Hermitian Quaternionic Analysis and the Conjugate Cauchy–Riemann Operator

Introduction

The Hermitian refinement of Clifford analysis has, besides the complex case of Hermitian Clifford Analysis and the Hermitian Monogenic Functions, a quaternionic case in which the ambient dimension is a multiple of four and the single Cauchy–Riemann operator is split into four operators instead of two. The number is dictated by the algebra — two Hermitian Dirac operators for $\mathbb{C}$, four for $\mathbb{H}$ — and the four-operator case is the one in which the full Hermitian structure of a quaternionic Clifford module is visible: the four operators are the four $q$-components of the Euclidean operator, their symbols pair through the quaternions, and the conjugate Cauchy–Riemann operator is the Hermitian adjoint of the whole construction.

The setting is that of $4n$ Euclidean dimensions with quaternion coefficients. Let $N=4n$, let $\mathrm{Cl}_{0,N}$ be generated by $e_1,\dots,e_N$ with $e_ie_j+e_je_i=-2\delta_{ij}$, and let the quaternions act as central scalars, so that the values lie in

$$ \mathbb{H}_N = \mathbb{H}\otimes_{\mathbb{R}}\mathrm{Cl}_{0,N} ; $$

the quaternion units $i,j,k$ commute with the generators and are the coefficients of the Hermitian structure. The Euclidean variable is the pure vector $X=\sum_{a=1}^{N}e_ax_a$ and the Euclidean operator is $\partial_X=\sum_{a=1}^{N}e_a\partial_{x_a}$, with $\partial_X^2=-\Delta_N$.

This article fixes the four twisted Euclidean vectors, the four Hermitian variables and the four Hermitian Dirac operators, defines the Hermitian conjugation and states the split of the Laplacian, and describes the conjugate Cauchy–Riemann operator — the Hermitian adjoint of the Hermitian operators, and the operator that carries the second half of the Cauchy pairing. The complex case is the preceding article; the operator and the Fischer decomposition are The Hermitian Dirac Operator and the Fischer Decomposition; the integral theory, with the circulant matrix that carries the four Cauchy kernels, is The Hermitian Cauchy Integral and the Boundary Values and The Hermitian Cauchy Kernel as an Adjoint; the general construction is the Hermitian section of Clifford Analysis, and the spinor structure of the module $\mathbb{H}_N$ is Part II's, in Spin Representations and Clifford Modules with Inner Conjugation.

The Four Twisted Vectors and Operators

The Euclidean Data

Definition. The Euclidean operator is $\partial_X=\sum_{a=1}^{N}e_a\partial_{x_a}$, the vector part of the Cauchy–Riemann operator of The Dirac Operator; it is elliptic, with

$$ \partial_X^2 = -\Delta_N , \qquad \Delta_N = \sum_{a=1}^{N}\partial_{x_a}^2 , $$

the sign being the one of the corpus for the part of the operator assembled from the generators (the full operator $D$ of The Dirac Operator, with its scalar direction, satisfies $D\bar D=\Delta$; the squared vector part is $-\Delta_N$).

Definition (the twisted vectors). Group the generators in fours and put, for $l=1,\dots,n$,

$$ \begin{aligned} X_0&=\sum_{l=1}^{n}\bigl(e_{4l-3}x_{4l-3}+e_{4l-2}x_{4l-2}+e_{4l-1}x_{4l-1}+e_{4l}x_{4l}\bigr),\\ X_1&=\sum_{l=1}^{n}\bigl(e_{4l-3}x_{4l-2}-e_{4l-2}x_{4l-3}-e_{4l-1}x_{4l}+e_{4l}x_{4l-1}\bigr),\\ X_2&=\sum_{l=1}^{n}\bigl(e_{4l-3}x_{4l-1}+e_{4l-2}x_{4l}-e_{4l-1}x_{4l-3}-e_{4l}x_{4l-2}\bigr),\\ X_3&=\sum_{l=1}^{n}\bigl(e_{4l-3}x_{4l}-e_{4l-2}x_{4l-1}+e_{4l-1}x_{4l-2}-e_{4l}x_{4l-3}\bigr). \end{aligned} $$

The first is the ordinary Clifford vector; the other three are its companions under the quaternionic sign changes, the twisted vectors of the Hermitian literature (a different use of the word twisted from the twisted operator of Clifford Modules and the Twisted Cauchy–Riemann Operator). The four operators $\partial_{X_r}$ are formed by the same patterns.

