Clifford analysis

Clifford analysis,usingClifford algebrasnamed afterWilliam Kingdon Clifford,is the study ofDirac 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,$d+{\star }d{\star }$on aRiemannian manifold,the Dirac operator in euclidean space and its inverse on$C_{0}^{\infty }(\mathbf {R} ^{n})$and their conformal equivalents on the sphere, theLaplacianin euclideann-space and theAtiyah–Singer–Dirac operator on aspin manifold,Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators onSpin^Cmanifolds, systems of Dirac operators, thePaneitz operator,Dirac operators onhyperbolic space,the hyperbolic Laplacian and Weinstein equations.

Euclidean space

In Euclidean space the Dirac operator has the form

$D=\sum _{j=1}^{n}e_{j}{\frac {\partial }{\partial x_{j}}}$

wheree₁,...,e_nis an orthonormal basis forRⁿ,andRⁿis considered to be embedded in a complexClifford algebra,Cl_n(C) so thate_j²= −1.

This gives

$D^{2}=-\Delta _{n}$

where Δ_nis theLaplacianinn-euclidean space.

Thefundamental solutionto the euclidean Dirac operator is

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

where ω_nis the surface area of the unit sphereSⁿ⁻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 thefundamental solutiontoLaplace's equationforn≥ 3.

The most basic example of a Dirac operator is theCauchy–Riemann operator

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

in the complex plane. Indeed, many basic properties of one variablecomplex analysisfollow through for many first order Dirac type operators. In euclidean space this includes aCauchy Theorem,aCauchy integral formula,Morera's theorem,Taylor series,Laurent seriesandLiouville Theorem.In this case theCauchy kernelisG(x−y). The proof of theCauchy integral formulais the same as in one complex variable and makes use of the fact that each non-zero vectorxin 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 theKelvin inverseofx.Solutions to the euclidean Dirac equationDf= 0 are called (left) monogenic functions. Monogenic functions are special cases ofharmonic spinorson aspin manifold.

In 3 and 4 dimensions Clifford analysis is sometimes referred to asquaternionicanalysis. Whenn= 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 ofCauchy transforms,Bergman kernels,Szegő kernels,Plemelj operators,Hardy spaces,aKerzman–Stein formulaand a Π, orBeurling–Ahlfors,transform. These have all found applications in solvingboundary value problems,including moving boundary value problems,singular integralsandclassic harmonic analysis.In particular Clifford analysis has been used to solve, in certainSobolev 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 complexClifford algebraby a realClifford algebra,Cl_n.This is not the case though when we need to deal with the interaction between theDirac operatorand theFourier transform.

The Fourier transform

When we consider upper half spaceR^n,+with boundaryRⁿ⁻¹,the span ofe₁,...,e_n−1,under theFourier transformthe symbol of the Dirac operator

$D_{n-1}=\sum _{j=1}^{n-1}{\frac {\partial }{\partial x_{j}}}$

isiζwhere

$\zeta =\zeta _{1}e_{1}+\cdots +\zeta _{n-1}e_{n-1}.$

In this setting thePlemelj formulasare

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

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 onRⁿ⁻¹.

Note that

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

whereR_jis thej-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 theconvolution operator$G|_{\mathbf {R} ^{n}}$is a natural generalization to euclidean space of theHilbert transform.

SupposeU′ is a domain inRⁿ⁻¹andg(x) is a Cl_n(C) valuedreal analytic function.Thenghas aCauchy–Kovalevskaia extensionto theDirac equationon some neighborhood ofU′ inRⁿ.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 variablexin

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

we get that

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

is the restriction toRⁿ⁻¹ofE₊+E₋whereE₊is a monogenic function in upper half space andE₋is a monogenic function in lower half space.

There is also aPaley–Wiener theoreminn-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 onconformally flat manifoldsandconformal manifoldswhich are simultaneouslyspin manifolds.

Cayley transform (stereographic projection)

TheCayley transformorstereographic projectionfromRⁿto the unit sphereSⁿtransforms the euclidean Dirac operator to a spherical Dirac operatorD_S.Explicitly

$D_{S}=x\left(\Gamma _{n}+{\frac {n}{2}}\right)$

where Γ_nis the spherical Beltrami–Dirac operator

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

andxinSⁿ.

TheCayley transformovern-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 functionf(x) defined on a domainUinn-euclidean space and a solution to theDirac equation,then

$J(C^{-1},y)f(C^{-1}(y))$

is annihilated byD_S,onC(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 onSⁿ.Explicitly

$\triangle _{S}=-\triangle _{LB}+{\tfrac {1}{4}}n(n-2)$

where$\triangle _{LB}$is theLaplace–Beltrami operatoronSⁿ.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 then-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

AMöbius transformovern-euclidean space can be expressed as

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

wherea,b,candd∈ Cl_nand satisfy certain constraints. The associated2 × 2matrix is called an Ahlfors–Vahlen matrix. If

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

andDf(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 basicantiautomorphismacting on theClifford algebra.The operatorsD^k,or Δ_n^k/2whenkis even, exhibit similar covariances underMöbius transformincluding theCayley transform.

