The fundamental solution to the euclidean Dirac operator is where ωn is the surface area of the unit sphere Sn−1.
Indeed, many basic properties of one variable complex analysis follow through for many first order Dirac type operators.
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.
In particular Clifford analysis has been used to solve, in certain Sobolev spaces, the full water wave problem in 3D.
This is not the case though when we need to deal with the interaction between the Dirac operator and the Fourier transform.
When we consider upper half space Rn,+ with boundary Rn−1, the span of e1, ..., en−1, under the Fourier transform the symbol of the Dirac operator is iζ where In this setting the Plemelj formulas are and the symbols for these operators are, up to a sign, These are projection operators, otherwise known as mutually annihilating idempotents, on the space of Cln(C) valued square integrable functions on Rn−1.
The extension is explicitly given by When this extension is applied to the variable x in we get that is the restriction to Rn−1 of E+ + E− where E+ is a monogenic function in upper half space and E− is a monogenic function in lower half space.
The Cayley transform over n-space is Its inverse is For a function f(x) defined on a domain U in n-euclidean space and a solution to the Dirac equation, then is annihilated by DS, on C(U) where Further the conformal Laplacian or Yamabe operator on Sn.
A Möbius transform over n-euclidean space can be expressed as where a, b, c and d ∈ Cln and satisfy certain constraints.
is a solution to the Dirac equation where and ~ is a basic antiautomorphism acting on the Clifford algebra.
This means that for a conformally flat manifold M we need a spin structure on M in order to define a spinor bundle on whose sections we can allow a Dirac operator to act.
Explicit simple examples include the n-cylinder, the Hopf manifold obtained from n-euclidean space minus the origin, and generalizations of k-handled toruses obtained from upper half space by factoring it out by actions of generalized modular groups acting on upper half space totally discontinuously.
From an Atiyah–Singer–Dirac operator D we have the Lichnerowicz formula where τ is the scalar curvature on the manifold, and Γ∗ is the adjoint of Γ.
This allows us to note that over the space of smooth spinor sections the operator D is invertible such a manifold.
In the cases where the Atiyah–Singer–Dirac operator is invertible on the space of smooth spinor sections with compact support one may introduce where δy is the Dirac delta function evaluated at y.
All of this provides potential links to Atiyah–Singer index theory and other aspects of geometric analysis involving Dirac type operators.
In Clifford analysis one also considers differential operators on upper half space, the disc, or hyperbola with respect to the hyperbolic, or Poincaré metric.
In representation theory for the orthogonal group, O(n) it is common to consider functions taking values in spaces of homogeneous harmonic polynomials.
Now as the Clifford algebra is not commutative Dxf(x, u) then this function is no longer k monogenic but is a homogeneous harmonic polynomial in u.
Let P be the projection of hk to pk then the Rarita–Schwinger operator is defined to be PDk, and it is denoted by Rk.
Using Euler's Lemma one may determine that So There is a vibrant and interdisciplinary community around Clifford and Geometric Algebras with a wide range of applications.
The main conferences in this subject include the International Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA) and Applications of Geometric Algebra in Computer Science and Engineering (AGACSE) series.
A main publication outlet is the Springer journal Advances in Applied Clifford Algebras.