Proposition (the relations of the twisted data). The twisted vectors and their operators satisfy

$$ X_r^2=-|X|^2 , \qquad \{X_r,X_s\}=X_rX_s+X_sX_r=0 \quad (r\neq s), $$ $$ \partial_{X_r}^2=-\Delta_N , \qquad \{\partial_{X_r},\partial_{X_s}\}=0 \quad (r\neq s). $$

Proof. The coefficient matrices of the four vectors are column-orthonormal, as the Hadamard sign patterns show, whence $X_r^2=-|X|^2$ and the vanishing of the six anticommutators; the corresponding operator identities follow by applying the operators to the scalar coefficients, the relations being those of the same sign patterns. The verification is carried out in $\mathbb{H}\otimes_\mathbb{R} \mathrm{Cl}_{0,4}$ in Clifford Analysis. $\square$

The Hermitian Variables and Operators

Definition. The Hermitian variables and the Hermitian Dirac operators are the four invertible combinations built from the twisted data by the same four sign patterns:

$$ \begin{aligned} \partial_{Z_0}&=\tfrac{1}{16}\bigl(\partial_{X_0}+i\,\partial_{X_1}+j\,\partial_{X_2}+k\,\partial_{X_3}\bigr),& \partial_{Z_1}&=\tfrac{1}{16}\bigl(\partial_{X_0}+i\,\partial_{X_1}-j\,\partial_{X_2}-k\,\partial_{X_3}\bigr),\\ \partial_{Z_2}&=\tfrac{1}{16}\bigl(\partial_{X_0}-i\,\partial_{X_1}+j\,\partial_{X_2}-k\,\partial_{X_3}\bigr),& \partial_{Z_3}&=\tfrac{1}{16}\bigl(\partial_{X_0}-i\,\partial_{X_1}-j\,\partial_{X_2}+k\,\partial_{X_3}\bigr), \end{aligned} $$

with the variables $Z_r$ built from the $X_r$ by the same four sign patterns. The factor $\tfrac1{16}$ is the normalisation that makes the Laplacian split below; the patterns are the rows of the Hadamard matrix with $SS^{\mathsf T}=4I_4$ and $\det S=-16$, which is what makes the Hermitian and the Euclidean descriptions equivalent.

Remark (why four, and why these). The four combinations are the four irreducible components of the Euclidean operator under the quaternionic central structure: the quaternions act on the four twisted directions, and the Hadamard patterns diagonalise that action. The complex case uses two combinations because the central structure there is $\mathbb{C}$, with a two-row Hadamard matrix; the number of Hermitian operators equals the dimension of the central division algebra over which the value module is built.

The Hermitian Conjugation and the Split of the Laplacian

Definition. The Hermitian conjugation of $\mathbb{H}_N$ is the composition of the $\mathbb{H}$-conjugation with the Clifford conjugation:

$$ \lambda^{*} = \sum_A \bar\lambda_A\,e_A^{\natural} , $$

the sum running over the blades $e_A$ of $\mathrm{Cl}_{0,N}$, where $\bar\lambda_A$ is the quaternionic conjugate and $e_A^{\natural}$ the Clifford conjugate. Under it a vector is anti-self-adjoint, $X_r^{*}=-X_r$, and consequently $\partial_{X_r}^{\dagger}=-\partial_{X_r}$.

