable systems. Thus one…
We investigate using Clifford algebra methods the theory of algebraic dotted and undotted spinor fields over a Lorentzian spacetime and their realizations as matrix spinor fields, which are the usual dotted and undotted two component spinor…
Some notions of algebraic geometry can be defined for arbitrary varieties of algebras. This leads to universal algebraic geometry. The main idea of the presented theory is to consider interactions between algebra, logic and geometry in…