Hermitian Clifford Analysis and the Hermitian Monogenic Functions

Introduction

Hermitian Clifford analysis is the refinement of Clifford analysis in which the ambient space is an even-dimensional Euclidean space $\mathbb{R}^{2n}$ read as the complex space $\mathbb{C}^n$, the single Cauchy–Riemann operator is split into two operators invariant under the unitary group, and the functions of the theory are the simultaneous null solutions of the two. The refinement is the complex case of the Hermitian construction that Clifford Analysis records in its section on the Hermitian (there spelled Hermitean) refinement; the quaternionic case, with four operators in $\mathbb{R}^{4n}$, is Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator, and the operator-algebraic reading of the split is The Hermitian Dirac Operator and the Fischer Decomposition.

The setting is the one of The Dirac Operator: the Clifford algebra $A=\mathrm{Cl}_{0,2n}$, generated by $e_1,\dots,e_{2n}$ with $e_ae_b+e_be_a=-2\delta_{ab}$, and a left Clifford module $\mathcal{S}$ of values, extended to a complex module when needed. What distinguishes the Hermitian refinement is an extra structure: the complex structure of $\mathbb{R}^{2n}$, which selects the two Hermitian Dirac operators out of the one Euclidean operator $D$ and thereby selects the unitary group out of the orthogonal group. The class of functions is therefore smaller — it is a simultaneous kernel — and its symmetry group is larger than that of a single operator of the Clifford theory.

The article fixes the Witt basis and the two operators, defines the Hermitian monogenic functions, records the splitting of the Euclidean operator and of the Laplacian, and states the invariance and the relation to holomorphy in the underlying complex variables. The integral theory built on the split Cauchy kernel is The Hermitian Cauchy Integral and the Boundary Values; the module and form structures are Hermitian Hilbert Modules over a Clifford Algebra; the operator-theoretic and adjoint reading is The Hermitian Dirac Operator and The Hermitian Cauchy Kernel as an Adjoint.

The Complex Structure and the Witt Basis

The Complexification

Definition. Let $N=2n$ and let $A=\mathrm{Cl}_{0,N}$ be the Clifford algebra of a negative-definite form of rank $N$. The complexification $A_{\mathbb{C}}=A\otimes_{\mathbb{R}}\mathbb{C}$ is the Clifford algebra of the complexified quadratic space; the imaginary unit $i$ is central in $A_{\mathbb{C}}$ and commutes with every generator. The ambient space $\mathbb{R}^{2n}$ is identified with $\mathbb{C}^n$ by

$$ z_j = x_{2j-1}+i\,x_{2j} , \qquad \bar z_j = x_{2j-1}-i\,x_{2j} , \qquad j=1,\dots,n , $$

and the corresponding Wirtinger derivatives are

$$ \partial_{z_j} = \tfrac12\bigl(\partial_{2j-1}-i\,\partial_{2j}\bigr) , \qquad \partial_{\bar z_j} = \tfrac12\bigl(\partial_{2j-1}+i\,\partial_{2j}\bigr) . $$

The Witt Basis

Definition. The Witt basis of $A_{\mathbb{C}}$ is the $2n$ elements

$$ f_j = \tfrac12\bigl(e_{2j-1}+i\,e_{2j}\bigr) , \qquad f_j^{*} = -\tfrac12\bigl(e_{2j-1}-i\,e_{2j}\bigr) , \qquad j=1,\dots,n . $$

Proposition (the anticommutation relations). The Witt basis consists of isotropic elements that pair to the identity:

$$ f_j^2 = 0 , \qquad (f_j^{*})^2 = 0 , \qquad f_jf_k+f_kf_j = 0 , \qquad f_j^{*}f_k^{*}+f_k^{*}f_j^{*} = 0 , $$

$$ f_jf_k^{*}+f_k^{*}f_j = \delta_{jk} , \qquad j,k=1,\dots,n . $$

Proof. Each is a direct computation in the relations $e_ae_b+e_be_a=-2\delta_{ab}$, with $i$ central. For example $f_j^2=\tfrac14(e_{2j-1}^2+i(e_{2j-1}e_{2j}+e_{2j}e_{2j-1})+i^2e_{2j}^2) =\tfrac14(-1+0+1)=0$; the pairing $\{f_j,f_j^*\}=1$ and the vanishing of all other anticommutators follow the same way, the cross terms cancelling because distinct generators anticommute and the $i$ factors are central. $\square$

Remark (the reading). The relations are those of $n$ creation and $n$ annihilation operators; the Witt basis presents the complexified Clifford algebra as a fermionic Fock algebra, and the spaces $\bigwedge^k\mathbb{C}^n$ built from the $f_j^{*}$ acting on the vacuum are the spinor modules of the Hermitian theory. The presentation belongs to the representation theory of the Clifford algebra and is developed in Spin Representations and Clifford Modules with Inner Conjugation; here it is the algebraic skeleton of the split.

