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, $d+*d*$ on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on $C_{0}^{\infty}(\mathbf{R}^{n})$ 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 Spin^c 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

D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}

where e₁, ..., e_n is an orthonormal basis for Rⁿ, and Rⁿ is considered to be embedded in a complex Clifford algebra, Cℓ_n(C) so that e_j² = −1.

This gives

D^{2} = -\Delta_{n}

where Δ_n is the Laplacian in n-euclidean space.

The fundamental solution to the euclidean Dirac operator is

G(x-y):=\frac{1}{\omega_{n}}\frac{x-y}{\|x-y\|^n}

where ω_n is the surface area of the unit sphere Sⁿ⁻¹.

Note that

D\frac{1}{(n-2)\omega_{n}\|x-y\|^{n-2}}=G(x-y)

where

\frac{1}{(n-2)\;\omega_{n}\;\|x-y\|^{n-2}}

is the fundamental solution to Laplace's equation for n ≥ 3.

The most basic example of a Dirac operator is the Cauchy–Riemann operator

\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}

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

-\frac{x}{\|x\|^{2}}\in\mathbf{R}^{n}.

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.

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 3D. 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, Cℓ_n. 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 R^n,+ with boundary Rⁿ⁻¹, the span of e₁, ..., e_n−1, under the Fourier transform the symbol of the Dirac operator

D_{n-1}=\sum_{j=1}^{n-1}\frac{\partial}{\partial x_{j}}

is iζ where

\zeta=\zeta_{1}e_{1}+\ldots+\zeta_{n-1}e_{n-1}.

In this setting the Plemelj formulas are

\pm\tfrac{1}{2}+G(x-y)|_{\mathbf{R}^{n-1}}

and the symbols for these operators are, up to a sign,

\frac{1}{2} \left (1\pm i\frac{\zeta}{\|\zeta\|} \right ).

These are projection operators, otherwise known as mutually annihilating idempotents, on the space of Cℓ_n(C) valued square integrable functions on Rⁿ⁻¹.

Note that

G|_{\mathbf{R}^{n}}=\sum_{j=1}^{n-1}e_{j}R_{j}

where R_j is the j-th Riesz potential,

\frac{x_{j}}{\|x\|^{n}}.

As the symbol of $G|_{\mathbf{R}^{n}}$ is

\frac{i\zeta}{\|\zeta\|}

it is easily determined from the Clifford multiplication that

\sum_{j=1}^{n-1}R_{j}^{2}=1.

So the convolution operator $G|_{\mathbf{R}^{n}}$ is a natural generalization to euclidean space of the Hilbert transform.

Suppose U′ is a domain in Rⁿ⁻¹ and g(x) is a Cℓ_n(C) valued real analytic function. Then g has a Cauchy–Kovalevskaia extension to the Dirac equation on some neighborhood of U′ in Rⁿ. The extension is explicitly given by

\sum_{j=0}^{\infty} \left (x_{n}e_{n}^{-1}D_{n-1} \right )^{j}g(x).

When this extension is applied to the variable x in

e^{-i\langle x,\zeta\rangle} \left (\tfrac{1}{2} \left (1\pm i\frac{\zeta}{\|\zeta\|} \right ) \right )

we get that

e^{-i\langle x,\zeta\rangle}

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.

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)

The Cayley transform or stereographic projection from Rⁿ to the unit sphere Sⁿ transforms the euclidean Dirac operator to a spherical Dirac operator D_S. Explicitly

D_{S}=x(\Gamma_{n}+\frac{n}{2})

where Γ_n is the spherical Beltrami–Dirac operator

\sum\nolimits_{1\leq i<j\leq n+1}e_{i}e_{j} \left (x_{i}\frac{\partial}{\partial x_{j}}-x_{j}\frac{\partial}{\partial x_{i}} \right )

and x in Sⁿ.

The Cayley transform over n-space is

y=C(x)=(e_{n+1}x+1)(x+e_{n+1})^{-1}, \qquad x \in \mathbf{R}^n.

Its inverse is

x=(-e_{n+1}+1)(y-e_{n+1})^{-1}.

For a function f(x) defined on a domain U in n-euclidean space and a solution to the Dirac equation, then

J \left (C^{-1},y \right ) f \left (C^{-1}(y) \right )

is annihilated by D_S, on C(U) where

J(C^{-1},y)=\frac{y-e_{n+1}}{\|y-e_{n+1}\|^{n}}.

Further

D_{S}(D_{S}-x)=\triangle_{S},

the conformal Laplacian or Yamabe operator on Sⁿ. Explicitly

\triangle_{S}=-\triangle_{LB}+\tfrac{1}{4}n(n-2)

where $\triangle_{LB}$ is the Laplace–Beltrami operator on Sⁿ. The operator $\triangle_{S}$ is, via the Cayley transform, conformally equivalent to the euclidean Laplacian. Also

D_s(D_S-x)(D_S-x)(D_S-2x)

is the Paneitz operator,

-\triangle_{S}(\triangle_{S}+2),

on the n-sphere. Via the Cayley transform this operator is conformally equivalent to the bi-Laplacian, $\triangle_{n}^{2}$ . These are all examples of operators of Dirac type.

