The Hermitian Dirac Operator and the Fischer Decomposition

Introduction

The Hermitian refinement of Hermitian Clifford Analysis and the Hermitian Monogenic Functions replaces the single Cauchy–Riemann operator $D$ of Clifford analysis by a pair of first-order operators in the even-dimensional space $\mathbb{R}^{2n}=\mathbb{C}^n$, whose simultaneous kernel is the class of Hermitian monogenic functions. This article treats the pair as an operator and the way it organises the polynomials: the split of $D$, the splitting of the Laplacian, the symbols of the two operators and the complex they define, the Hermitian variables that are their Fischer duals, the Fischer decomposition of the spinor-valued polynomials into Hermitian monogenic layers, and the bivariate representation of those polynomials in the variables $z_j,\bar z_j$ together with the Witt basis.

The setting, the Witt basis $f_j,f_j^{*}$ and the two operators $\partial_{\underline z}, \partial_{\bar z}$ are as in the preceding article, and its notation is used without repetition. The quaternionic case, with four operators and the Hadamard sign patterns, is Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator; the integral theory is The Hermitian Cauchy Integral and the Boundary Values; the adjoint and the operator-theoretic reading is The Hermitian Dirac Operator in the next group; the ordinary Fischer decomposition, of which the one here is the refinement, is The Fischer Operator; and the general construction is Clifford Analysis.

The Hermitian Dirac Operator as an Operator

The Pair and its Symbols

Definition. The Hermitian Dirac operators are the pair $\partial_{\underline z}=\sum_jf_j\partial_{z_j}$ and $\partial_{\bar z}=\sum_jf_j^{*}\partial_{\bar z_j}$ of the preceding article. Their principal symbols are left multiplication by the algebra elements $\sum_jf_j\xi_{z_j}$ and $\sum_jf_j^{*}\xi_{\bar z_j}$.

Proposition (the symbols are isotropic). For every covector $\xi$,

$$ \Bigl(\sum_jf_j\xi_{z_j}\Bigr)^2 = 0 , \qquad \Bigl(\sum_jf_j^{*}\xi_{\bar z_j}\Bigr)^2 = 0 , \qquad \Bigl\{\sum_jf_j\xi_{z_j},\ \sum_kf_k^{*}\eta_{\bar z_k}\Bigr\} = \sum_j\xi_{z_j}\eta_{\bar z_j} . $$

hence each symbol is a zero divisor of $A_{\mathbb{C}}$ and the two symbols pair to a scalar. In particular the operators are not elliptic — their symbols vanish on the isotropic covectors — while their anticommutator is the Laplacian and therefore elliptic.

Proof. The squares vanish termwise because $f_j^2=(f_j^{*})^2=0$ and the cross terms cancel between $j\neq k$ by the anticommutation of the Witt basis against the commutativity of the coefficients; the pairing is the diagonal anticommutator $\{f_j,f_j^{*}\}=1$. The failure of ellipticity is the vanishing of the symbol on the covectors with $\xi_{z_j}\neq0$ for some $j$ (for $\partial_{\bar z}$) respectively $\xi_{\bar z_j}\neq0$ (for $\partial_{\underline z}$), which is the isotropy of the Witt basis. $\square$

Remark (ellipticity recovered by the pair). Neither operator is elliptic, but the operator $\partial_{\underline z}-\partial_{\bar z}=\tfrac12D$ is, and the pair has the elliptic Laplacian as its anticommutator. The refinement substitutes a non-elliptic pair with an elliptic second-order invariant for a single elliptic first-order operator; this is the operator content of the strictness recorded in the preceding article.

The Differential Complex and the Split

Theorem (the split of the operator and of the Laplacian). $D=2(\partial_{\underline z}-\partial_{\bar z})$, $\{\partial_{\underline z},\partial_{\bar z}\}=\tfrac14\Delta_{2n}$, and $\partial_{\underline z}^2=\partial_{\bar z}^2=0$.

Proof. The identities are the theorem of the preceding article; they are restated here because they are the two structural facts on which the Fischer decomposition rests. $\square$

Corollary (the complex). The graded algebra generated by the two operators and their symbols is the exterior algebra $\bigwedge(\mathbb{C}^n\oplus\overline{\mathbb{C}^n})$: the Witt basis is a fermionic basis, the two operators are the two odd differentials, and the split of the Laplacian is the statement that their anticommutator is a scalar second-order operator.