The Hermitian Dirac Operators

Definition. The Hermitian Dirac operators are

$$ \partial_{\underline{z}} = \sum_{j=1}^{n} f_j\,\partial_{z_j} , \qquad \partial_{\bar z} = \sum_{j=1}^{n} f_j^{*}\,\partial_{\bar z_j} . $$

Theorem (the two operators split the Euclidean one and the Laplacian). The Euclidean operator $D=\sum_{a=1}^{2n}e_a\partial_a$ and the two Hermitian operators satisfy

$$ D = 2\bigl(\partial_{\underline{z}}-\partial_{\bar z}\bigr) , \qquad \{\partial_{\underline{z}},\partial_{\bar z}\} = \tfrac14\,\Delta_{2n} . $$

Moreover each operator is isotropic,

$$ \partial_{\underline{z}}^2 = 0 , \qquad \partial_{\bar z}^2 = 0 , $$

so that the pair $(\partial_{\underline{z}},\partial_{\bar z})$ is a complex of first-order operators whose anticommutator is the Laplacian.

Proof. Write $e_{2j-1}=f_j-f_j^{*}$ and $e_{2j}=-i(f_j+f_j^{*})$, and $\partial_{2j-1}=\partial_{z_j}+\partial_{\bar z_j}$, $\partial_{2j}=i(\partial_{z_j}-\partial_{\bar z_j})$; then

$$ e_{2j-1}\partial_{2j-1}+e_{2j}\partial_{2j} =(f_j-f_j^{*})\bigl(\partial_{z_j}+\partial_{\bar z_j}\bigr) +(f_j+f_j^{*})\bigl(\partial_{z_j}-\partial_{\bar z_j}\bigr) =2\bigl(f_j\partial_{z_j}-f_j^{*}\partial_{\bar z_j}\bigr), $$

and summing over $j$ gives the first identity. For the second, the anticommutator sums the diagonal pairing of the Witt basis against the mixed Wirtinger derivatives: $\{\partial_{\underline z},\partial_{\bar z}\}=\sum_j\partial_{z_j}\partial_{\bar z_j} =\tfrac14\sum_j(\partial_{2j-1}^2+\partial_{2j}^2)=\tfrac14\Delta_{2n}$. The squares vanish because $f_jf_k=-f_kf_j$ while $\partial_{z_j}\partial_{z_k}=\partial_{z_k}\partial_{z_j}$, and the same with daggers. $\square$

Corollary (the differential complex). The two operators make $\mathcal{E}(\Omega;\mathcal{S})\xrightarrow{\ \partial_{\bar z}\ }\mathcal{E}\xrightarrow{\ \partial_{\underline z}\ }$ a complex, and the associated graded of the algebra they generate is the exterior algebra on the $f_j,f_j^{*}$; the class of functions annihilated by both is the kernel of the Laplacian by the anticommutation relation.

Hermitian Monogenic Functions

Definition. A $C^1$ function $f:\Omega\to\mathcal{S}$, $\Omega\subseteq\mathbb{R}^{2n}$, is Hermitian monogenic when

$$ \partial_{\underline{z}}f = 0 \quad\text{and}\quad \partial_{\bar z}f = 0 . $$

Proposition (Hermitian monogenicity implies monogenicity). Every Hermitian monogenic function is monogenic and harmonic: $\ker\partial_{\underline z}\cap\ker\partial_{\bar z}\subseteq\ker D\subseteq \ker\Delta_{2n}$. The converse fails, and the class of Hermitian monogenic functions is strictly smaller than the class of monogenic functions.