Theorem (the split of the Laplacian). With the Hermitian conjugation,

$$ \sum_{r=0}^{3}Z_rZ_r^{*} = \sum_{r=0}^{3}Z_r^{*}Z_r = 16\,|X|^2 , \qquad \Delta_N = 16\sum_{r=0}^{3}\partial_{Z_r}\partial_{Z_r}^{\dagger} = 16\sum_{r=0}^{3}\partial_{Z_r}^{\dagger}\partial_{Z_r} . $$

Proof. The Hadamard patterns are orthogonal with $SS^{\mathsf T}=4I_4$; combining the four imaginary-quaternion combinations with their conjugates pairs each pattern with itself and cancels the cross patterns, leaving $4\sum_r(\text{pattern})^2=4\cdot4|X|^2$ in the variables and the corresponding operator identity. The full verification in $\mathbb{H}\otimes_\mathbb{R}\mathrm{Cl}_{0,4}$ is in Clifford Analysis: the norm identity holds, and the split of the Laplacian holds only when the involution is the composition of the two conjugations, as stated. $\square$

Remark (the involution is not the $L^2$ adjoint). The involution entering the split is an algebra anti-involution, under which a $1$-vector changes sign; it is not the adjoint with respect to the $L^2$ inner product of functions. The two agree only in the definite case and with a suitable choice of the module form, and the distinction is the one developed in The Hermitian Dirac Operator and Adjoints on a Clifford Module; the identity above is an operator identity in the algebra, and its sign is fixed by the stated convention.

The Conjugate Cauchy–Riemann Operator

Definition. The conjugate Cauchy–Riemann operator of the Hermitian theory is the Hermitian adjoint $\partial_{Z_r}^{\dagger}$ of the Hermitian Dirac operator, the operator obtained by applying the Hermitian conjugation to the coefficients:

$$ \partial_{Z_r}^{\dagger} = \tfrac{1}{16}\sum_{a=0}^{3}\overline{\varepsilon_{ra}}\;\partial_{X_a}^{\dagger} = -\tfrac{1}{16}\sum_{a=0}^{3}\overline{\varepsilon_{ra}}\;\partial_{X_a}, $$

where $\varepsilon_{ra}$ are the entries of the Hadamard matrix and $\overline{\varepsilon_{ra}}$ the quaternionic conjugates of the coefficients.

Theorem (the pair of Cauchy–Riemann operators). The conjugate Cauchy–Riemann operator is the partner of $\partial_{Z_r}$ in the split of the Laplacian:

$$ \Delta_N = 16\sum_r\partial_{Z_r}\partial_{Z_r}^{\dagger} = 16\sum_r\partial_{Z_r}^{\dagger}\partial_{Z_r}, $$

and the two operators $\partial_{Z_r}$ and $\partial_{Z_r}^{\dagger}$ are the two halves of the Cauchy pairing: the kernel of $\partial_{Z_r}$ is what the Cauchy integral of The Hermitian Cauchy Integral and the Boundary Values reproduces, and the conjugate operator is what turns the Hermitian variables into the Cauchy kernel.

Proof. The split is the theorem above; the role of the conjugate operator in the kernel computation is the computation $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$, in which the conjugate variable appears in the numerator and the kernel satisfies the Euclidean identities $\partial_{X_r}F_r=\delta$ and $\partial_{X_r}F_s+\partial_{X_s}F_r=0$ for $r\neq s$; these are stated in The Hermitian Cauchy Integral and the Boundary Values. $\square$

Remark (the complex reduction). In the complex case $n=1$ of the preceding article the two Hermitian Dirac operators are conjugates in the same sense: the conjugate of $\partial_{\underline z}=\sum_jf_j\partial_{z_j}$ is the anti-holomorphic $\partial_{\bar z}=\sum_jf_j^{*}\partial_{\bar z_j}$, and the split of the Laplacian reads $\Delta_{2n}=4\{\partial_{\underline z},\partial_{\bar z}\}$, the two-operator case of the identity above. The quaternionic case is the four-operator case of the same statement, and the conjugate operator is the one that carries the second half of every Hermitian Cauchy formula.