The Hermitian Variables and Fischer Duality

Definition. The Hermitian variables are

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

Theorem (the relations of the Hermitian variables). The Hermitian variables are isotropic and pair to the norm:

$$ Z^2 = 0 , \qquad (Z^{*})^2 = 0 , \qquad Z Z^{*}+Z^{*}Z = \sum_{j=1}^{n}|z_j|^2 = |z|^2 . $$

Moreover $Z$ and $Z^{*}$ are Hermitian monogenic, $\partial_{\underline z}Z=\partial_{\bar z}Z=\partial_{\underline z}Z^{*} =\partial_{\bar z}Z^{*}=0$.

Proof. The first display is the computation of the Witt relations against the scalar coefficients: the cross terms cancel by $f_jf_k=-f_kf_j$ with $z_jz_k=z_kz_j$, and the pairing is diagonal. For the second, each operator acts termwise; the diagonal terms are killed by $f_j^2=0$ respectively $(f_j^{*})^2=0$, and the cross terms vanish because $\partial_{z_j}z_k=0$ and $\partial_{\bar z_j}\bar z_k=0$ for $j\neq k$ while $\partial_{\bar z_j}z_k=\partial_{z_j}\bar z_k=0$ for all $j,k$. Hence all four derivatives vanish on $Z$ and on $Z^{*}$. $\square$

Remark (the Fischer duality). The relations $Z^2=0$, $\{Z,Z^{*}\}=|z|^2$ are the Hermitian form of the Fischer duality of The Fischer Operator: the Hermitian variable is the multiplier whose Clifford norm is the radial factor, and the Laplacian is recovered as the anticommutator of the two operators of which $Z$ and $Z^{*}$ are the respective dual variables. In the quaternionic case the four Hermitian variables $Z_r$ play the same role for the four operators, with the norms pairing by the Hadamard matrix; Clifford Analysis records the construction there.

The Hermitian Fischer Decomposition

Definition. Let $\mathcal{P}_k$ be the space of $\mathcal{S}$-valued homogeneous polynomials of degree $k$ on $\mathbb{C}^n$, and let

$$ \mathcal{M}_k^{\mathrm{H}} = \{P\in\mathcal{P}_k : \partial_{\underline z}P=\partial_{\bar z}P=0\} $$

be the space of Hermitian monogenic homogeneous polynomials of degree $k$.

Theorem (the Hermitian Fischer decomposition; standard). Every homogeneous polynomial has a unique expansion

$$ \mathcal{P}_k = \bigoplus_{j=0}^{k}\, Z^j\,\mathcal{M}_{k-j}^{\mathrm{H}} , $$

with $Z=\sum_jz_jf_j$ the Hermitian variable of degree $1$; more generally the multiplication by the Hermitian variables $Z$ and $Z^{*}$ generates the complement of the Hermitian monogenic part, and

$$ \dim\mathcal{M}_k^{\mathrm{H}} = \dim\mathcal{S}\cdot\text{(the number of Hermitian monogenic monomials of degree }k\text{)} , $$

computed from the bivariate structure of the following section.

Proof. Quoted as standard from the Hermitian Fischer theory. The proof is the triangular induction of The Fischer Operator with the pair $(\partial_{\underline z},\partial_{\bar z})$ in place of $D$ and the Hermitian variables in place of $x^{\natural}$: the operators lower the degree, the multipliers raise it, the Fischer lemma holds for the pair, and the layers are the simultaneous kernels. The decomposition is finer than the monogenic one because the kernel of a pair is smaller than the kernel of one operator. $\square$

Remark (comparison with the ordinary decomposition). The ordinary monogenic decomposition $\mathcal{P}_k=\bigoplus_j(x^{\natural})^j\mathcal{M}_{k-j}$ of The Fischer Operator has a single multiplier and a single kernel; the Hermitian decomposition has the multiplier $Z$ and the smaller kernel $\mathcal{M}^{\mathrm H}$, so its layers are smaller and its multiplier series is shorter. Since $\mathcal{M}^{\mathrm H}_k\subseteq\mathcal{M}_k\subseteq\mathcal{H}_k$, the three decompositions — Hermitian, monogenic, harmonic — refine one another, and the Hermitian one is the finest of the three.

The Bivariate Representation

Remark (the two sets of variables). The space of polynomials in the $2n$ real coordinates is the space of polynomials in the bivariate data $(z_1,\dots,z_n;\bar z_1,\dots,\bar z_n)$, and a Hermitian monogenic polynomial is a bivariate polynomial whose two sets of variables are coupled through the Witt basis. The representation is canonical: the module $\mathcal{S}$ is identified with the exterior algebra $\bigwedge\mathbb{C}^n$ built from the creation operators $f_j^{*}$, the value of a Hermitian monogenic function is a sum of holomorphic-coefficient terms $\sum_{A}c_A(z)f_A^{*}$ and anti-holomorphic-coefficient terms, and the two operators act on the two halves as against the bidegree. The identification of the spinor module with the exterior algebra and the action of the Witt basis on it are Part II's, in Spin Representations and Clifford Modules with Inner Conjugation.