Proof. The two equations give $Df=2(\partial_{\underline z}-\partial_{\bar z})f=0$, so $f$ is monogenic; $\partial_{\underline z}f=\partial_{\bar z}f=0$ gives $\Delta_{2n}f=4(\partial_{\underline z}\partial_{\bar z}+\partial_{\bar z}\partial_{\underline z})f=0$. For the converse: the Hermitian system is a pair of equations where monogenicity is one, so the simultaneous kernel is in general strictly smaller, the two coinciding only when the pair assembles a single equation. The strictness is recorded for the quaternionic refinement in Clifford Analysis, and it occurs in the complex refinement as soon as the value module is not scalar; the reason is visible in the algebra, since the Witt basis elements $f_j$ are zero divisors, so $\partial_{\underline z}f=0$ does not force the vanishing of each component $\partial_{z_j}f$. $\square$

Remark (the class is a system, not an equation). The vanishing of each of the two operators alone defines a class in which the Laplacian does not vanish — $\partial_{\underline z}f=0$ alone gives $0=\Delta_{2n}f=4\partial_{\bar z}\partial_{\underline z}f$, so only $\partial_{\bar z}\partial_{\underline z}f=0$, and the second derivative is not required to vanish. The simultaneous system is what makes the class a theory: Hermitian monogenic functions are the solutions of an overdetermined elliptic system, and their rigidity — they satisfy the maximum principle, the mean value property, Liouville's theorem and the identity theorem as harmonic functions — is inherited from the Laplacian.

Invariance and the Relation to Holomorphy

The Unitary Group

Theorem (invariance). The two Hermitian Dirac operators, and hence the class of Hermitian monogenic functions, are invariant under the natural action of the unitary group $U(n)$ on $\mathbb{C}^n$: a $U(n)$-transformation of the variables lifts to a Clifford-algebra automorphism (an element of $\mathrm{Spin}(2n)$) under which the Witt basis is transformed among itself, and $(\partial_{\underline z},\partial_{\bar z})$ is carried to itself.

Proof. The unitary group is the stabiliser of the complex structure in the orthogonal group; an element $g\in U(n)$ acts on $\mathbb{C}^n=\mathbb{R}^{2n}$ by an orthogonal transformation commuting with the complex structure, and its lift into $\mathrm{Spin}(2n)$ acts on the vectors $e_{2j-1},e_{2j}$ by the same orthogonal map. Such a map carries the $(1,0)$-vectors $f_j$ to linear combinations of the $(1,0)$-vectors and preserves the splitting, so it preserves the two operators. $\square$

Remark (what the split buys). The Euclidean operator $D$ alone is invariant only under $SO(2n)$; the Hermitian pair is invariant under the larger unitary structure of the space, and the class of Hermitian monogenic functions is the smallest class carrying that invariance. This is the reason the refinement is a statement about the operator: the single Cauchy–Riemann equation is replaced by the pair, and the symmetry group grows to $U(n)$.

The Relation to Holomorphy

Remark (the underlying complex variables). The operator $\partial_{\bar z}$ alone is the anti-holomorphic Cauchy–Riemann operator of the variables $z_1,\dots,z_n$, acting on the values by $f_j^{*}$, and its vanishing alone defines a class of functions that are holomorphic in the $z_j$ in the Clifford-module sense. The Hermitian system adds the vanishing of $\partial_{\underline z}$. In the special cases in which the value module and the two operators assemble the same equation, Hermitian monogenicity is equivalent to holomorphy in the $z_j$; in the general case it is strictly stronger, and the extra rigidity is exactly what the unitary invariance demands. The general several-variable theory — the Hartogs phenomenon, the domains of holomorphy, the Stein theory — is Several Complex Variables's; the Clifford-module refinement is the subject here.

Example (the scalar case $n=1$). For $n=1$ the pair reduces to two operators in $\mathbb{C}$, $\partial_{\bar z}=\tfrac12(\partial_1+i\partial_2)f_1^{*}$ and its conjugate; in the scalar model $f_1,f_1^{*}$ act as the idempotents of a minimal left ideal, and the Hermitian monogenic functions are the holomorphic functions of $z$ valued in the ideal. This is the case in which the refinement is holomorphy itself, recorded in Complex Analysis.