Whenax+bandcx+dare non-zero they are both members of theClifford group.

As

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

then we have a choice in sign in definingJ(M,x). This means that for aconformally flat manifoldMwe need aspin structureonMin order to define aspinor bundleon whose sections we can allow a Dirac operator to act. Explicit simple examples include then-cylinder, theHopf manifoldobtained fromn-euclidean space minus the origin, and generalizations ofk-handled toruses obtained from upper half space by factoring it out by actions of generalized modular groups acting on upper half space totally discontinuously. ADirac operatorcan be introduced in these contexts. These Dirac operators are special examples of Atiyah–Singer–Dirac operators.

Atiyah–Singer–Dirac operator

Given aspin manifoldMwith aspinor bundleSand a smooth sections(x) inSthen, in terms of a local orthonormal basise₁(x),...,e_n(x) of the tangent bundle ofM,the Atiyah–Singer–Dirac operator acting onsis defined to be

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

where${\widetilde {\Gamma }}$is thespin connection,the lifting toSof theLevi-Civita connectiononM.WhenMisn-euclidean space we return to the euclideanDirac operator.

From an Atiyah–Singer–Dirac operatorDwe have theLichnerowicz formula

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

whereτis thescalar curvatureon themanifold,and Γ^∗is the adjoint of Γ. The operatorD²is known as the spinorial Laplacian.

IfMis compact andτ≥ 0andτ> 0somewhere then there are no non-trivialharmonic spinorson the manifold. This is Lichnerowicz' theorem. It is readily seen that Lichnerowicz' theorem is a generalization ofLiouville's theoremfrom one variable complex analysis. This allows us to note that over the space of smooth spinor sections the operatorDis 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

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

whereδ_yis theDirac delta functionevaluated aty.This gives rise to aCauchy kernel,which is thefundamental solutionto this Dirac operator. From this one may obtain aCauchy integral formulaforharmonic spinors.With this kernel much of what is described in the first section of this entry carries through for invertible Atiyah–Singer–Dirac operators.

UsingStokes' 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, orPoincaré metric.

For upper half space one splits theClifford algebra,Cl_ninto Cl_n−1+ Cl_n−1e_n.So forain Cl_none may expressaasb+ce_nwitha,bin Cl_n−1.One then has projection operatorsPandQdefined as followsP(a) =bandQ(a) =c.The Hodge–Dirac operator acting on a functionfwith 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 theLaplacianwith respect to thePoincaré metricwhile the other operator is an example of a Weinstein operator.

Thehyperbolic Laplacianis 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 andPin groups.The operatorR_kis aconformally covariantfirst order differential operator. Herek= 0, 1, 2,.... Whenk= 0, the Rarita–Schwinger operator is just the Dirac operator. In representation theory for theorthogonal group,O(n) it is common to consider functions taking values in spaces of homogeneousharmonic polynomials.When one refines thisrepresentation theoryto the double covering Pin(n) of O(n) one replaces spaces of homogeneous harmonic polynomials by spaces ofkhomogeneous polynomialsolutions to the Dirac equation, otherwise known askmonogenic polynomials. One considers a functionf(x,u) wherexinU,a domain inRⁿ,anduvaries overRⁿ.Furtherf(x,u) is ak-monogenic polynomial inu.Now apply the Dirac operatorD_xinxtof(x,u). Now as the Clifford algebra is not commutativeD_xf(x,u) then this function is no longerkmonogenic but is a homogeneous harmonic polynomial inu.Now for each harmonic polynomialh_khomogeneous of degreekthere is anAlmansi–Fischer decomposition

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

wherep_kandp_k−1are respectivelykandk−1monogenic polynomials.LetPbe the projection ofh_ktop_kthen the Rarita–Schwinger operator is defined to bePD_k,and it is denoted byR_k.Using Euler's Lemma one may determine that

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

So

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

Conferences and Journals

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 theInternational Conference on Clifford Algebras and their Applications in Mathematical Physics (ICCA)andApplications of Geometric Algebra in Computer Science and Engineering (AGACSE)series. A main publication outlet is the Springer journalAdvances in Applied Clifford Algebras.

See also

• Clifford algebra
• Complex spin structure
• Conformal manifold
• Conformally flat manifold
• Dirac operator
• Poincaré metric
• Spin group
• Spin structure
• Spinor bundle

References

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External links

• Lecture notes on Dirac operators in analysis and geometry
• Calderbank, David M.J. (1997-12-19),Dirac operators and Clifford analysis on manifolds with boundary,Danish Mathematical Society, DMF-1997-12-007 PP-1997-53, archived fromthe originalon 2009-08-13