Remark (what the bivariate representation gives). In the bivariate representation the Hermitian system becomes a pair of commuting Cauchy–Riemann systems, one in each set of variables, coupled by a finite-dimensional algebraic factor; the Hermitian monogenic polynomials are those whose bivariate coefficients lie in the joint kernel of the two algebraic actions. This is the reason the Hermitian Fischer decomposition is computed by counting bivariate monomials and applying the algebraic anticommutation relations, and the reason the integral theory of The Hermitian Cauchy Integral and the Boundary Values can be written as a matrix device: the two operators are two bidegrees of one object, and the kernel that inverts the pair is a matrix whose entries are the two Cauchy kernels and their partners.

Example (the case $n=1$). For $n=1$ the bivariate data are $(z,\bar z)$, the Witt basis is $f,f^{*}$ with $f^2=(f^{*})^2=0$, $\{f,f^{*}\}=1$, and the Hermitian monogenic polynomials of degree $k$ are spanned by $z^a\bar z^b$ with the value in the appropriate ideal of the Clifford algebra, subject to the pair of equations. The count of the Hermitian monogenic pieces and the comparison with the monogenic and harmonic pieces is the explicit low-dimensional check of the refinement, and the model is the repère for the general case.

Summary

The Hermitian Dirac operators $\partial_{\underline z}=\sum_jf_j\partial_{z_j}$ and $\partial_{\bar z}=\sum_jf_j^{*}\partial_{\bar z_j}$ have isotropic symbols, $(\sum_jf_j\xi_{z_j})^2=(\sum_jf_j^{*}\xi_{\bar z_j})^2=0$, pairing to the scalar $\sum_j\xi_{z_j}\eta_{\bar z_j}$; neither is elliptic, but $D=2(\partial_{\underline z}-\partial_{\bar z})$ is, and $\{\partial_{\underline z},\partial_{\bar z}\}=\tfrac14\Delta_{2n}$. The algebra they generate is the exterior algebra on the Witt basis. The Hermitian variables $Z=\sum_jz_jf_j$ and $Z^{*}=\sum_j\bar z_jf_j^{*}$ are isotropic, $Z^2=(Z^{*})^2=0$, pair to the norm, $\{Z,Z^{*}\}=|z|^2$, and are themselves Hermitian monogenic; they are the Fischer duals of the two operators, the Hermitian form of the duality of The Fischer Operator. The Hermitian Fischer decomposition writes every homogeneous polynomial as $\bigoplus_jZ^j\mathcal{M}^{\mathrm H}_{k-j}$, with $\mathcal{M}^{\mathrm H}_k$ the joint kernel of the pair; the expansion is finer than the monogenic and the harmonic ones, the three refining one another as $\mathcal{M}^{\mathrm H}_k\subseteq\mathcal{M}_k\subseteq\mathcal{H}_k$. In the bivariate representation the variables are the two sets $z_j$ and $\bar z_j$, the module is the exterior algebra $\bigwedge\mathbb{C}^n$, and the Hermitian system is a pair of commuting Cauchy–Riemann systems coupled by the finite-dimensional Witt algebra; the general construction is Clifford Analysis, the quaternionic case is Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator, and the integral theory is The Hermitian Cauchy Integral and the Boundary Values.

Summary of Notation

Symbol	Meaning
$\partial_{\underline z},\partial_{\bar z}$	Hermitian Dirac operators; isotropic, pair to $\tfrac14\Delta_{2n}$
$Z=\sum_jz_jf_j$, $Z^{}=\sum_j\bar z_jf_j^{}$	Hermitian variables; Fischer duals
$Z^2=(Z^{})^2=0$, $\{Z,Z^{}\}=\|z\|^2$	Relations of the Hermitian variables
$\mathcal{M}^{\mathrm H}_k$	Hermitian monogenic homogeneous polynomials of degree $k$
$\mathcal{P}_k=\bigoplus_jZ^j\mathcal{M}^{\mathrm H}_{k-j}$	Hermitian Fischer decomposition
$\mathcal{M}^{\mathrm H}_k\subseteq\mathcal{M}_k\subseteq\mathcal{H}_k$	Refinement of the three kernels
$\bigwedge\mathbb{C}^n$	Spinor module, identified with the exterior algebra