Fueter Theory for Split-Quaternions

Introduction

This article develops Fueter theory in the split-quaternion setting: the Fueter operator, the Fueter-regular functions it defines, the axial and slice approach, and the power-series description. In the quaternion and biquaternion theories the Fueter operator is a square root of the definite Laplacian on the quaternion subspace, and the Fueter construction converts holomorphic functions of one complex variable into regular functions by applying a power of the Laplacian to their axial extension. In the split signature the second-order operator is not a Laplacian but the ultrahyperbolic operator $\Box = \partial_{q_0}^2+\partial_{q_1}^2-\partial_{q_2}^2-\partial_{q_3}^2$ of signature $(2,2)$, and the spherical symmetry on which the classical scalar case rests is replaced by hyperbolic symmetry. The slice results survive verbatim; the axial construction does not, and the reason is the indefinite form together with the zero divisors.

The article owns the Fueter theory of the category. It relies on Split-Quaternion Regular Functions for the Cauchy–Riemann operator and its factorization, on Split-Quaternion Roots of Minus One for the imaginary units, and on the power-series material developed in Split-Quaternion Analysis and Split-Quaternion Elementary Functions. It is compared throughout with the biquaternion counterpart Fueter Theory for Biquaternions and with the quaternion theory. No physics is invoked, and the first-order operator is called the Cauchy–Riemann operator, not the Dirac operator.

Conventions. Coordinates are $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with $N(\tilde q) = q_0^2+q_1^2-q_2^2-q_3^2$; the vector part is $\mathbf v = q_1 e_1+q_2 e_2+q_3 e_3 \in V$ with $N(\mathbf v) = q_1^2-q_2^2-q_3^2$. Partial derivatives are $\partial_{q_0},\partial_{q_1},\partial_{q_2},\partial_{q_3}$.

The Fueter Operator in the Split Signature

Definition and Conjugate

Definition. The Fueter operator (the Cauchy–Riemann–Fueter operator) and its conjugate are

$$ \nabla = e_0\partial_{q_0} + e_1\partial_{q_1} + e_2\partial_{q_2} + e_3\partial_{q_3} = \partial_{q_0} + \mathbf{D}, \qquad \bar{\nabla} = \partial_{q_0} - \mathbf{D}, \qquad \mathbf{D} = e_1\partial_{q_1} + e_2\partial_{q_2} + e_3\partial_{q_3}, $$

acting on the left. The operators are the same expressions as in the quaternion and biquaternion theories; only the squares of the generators differ, $e_1^2 = -1$, $e_2^2 = e_3^2 = +1$.

Factorization of the Ultrahyperbolic Operator

Proposition. $\nabla\bar{\nabla} = \bar{\nabla}\nabla = \Box e_0$, where

$$ \Box = \partial_{q_0}^2 + \partial_{q_1}^2 - \partial_{q_2}^2 - \partial_{q_3}^2 $$

is the ultrahyperbolic operator of signature $(2,2)$.

Proof. Expand $\nabla\bar{\nabla} = \sum_{\mu,\nu}e_\mu e_\nu^{\natural}\partial_\mu\partial_\nu$. The diagonal coefficients are $e_0e_0^{\natural} = e_0$, $e_1e_1^{\natural} = -e_1^2 = e_0$, $e_2e_2^{\natural} = -e_2^2 = -e_0$, $e_3e_3^{\natural} = -e_0$; the off-diagonal coefficients vanish by the Clifford relations $e_\mu e_\nu^{\natural} + e_\nu e_\mu^{\natural} = 0$ for $\mu\neq\nu$.

Thus the Fueter operator is a square root not of the Laplacian but of a wave operator: it is the Cauchy–Riemann operator of the Clifford algebra $\mathrm{Cl}_{2,2}$, of signature $(2,2)$. This is the single structural sign that separates the theory from the quaternion ($\mathrm{Cl}_{0,3}$) and biquaternion ($\mathrm{Cl}_{1,3}$) cases.

Relation to the Cauchy–Riemann Operator

On the full algebra the Fueter operator is exactly the Cauchy–Riemann operator of Split-Quaternion Regular Functions: its symbol $s(\xi) = \xi_0 + \xi_1e_1+\xi_2e_2+\xi_3e_3$ satisfies $s(\xi)s^{\natural}(\xi) = N(\xi)e_0$, so it is not elliptic and its characteristic set is the null cone $\mathcal{N}$. In Clifford language, $\mathbb{H}_{\mathrm{s}}$ is the even subalgebra of $\mathrm{Cl}_{2,2}$ up to the standard identification, and the Fueter-regular functions are its monogenic functions.