The Fischer Decomposition in the Hermitian Setting

Remark (the Hermitian Fischer decomposition). The Hermitian system has its own Fischer decomposition: the space of homogeneous polynomials is decomposed into the simultaneous kernels of $\partial_{\underline z}$ and $\partial_{\bar z}$, multiplied by the appropriate powers of the Hermitian variables. The decomposition is finer than the monogenic one of The Fischer Operator, because the simultaneous kernel is smaller, and its layers are indexed by pairs of multi-indices rather than by a single degree. The operator-algebraic form of the decomposition, and the bivariate representation of Hermitian monogenic polynomials, is The Hermitian Dirac Operator and the Fischer Decomposition; the dimension count of the Hermitian monogenic pieces is recorded there and in Clifford Analysis.

Remark (the operator, split). Read as an operator, the Hermitian refinement replaces the single operator $D$ by the pair, replaces the single symbol by the pair of symbols $c(\sum_jf_j\xi_{z_j})$ and $c(\sum_jf_j^{*}\xi_{\bar z_j})$, and replaces the factorisation $D^2=-\Delta$ by the anticommutation relation $\{\partial_{\underline z},\partial_{\bar z}\}=\tfrac14\Delta$. The whole of the refinement — integral formulae, boundary values, module structure — rests on the last identity, and it is the two-operator case of the matrix device that Clifford Analysis records for the four-operator quaternionic theory.

Summary

In $\mathbb{R}^{2n}$ read as $\mathbb{C}^n$, with $z_j=x_{2j-1}+ix_{2j}$, the Witt basis $f_j=\tfrac12(e_{2j-1}+ie_{2j})$, $f_j^{*}=-\tfrac12(e_{2j-1}-ie_{2j})$ consists of isotropic elements with $\{f_j,f_k^{*}\}=\delta_{jk}$ and all other anticommutators zero. The Hermitian Dirac operators $\partial_{\underline z}=\sum_jf_j\partial_{z_j}$ and $\partial_{\bar z}=\sum_jf_j^{*}\partial_{\bar z_j}$ are isotropic, $\partial_{\underline z}^2=\partial_{\bar z}^2=0$, and satisfy $\{\partial_{\underline z},\partial_{\bar z}\}=\tfrac14\Delta_{2n}$; the Euclidean operator splits as $D=2(\partial_{\underline z}-\partial_{\bar z})$. A function is Hermitian monogenic when both operators annihilate it; such a function is monogenic and harmonic, and the class is strictly smaller than the monogenic class, the system being overdetermined and elliptic. The pair and the class are invariant under the unitary group $U(n)$, which acts by the lift into $\mathrm{Spin}(2n)$; the single operator is invariant only under $SO(2n)$, so the refinement trades one equation for a pair and gains the unitary structure. The operator $\partial_{\bar z}$ alone is the anti-holomorphic Cauchy–Riemann operator in the $z_j$, and in special cases Hermitian monogenicity is holomorphy. The quaternionic case with four operators is Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator; the Fischer decomposition and the bivariate representation are The Hermitian Dirac Operator and the Fischer Decomposition; the integral theory is The Hermitian Cauchy Integral and the Boundary Values; and the general Hermitian construction is Clifford Analysis.

Summary of Notation

Symbol	Meaning
$N=2n$, $\mathbb{R}^{2n}\cong\mathbb{C}^n$	Ambient space and its complex structure
$z_j=x_{2j-1}+ix_{2j}$, $\bar z_j$	Complex coordinates
$\partial_{z_j},\partial_{\bar z_j}$	Wirtinger derivatives
$f_j=\tfrac12(e_{2j-1}+ie_{2j})$	Witt basis, $(1,0)$ part
$f_j^{*}=-\tfrac12(e_{2j-1}-ie_{2j})$	Witt basis, $(0,1)$ part; $\{f_j,f_k^{*}\}=\delta_{jk}$
$\partial_{\underline z}=\sum_jf_j\partial_{z_j}$	Hermitian Dirac operator, $(1,0)$
$\partial_{\bar z}=\sum_jf_j^*\partial_{\bar z_j}$	Hermitian Dirac operator, $(0,1)$
$\partial_{\underline z}^2=\partial_{\bar z}^2=0$, $\{\partial_{\underline z},\partial_{\bar z}\}=\tfrac14\Delta_{2n}$	Isotropy and the split of the Laplacian
$D=2(\partial_{\underline z}-\partial_{\bar z})$	Splitting of the Euclidean operator
$\partial_{\underline z}f=\partial_{\bar z}f=0$	Hermitian monogenicity
$U(n)$	Invariance group of the pair