Möbius transform

A Möbius transform over n-euclidean space can be expressed as

\frac{ax+b}{cx+d},

where a, b, c and d ∈ Cℓ_n and satisfy certain constraints. The associated 2 × 2 matrix is called an Ahlfors–Vahlen matrix. If

y=M(x)+\frac{ax+b}{cx+d}

and Df(y) = 0 then $J(M,x)f(M(x))$ is a solution to the Dirac equation where

J(M,x)=\frac{\widetilde{cx+d}}{\|cx+d\|^{n}}

and ~ is a basic antiautomorphism acting on the Clifford algebra. The operators D^k, or Δ_n^k/2 when k is even, exhibit similar covariances under Möbius transform including the Cayley transform.

When ax+b and cx+d are non-zero they are both members of the Clifford group.

\frac{ax+b}{cx+d}=\frac{-ax-b}{-cx-d}

then we have a choice in sign in defining J(M, x). 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. 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, M, with a spinor bundle S then given a smooth section s(x) in S then in terms of a local orthonormal basis e₁(x), ..., e_n(x) of the tangent bundle of M the Atiyah–Singer–Dirac operator acting on s is defined to be

Ds(x)=\sum_{j=1}^{n}e_{j}(x)\tilde{\Gamma}_{e_{j}(x)}s(x)

where $\widetilde{\Gamma}$ is the lifting to S of the Levi-Civita connection on M. When M is n-euclidean space we return to the euclidean Dirac operator.

From an Atiyah–Singer–Dirac operator D we have the Lichnerowicz formula

D^{2}=\Gamma^{*}\Gamma+\tfrac{\tau}{4}

where τ is the scalar curvature on the manifold, and Γ* is the adjoint of Γ. The operator D² is known as the spinorial Laplacian.

If M is compact and τ ≥ 0 and τ > 0 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 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

C(x,y):=D^{-1}*\delta_{y}, \qquad x \neq y \in M,

where δ_y is the Dirac delta function evaluated at y. 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, Cℓ_n into Cℓ_n−1+Cℓ_n-1e_n. So for a in Cℓ_n one may express a as b+ce_n with a, b in Cℓ_n−1. One then has projection operators P and Q defined as follows P(a) = b and Q(a) = c. The Hodge–Dirac operator acting on a function f with respect to the hyperbolic metric in upper half space is now defined to be

Mf=Df+\frac{n-2}{x_{n}}Q(f)

In this case

M^{2}f=-\triangle_{n}P(f)+\frac{n-2}{x_{n}}\frac{\partial P(f)}{\partial x_{n}}- \left (\triangle_{n}Q(f)-\frac{n-2}{x_{n}}\frac{\partial Q(f)}{\partial x_{n}}+ \frac{n-2}{x_{n}^{2}}Q(f) \right )e_{n}

The operator

\triangle_{n}-\frac{n-2}{x_{n}}\frac{\partial}{\partial x_{n}}

is the Laplacian with respect to the Poincaré metric while the other operator is an example of a Weinstein operator.

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 R_k is a conformally covariant first order differential operator. Here k = 0, 1, 2, .... When k = 0, the Rarita–Schwinger operator is just the Dirac operator. In representation theory for the orthogonal group, O(n) it is common to consider functions taking values in spaces of homogeneous harmonic polynomials. When one refines this representation theory to the double covering Pin(n) of O(n) one replaces spaces of homogeneous harmonic polynomials by spaces of k homogeneous polynomial solutions to the Dirac equation, otherwise known as k monogenic polynomials. One considers a function f(x, u) where x in U, a domain in Rⁿ, and u varies over Rⁿ. Further f(x, u) is a k-monogenic polynomial in u. Now apply the Dirac operator D_x in x to f(x, u). Now as the Clifford algebra is not commutative D_xf(x, u) then this function is no longer k monogenic but is a homogeneous harmonic polynomial in u. Now for each harmonic polynomial h_k homogeneous of degree k there is an Almansi–Fischer decomposition

h_{k}(x)=p_{k}(x)+xp_{k-1}(x)

where p_k and p_k−1 are respectively k and k−1 monogenic polynomials. Let P be the projection of h_k to p_k then the Rarita–Schwinger operator is defined to be PD_k, and it is denoted by R_k. Using Euler's Lemma one may determine that

D_{u}up_{k-1}(u)=(-n-2k+2)p_{k-1}.

R_{k}=\left(I+\frac{1}{n+2k-2}uD_{u}\right)D_{x}.

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