Fueter-Regular Functions

Left and Right Regularity

Definition. Let $\Omega$ be open in $\mathbb{R}^4$, and let $F : \Omega \to \mathbb{H}_{\mathrm{s}}$ be continuously differentiable. Then $F$ is left-Fueter-regular if $\nabla F = 0$; right-Fueter-regular if $F\nabla := \sum_\mu\partial_\mu F\,e_\mu = 0$; and anti-regular if $\bar\nabla F = 0$. As in the rest of the category, regular means left-regular, and the first-order operator $\nabla$ is the one inverted.

The Componentwise System

Writing $F = F_0 + \mathbf F$ with $\mathbf F = F_1e_1+F_2e_2+F_3e_3$, the regularity condition is the split Cauchy–Riemann–Fueter system of Split-Quaternion Regular Functions,

$$ \partial_{q_0} F_0 = \mathrm{div}_{\mathrm{s}}\,\mathbf F, \qquad \partial_{q_0}\mathbf F + \operatorname{grad}_{\mathrm{s}}F_0 + \operatorname{rot}_{\mathrm{s}}\mathbf F = 0, $$

with the divergence, gradient and curl taken for the indefinite form $N|_V$. Left- and right-regularity differ only in the sign of the curl term, exactly as in the quaternion case.

Proposition (harmonicity). Every Fueter-regular function is $\Box$-harmonic: $\nabla F = 0$ implies $\Box F = \bar\nabla\nabla F = 0$. The converse is false.

Proof. Immediate from the factorization; the function $F_0 = q_0$ is $\Box$-harmonic but $\nabla q_0 = e_0 \neq 0$.

The Axial and Slice Approach

Imaginary Units and Slices

Definition. An imaginary unit is an element $I$ with $I^2 = -1$, and a slice is a real plane $\mathbb{C}_I = \mathbb{R} + I\mathbb{R}$ spanned by $1$ and an imaginary unit.

Theorem. The imaginary units of $\mathbb{H}_{\mathrm{s}}$ are the elements of the vector subspace

$$ \{ I \in V : I^2 = -1 \} = \{ I \in V : N(I) = 1 \} = \{ q_1 e_1 + q_2 e_2 + q_3 e_3 : q_1^2 - q_2^2 - q_3^2 = 1 \}, $$

a two-sheeted hyperboloid of dimension $2$ in $V$, with the two sheets distinguished by the sign of the coefficient $q_1$. Each imaginary unit spans a slice $\mathbb{C}_I \cong \mathbb{C}$, and two slices $\mathbb{C}_I$, $\mathbb{C}_J$ coincide when $I = \pm J$ and otherwise meet only in $\mathbb{R}$.

Proof. $I \in V$ satisfies $I^2 = -N(I)$ (the vector part squares to minus its split-quaternion norm), so $I^2 = -1$ iff $N(I) = 1$, which is the displayed hyperboloid, two-sheeted because $q_1^2 = 1+q_2^2+q_3^2\geq1$; the slice intersection statement is linear algebra.

This is the first departure from the quaternion theory, where the imaginary units form the two-sphere $S^2$. Here there is no compact imaginary sphere: the imaginary units are unbounded and split into two sheets, so an "imaginary direction" must be selected one root at a time, as in the biquaternion case but for a different reason.

Holomorphic Functions on a Slice Are Regular

Theorem. Fix an imaginary unit $I$, with slice coordinate $A = q_0 + I\rho$ on $\mathbb{C}_I$, and let $F$ depend only on $q_0$ and $\rho$ (not on the two directions of $V$ orthogonal to $I$). Then

$$ \nabla F = (\partial_{q_0} + I\partial_\rho)F = 2\partial_{\bar A}F, $$

so $F$ is regular if and only if $F$ is holomorphic in $A$ in the classical sense. Hence every classical holomorphic function of $A = q_0+I\rho$, extended by constancy in the orthogonal directions, is Fueter-regular.