q-Hermitian Monogenic Functions

Definition. Let $\Omega\subseteq\mathbb{R}^{4n}$ be open. A $C^1$ function $f:\Omega\to\mathbb{H}_N$ is $q$-Hermitian monogenic in $\Omega$ when

$$ \partial_{Z_0}f=\partial_{Z_1}f=\partial_{Z_2}f=\partial_{Z_3}f=0 , $$

equivalently, by the same four sign patterns, when $\partial_{X_0}f=\partial_{X_1}f=\partial_{X_2}f=\partial_{X_3}f=0$. The two systems are equivalent because the sign matrix relating them is invertible; neither equation alone defines the class.

Proposition (the class is monogenic and smaller). Every $q$-Hermitian monogenic function is monogenic for the Euclidean operator $\partial_X$ and harmonic; the class is strictly smaller than the monogenic one, the four equations being one equation read four times through the sign patterns.

Proof. The Euclidean operator is a linear combination of the four Hermitian operators — its coefficients are the signs of the Hadamard matrix with quaternionic weights — so its vanishing follows from theirs; the Laplacian is the sum of the four products $\partial_{Z_r}\partial_{Z_r}^{\dagger}$, so it vanishes too. The strictness is that of a system over a single equation, and it is recorded in Clifford Analysis. $\square$

Remark (the axial and the slice structures). For $n=1$ the four-operator system reduces to a five-dimensional structure in $\mathbb{H}\otimes_\mathbb{R}\mathrm{Cl}_{0,4}$; the axial description of Fueter Theory is the description of the functions constant on the spheres about the real axis, and the Hermitian refinement of Fueter theory is Hermitian Fueter Theory. The slice description is Part V's, and the relation between the Hermitian operators and the quaternionic Fueter operator is the one in which the two indices — Hermitian and fueterian — meet.

The Anti-Monogenic Functions

Definition. A $C^1$ function $f$ is anti- $q$-Hermitian monogenic (or anti-monogenic) in $\Omega$ when it is annihilated by the conjugate operators,

$$ \partial_{Z_0}^{\dagger}f=\partial_{Z_1}^{\dagger}f=\partial_{Z_2}^{\dagger}f =\partial_{Z_3}^{\dagger}f=0 , $$

equivalently when it is $q$-Hermitian monogenic for the conjugate module structure $\mathcal{S}^{*}$, the values conjugated by the Clifford conjugation. The two classes $\mathfrak{M}$ and $\mathfrak{M}^{\dagger}$ are interchanged by the conjugation of the values, $f\in\mathfrak{M}\iff f^{\dagger}\in\mathfrak{M}^{\dagger}$, and neither contains the other.

Proposition (the anti-monogenic class and the adjoint). The anti-monogenic functions are the $\dagger$-images of the $q$-Hermitian monogenic ones, and the conjugate Cauchy–Riemann operator is the adjoint of the Hermitian one, so the two classes are the two kernels

$$ \ker\partial_{Z_r}\quad\text{and}\quad\ker\partial_{Z_r}^{\dagger}=\bigl(\ker\partial_{Z_r}\bigr)^{\dagger} ; $$

a function is both $q$-Hermitian monogenic and anti-monogenic exactly when it is annihilated by all eight operators, which is the kernel of the Hermitian Dirac operator $\mathcal{D}$ and is not reduced by the singular split.

Proof. The first identity is the definition of the anti-monogenic class read through the conjugation of the values; the relation between the kernel of an operator and the kernel of its adjoint is the one proved for $\partial_{Z_r}$ and $\partial_{Z_r}^{\dagger}$ in The Hermitian Dirac Operator, and the eight-operator kernel is the intersection, which the singular split does not preserve because the split $\mathcal{D}=\partial_{\underline z}+\partial_{\bar z}$ is a sum and not a direct sum on the kernel. $\square$

Remark (why the pair matters). The Hermitian refinement carries the monogenic and the anti-monogenic classes together: the Cauchy kernel of the Hermitian theory is monogenic in the first variable and anti-monogenic in the second, the positive definite kernels of the theory are built from such a pair, and the boundary theory pairs a monogenic function with an anti-monogenic density. The two classes are the two halves of the refined function theory, exactly as the two Hermitian Dirac operators are the two halves of the refined operator, and the refinement of the Cauchy integral is The Hermitian Cauchy Integral and the Boundary Values.

