Clifford analysis

Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on SpinC manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations. In Euclidean space the Dirac operator has the form where e1, ..., en is an orthonormal basis for Rn, and Rn is considered to be embedded in a complex Clifford algebra, Cln(C) so that ej2 = −1. This gives where Δn is the Laplacian in n-euclidean space. The fundamental solution to the euclidean Dirac operator is where ωn is the surface area of the unit sphere Sn−1. Note that where is the fundamental solution to Laplace's equation for n ≥ 3. The most basic example of a Dirac operator is the Cauchy–Riemann operator in the complex plane. Indeed, many basic properties of one variable complex analysis follow through for many first order Dirac type operators. In euclidean space this includes a Cauchy Theorem, a Cauchy integral formula, Morera's theorem, Taylor series, Laurent series and Liouville Theorem. In this case the Cauchy kernel is G(x−y). The proof of the Cauchy integral formula is the same as in one complex variable and makes use of the fact that each non-zero vector x in euclidean space has a multiplicative inverse in the Clifford algebra, namely Up to a sign this inverse is the Kelvin inverse of x. Solutions to the euclidean Dirac equation Df = 0 are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold.