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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.
Euclidean space 
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.
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.
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 . 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 in euclidean space has a multiplicative inverse in the Clifford algebra, namely . Up to a sign this inverse is the Kelvin inverse of . Solutions to the euclidean Dirac equation are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold.
In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When n = 4, the Dirac operator is sometimes referred to as the Cauchy–Riemann–Fueter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis.
Clifford analysis has analogues of Cauchy transforms, Bergman kernels, Szegő kernels, Plemelj operators, Hardy spaces, a Kerzman–Stein formula and a Π, or Beurling–Ahlfors, transform. These have all found applications in solving boundary value problems, including moving boundary value problems, singular integrals and classic harmonic analysis. In particular Clifford analysis has been used to solve, in certain Sobolev spaces, the full water wave problem in 3-D. This method works in all dimensions greater than 2.
Much of Clifford analysis works if we replace the complex Clifford algebra by a real Clifford algebra, . This is not the case though when we need to deal with the interaction between the Dirac operator and the Fourier transform.
The Fourier transform 
When we consider upper half space with boundary , the span of , under the Fourier transform the symbol of the Dirac operator is 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.
Note that where is the th Riesz potential, . As the symbol of is it is easily determined from the Clifford multiplication that . So the convolution operator is a natural generalization to euclidean space of the Hilbert transform.
When this extension is applied to the variable in we get that is the restriction to of where is a monogenic function in upper half space and is a monogenic function in lower half space.
There is also a Paley–Wiener theorem in n-euclidean space arising in Clifford analysis.
Conformal structure 
Many Dirac type operators have a covariance under conformal change in metric. This is true for the Dirac operator in euclidean space, and the Dirac operator on the sphere under Möbius transformations. Consequently this holds true for Dirac operators on conformally flat manifolds and conformal manifolds which are simultaneously spin manifolds.
Cayley transform (stereographic projection) 
where is the spherical Beltrami–Dirac operator
The Cayley transform over n-space is
where . Its inverse is
For a function defined on a domain in n-euclidean space and a solution to the Dirac equation, then is annihilated by , on where
Further , the conformal Laplacian or Yamabe operator on . Explicitly where is the Laplace–Beltrami operator on . The operator is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also is the Paneitz operator, , on the n-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, . These are all examples of operators of Dirac type.
Möbius transform 
A Möbius transform over n-euclidean space can be expressed as where , , and ∈ Cℓn and satisfy certain constraints. The associated 2 × 2 matrix is called an Ahlfors–Vahlen matrix. If and then is a solution to the Dirac equation where and is a basic antiautomorphism acting on the Clifford algebra. The operators , or Δnk/2 when is even, exhibit similar covariances under Möbius transform including the Cayley transform.
When and are non-zero they are both members of the Clifford group.
As then we have a choice in sign in defining . This means that for a conformally flat manifold we need a spin structure on 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. A Dirac operator can be introduced in these contexts. These Dirac operators are special examples of Atiyah–Singer–Dirac operators.
The Atiyah–Singer–Dirac operator 
Given a spin manifold, , with a spinor bundle then given a smooth section in then in terms of a local orthonormal basis e1(x)... en(x) of the tangent bundle of the Atiyah–Singer–Dirac operator acting on is defined to be
From an Atiyah–Singer–Dirac operator we have the Lichnerowicz formula
where is the scalar curvature on the manifold, and is the adjoint of . The operator is known as the spinorial Laplacian.
If is compact and and somewhere then there are no non-trivial harmonic spinors on the manifold. This is Lichnerowicz' Theorem. It is readily seen that Lichnerowicz' Theorem is a generalization of Liouville's Theorem from one variable complex analysis. This allows us to note that over the space of smooth spinor sections the operator is invertible for 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 with and is the Dirac delta function evaluated at . This gives rise to a Cauchy kernel, which is the fundamental solution to this Dirac operator. From this one may obtain a Cauchy integral formula for harmonic spinors. With this kernel much of what is described in the first section of this entry carries through for invertible Atiyah–Singer–Dirac operators.
Using Stokes' theorem, or otherwise, one can further determine that under a conformal change of metric the Dirac operators associated to each metric are proportional to each other, and consequently so are their inverses, if they exist.
All of this provides potential links to Atiyah–Singer index theory and other aspects of geometric analysis involving Dirac type operators.
Hyperbolic 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.
For upper half space one splits the Clifford algebra, into . So for one may express as with , . One then has projection operators and defined as follows and . The Hodge–Dirac operator acting on a function with respect to the hyperbolic metric in upper half space is now defined to be
In this case
The hyperbolic Laplacian is invariant under actions of the conformal group, while the hyperbolic Dirac operator is covariant under such actions.
Rarita–Schwinger/Stein–Weiss operators 
Rarita–Schwinger operators, also known as Stein–Weiss operators, arise in representation theory for the Spin and Pin groups. The operator is a conformally covariant first order differential operator. Here . When , the Rarita–Schwinger operator is just the Dirac operator. In representation theory for the orthogonal group, it is common to consider functions taking values in spaces of homogeneous harmonic polynomials. When one refines this representation theory to the double covering of one replaces spaces of homogeneous harmonic polynomials by spaces of k homogeneous polynomial solutions to the Dirac equation, otherwise known as monogenic polynomials. One considers a function where a domain in and varies over . Further is a k-monogenic polynomial in . Now apply the Dirac operator in to . Now as the Clifford algebra is not commutative then this function is no longer monogenic but is a homogeneous harmonic polynomial in . Now for each harmonic polynomial homogeneous of degree there is an Almansi–Fischer decomposition where and are respectively and monogenic polynomials. Let be the projection of to then the Rarita–Schwinger operator is defined to be , and it is denoted by . Using Euler's Lemma one may determine that . So .
See also 
- Clifford algebra
- Complex spin structure
- Conformal manifold
- Conformally flat manifold
- Dirac operator
- Poincaré metric
- Spin group
- Spin structure
- Spinor bundle
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