View source for Clifford analysis

Clifford analysis studies functions with values in a [[Clifford algebra|Clifford algebra]], and, as such, is a direct generalization to higher dimensions of the classical theory of functions of one complex variable (cf. [[Functions of a complex variable, theory of|Functions of a complex variable, theory of]]). It has its roots in quaternionic analysis, which was developed from the 1920s onwards as an, albeit modest, counterpart of the theory of two complex variables. The latter was to evolve into the vast and rich theory of several complex variables. The former gained renewed interest in the 1950s and led to hypercomplex function theory (cf. also [[Hypercomplex functions|Hypercomplex functions]]), renamed Clifford analysis in the 1980s, when it grew into an autonomous discipline.

Clifford algebra, also called geometric algebra, was introduced in 1878 by the English mathematician and philosopher W.K. Clifford, generalizing W.R. Hamilton's quaternions (cf. also [[Quaternion|Quaternion]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102501.png" /> is a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102502.png" />-dimensional (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102503.png" />) quadratic space with non-degenerate [[Bilinear form|bilinear form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102504.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102505.png" />, then, roughly speaking, the Clifford algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102506.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102507.png" /> is the real, associative and non-commutative algebra with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102508.png" />, containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c1102509.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025010.png" /> as subspaces, and with a multiplication governed by the rule

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025011.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025012.png" /> is an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025013.png" />, i.e.,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025014.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025015.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025016.png" /></td> </tr></table>

then for vectors

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025017.png" /></td> </tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025018.png" />, their (geometric) product splits into a scalar and a so-called bivector part:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025019.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025020.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025021.png" /></td> </tr></table>

Introducing conjugation as the anti-involution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025022.png" />, the squared length of a vector can be factorized as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025023.png" /></td> </tr></table>

This factorization implies the linearization of some fundamental partial differential operators of mathematical physics, such as the [[Laplace operator|Laplace operator]] and the d'Alembert operator, and turns out to be crucial to Clifford analysis.

A general reference for Clifford algebra is [[#References|[a4]]].

In the so-called Euclidean case, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025025.png" />, two fundamental differential operators are introduced: the Euclidean Dirac operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025026.png" /></td> </tr></table>

and the Euclidean Cauchy–Riemann operator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025027.png" /></td> </tr></table>

the latter being a generalization of the classical Cauchy–Riemann operator which is obtained for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025028.png" />, identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025029.png" /> with the imaginary unit (cf. also [[Cauchy–Riemann equations|Cauchy–Riemann equations]]). These operators linearize the Laplace operator in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025031.png" />.

Clifford analysis essentially deals with the study of null solutions of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025033.png" />, both cases showing many similarities, at least formally, as shown in [[#References|[a2]]].

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025034.png" />, continuously differentiable in an open domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025036.png" /> and taking values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025037.png" />, is said to be monogenic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025038.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025039.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025040.png" />.

Monogenicity thus leads to an overdetermined first-order system, generalizing the Cauchy–Riemann system. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025041.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025042.png" />-valued, the so-called Riesz system is obtained.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025043.png" /> is a [[Fundamental solution|fundamental solution]] of the Laplace operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025045.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025046.png" />-valued fundamental solution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025047.png" />.

The power of classical function theory of one complex variable is based on the Cauchy integral formula (cf. [[Cauchy integral theorem|Cauchy integral theorem]]) and the Weierstrass convergence theorem (cf. [[Weierstrass theorem|Weierstrass theorem]]) on the one hand, and on Riemann's concept of [[Conformal mapping|conformal mapping]] on the other; they constitute the nuclei of the so-called analytic and geometric function theories, respectively.

In their turn, the basic analytic properties of monogenic functions are based upon the Cauchy formula,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025048.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025049.png" /> is monogenic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025051.png" /> is a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025052.png" />-dimensional manifold with boundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025053.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025054.png" /> is the vector-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025055.png" />-dimensional surface element given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025056.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025057.png" /></td> </tr></table>

References for function-theoretic results in Clifford analysis are [[#References|[a1]]] and [[#References|[a2]]].

Among others, the Cauchy operator associated with the Dirac operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025058.png" />, as defined by the left-hand side of Cauchy's formula, is used in studying boundary value problems in domains of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025059.png" />.

The Cauchy operator associated with the Cauchy–Riemann operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025060.png" />, when applied to functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025061.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025062.png" />), gives rise to the Cauchy transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025063.png" /> and the singular Cauchy transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025064.png" /> on the Lipschitz surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025065.png" />, i.e. on a surface of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025066.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025067.png" /> is a Lipschitz function, possibly satisfying additional technical conditions (cf. also [[Lipschitz condition|Lipschitz condition]]). It turns out that the scalar part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025068.png" /> is the classical singular [[Double-layer potential|double-layer potential]] operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025069.png" />. As, moreover, the projection operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025070.png" /> determine the [[Hardy spaces|Hardy spaces]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025071.png" /> and the Hardy decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025072.png" />, the singular Cauchy transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025073.png" /> may be viewed as a generalization of the classical [[Hilbert transform|Hilbert transform]] on the real line. These results reveal the potentials of Clifford analysis for dealing with problems of [[Harmonic analysis|harmonic analysis]] in Euclidean space.

References for this area are [[#References|[a3]]] and [[#References|[a5]]].

Regarding the geometric viewpoint in Clifford analysis, Möbius transforms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025074.png" /> can be completely characterized by specific <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110250/c11025075.png" />-matrices over the Clifford group, the so-called Vahlen matrices. By means of this characterization, transforms of metrics and operators may be calculated. The introduction of Vahlen matrices has even inspired the recent development of Clifford analysis on hyperbolic spaces.

A reference for this area is [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Brackx, R. Delanghe, F. Sommen, "Clifford analysis" , Pitman (1982) {{MR|0697564}} {{ZBL|0529.30001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Delanghe, F. Sommen, V. Souček, "Clifford algebra and spinor valued functions: a function theory for the Dirac operator" , Kluwer Acad. Publ. (1992) {{MR|}} {{ZBL|0747.53001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Gilbert, M. Murray, "Clifford algebras and Dirac operators in harmonic analysis" , Cambridge Univ. Press (1991) {{MR|1130821}} {{ZBL|0733.43001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> I.R. Porteous, "Clifford algebras and the classical groups" , Cambridge Univ. Press (1995) {{MR|1369094}} {{ZBL|0855.15019}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> "Clifford algebras in analysis and related topics" J. Ryan (ed.) , CRC (1996) {{MR|1383097}} {{ZBL|0856.30035}} {{ZBL|0835.00014}} </TD></TR></table>