Summary

In $4n$ Euclidean dimensions with quaternions as central scalars, the four twisted vectors $X_0,\dots, X_3$ built from the generators by the Hadamard sign patterns satisfy $X_r^2=-|X|^2$ and $\{X_r,X_s\}=0$, and their operators satisfy the same relations with $-\Delta_N$. The four Hermitian variables $Z_r$ and Hermitian Dirac operators $\partial_{Z_r}$, built from the twisted data by the same patterns with the normalisation $\tfrac1{16}$, are the components of the Euclidean operator under the quaternionic central structure; the Hermitian conjugation — the composition of the quaternionic with the Clifford conjugation, under which a vector is anti-self-adjoint — gives the norm identity $\sum_rZ_rZ_r^{*}=16|X|^2$ and the split of the Laplacian $\Delta_N=16\sum_r\partial_{Z_r}\partial_{Z_r}^{\dagger}=16\sum_r\partial_{Z_r}^{\dagger}\partial_{Z_r}$, the four-operator form of $D\bar D=\Delta$. The conjugate Cauchy–Riemann operator $\partial_{Z_r}^{\dagger}$ is the Hermitian adjoint and the partner of $\partial_{Z_r}$ in the split; it is the operator that produces the Hermitian Cauchy kernel $E_r(Z)=Z_r^{*}/(a_{4n}|Z|^{4n})$ and carries the second half of the Hermitian Cauchy formulae, which are the subject of The Hermitian Cauchy Integral and the Boundary Values. The ** $q$-Hermitian monogenic functions are the simultaneous null solutions of the four operators, hence monogenic and harmonic, and strictly smaller than the monogenic class. The involution is an algebra anti-involution, not the $L^2$ adjoint — the distinction is The Hermitian Dirac Operator and Adjoints on a Clifford Module; the complex case is Hermitian Clifford Analysis and the Hermitian Monogenic Functions; the general construction is Clifford Analysis.

Summary of Notation

Symbol	Meaning
$N=4n$, $\mathbb{H}_N=\mathbb{H}\otimes_\mathbb{R}\mathrm{Cl}_{0,N}$	Ambient dimension and value algebra
$\partial_X=\sum_{a=1}^Ne_a\partial_{x_a}$	Euclidean operator; $\partial_X^2=-\Delta_N$
$X_0,\dots,X_3$	Twisted vectors; $X_r^2=-\|X\|^2$, $\{X_r,X_s\}=0$
$Z_0,\dots,Z_3$	Hermitian variables; Hadamard patterns
$\partial_{Z_0},\dots,\partial_{Z_3}$	Hermitian Dirac operators; normalisation $\tfrac1{16}$
$\lambda^{*}=\sum_A\bar\lambda_Ae_A^{\natural}$	Hermitian conjugation; $X_r^{*}=-X_r$
$\Delta_N=16\sum_r\partial_{Z_r}\partial_{Z_r}^{\dagger}$	Split of the Laplacian
$\partial_{Z_r}^{\dagger}$	Conjugate Cauchy–Riemann operator
$E_r(Z)=Z_r^{*}/(a_{4n}\|Z\|^{4n})$	Hermitian Cauchy kernels
$\partial_{Z_0}f=\cdots=\partial_{Z_3}f=0$	$q$-Hermitian monogenicity
$a_{4n}=\|S^{4n-1}\|=2\pi^{2n}/\Gamma(2n)$	Area of the unit sphere in $\mathbb{R}^{4n}$