---
tag: CD7JZ43JBA
kind: definition
name: Clifford analysis
msc: [30G35, 15A66]
defines: DKVSAZQWF7
uses: [A38B1QFK20, 646PNPT1R6, 0E0B6RA8K0]
sources:
  - url: https://en.wikipedia.org/wiki/Clifford_analysis
---

## Definition

Let $n \ge 1$ and let $\mathbb{R}_{0,n}$ be the real [[Clifford algebra|A38B1QFK20]] generated by $e_1, \ldots, e_n$ with the relations $e_j^2 = -1$ and $e_j e_k = -e_k e_j$ for $j \ne k$.
The *Dirac operator* on $\R^n$ is $D = \sum_{j=1}^{n} e_j \dfrac{\partial}{\partial x_j}$, acting on differentiable functions from open subsets of $\R^n$ to $\mathbb{R}_{0,n}$.
*Clifford analysis* is the study of functions from open subsets of $\R^n$ to $\mathbb{R}_{0,n}$ by means of $D$ and related [[Dirac operators|3DXGJTEQZH]].
Its main objects are the [[monogenic functions|646PNPT1R6]], the solutions of $Df = 0$.

## Remarks

The anticommutation relations give $D^2 = -\Delta$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] of $\R^n$: the mixed terms $e_j e_k \partial_j \partial_k + e_k e_j \partial_k \partial_j$ cancel.
So every monogenic function of class $C^2$ has harmonic components.

Example: for $n = 1$, $\mathbb{R}_{0,1} = \R + \R e_1$ with $e_1^2 = -1$ is a copy of $\C$, and $D = e_1 \partial_x$ only sees one real variable.
To recover complex analysis one uses instead the operator $\partial_{x_0} + e_1 \partial_{x_1}$ on $\R^2$, which is twice the [[Wirtinger derivative|WMXDVF36ZT]] $\partial_{\bar z}$; many authors work with this operator $\partial_{x_0} + \sum_j e_j \partial_{x_j}$ on $\R^{n+1}$.

Non-example: harmonic analysis of real-valued functions alone is not Clifford analysis, since it uses only $\Delta$ and not its first-order factor $D$.