Proof. Restricted to functions of $q_0,\rho$, the operator is $\nabla = e_0\partial_{q_0} + I\partial_\rho$, and $I^2 = -1$ makes $\mathbb{R}[I]$ a copy of $\mathbb{C}$ with $\bar A = q_0 - I\rho$; the classical Cauchy–Riemann operator of that copy is $\partial_{\bar A} = \tfrac12(\partial_{q_0} + I\partial_\rho)$.

This slice statement is exact and requires no modification for the split signature, because on a slice the form is definite (the slice is a copy of $\mathbb{C}$). It is the part of the theory that survives intact.

Axial Coordinates and the Obstruction by the Zero Divisors

For an element $\tilde q = q_0+\mathbf v$ with $N(\mathbf v) > 0$, the axial coordinates are

$$ \rho = \sqrt{N(\mathbf v)} > 0, \qquad \hat{\mathbf v} = \mathbf v/\rho, \qquad \hat{\mathbf v}^2 = -1, $$

so that $\hat{\mathbf v}$ is an imaginary unit and $\tilde q$ lies on the slice $\mathbb{C}_{\hat{\mathbf v}}$. On this region an axially symmetric function has the representation

$$ F(\tilde q) = A(q_0,\rho) + \hat{\mathbf v}\,B(q_0,\rho), $$

with axial coefficients $A, B$ that depend on the direction only through $\hat{\mathbf v}$. This is the exact analogue of the quaternion axial representation, and it is available precisely where $N(\mathbf v) > 0$.

Theorem (the obstruction). The axial representation is defined only on the timelike region $N(\mathbf v) > 0$. On the null cone $N(\mathbf v) = 0$ the axial direction is a zero divisor and no imaginary unit is defined; on the spacelike region $N(\mathbf v) < 0$ the normalised direction satisfies $\hat{\mathbf v}^2 = +1$, a root of $+1$, so the axial coefficient system is hyperbolic rather than complex. The region is the disjoint union of the timelike region $N(\mathbf v)>0$, on which the axial complex structure exists, the null cone $\mathcal{N}$, on which it degenerates, and the spacelike region $N(\mathbf v)<0$, on which it is replaced by a split-complex structure.

Proof. $\hat{\mathbf v}^2 = -N(\mathbf v)/\rho^2 = -\operatorname{sgn}N(\mathbf v)$, so $\hat{\mathbf v}^2 = -1$ on the timelike region, $\hat{\mathbf v}^2 = +1$ on the spacelike region, and $\hat{\mathbf v}$ is undefined on the null cone.

Thus the axial and slice approach is available on the whole timelike region and on a slice-by-slice basis, but it cannot be extended across the null cone, exactly as in the indefinite biquaternion case; and here, unlike the biquaternion case, there is no definite half to fall back on.

Why the Definite Fueter Construction Does Not Transfer

The classical Fueter construction sends a holomorphic $f_0$ to $F = \Delta_4\tilde f_0$, the Laplacian applied to the axial extension, and its proof uses the radial identity $\sum_{j,k}(\partial_{v_k}\hat v_j)e_ke_j = -2/\rho$ on the sphere of imaginary units. In the split signature the level sets of $N(\mathbf v)$ in the timelike region are two-sheeted hyperboloids, not spheres, and the corresponding identity acquires a vector part:

$$ \sum_{j,k}(\partial_{v_k}\hat v_j)\,e_k e_j = \frac{1}{\rho}\sum_k e_k^2 - \frac{1}{\rho^3}\sum_{j,k} g_{kk} v_j v_k\, e_k e_j, \qquad \sum_k e_k^2 = 1, $$

and the second sum is not a scalar multiple of $\hat{\mathbf v}$ but has the nonzero vector part $2v_1(-v_3e_2+v_2e_3)$ together with the Euclidean scalar part $-|\mathbf v|_E^2$. Hence the definite Fueter construction, with $\Box$ in place of $\Delta_4$, does not in general produce regular functions, and there is no direct radial/spherical version of Fueter's theorem in the split signature.

Remark. What remains is the slice construction of the previous subsection, which is exact and local to each slice, and the general Clifford-algebraic theory of monogenic functions on $\mathrm{Cl}_{2,2}$, which is signature-independent for the Fischer decomposition below. The axial radial construction is the part that fails, and its failure is measured by the vector part displayed above.

Power Series Representations

Theorem (right-coefficient series). A series $F(\tilde q) = \sum_{n\geq0} \tilde q^n a_n$ with coefficients $a_n \in \mathbb{H}_{\mathrm{s}}$ on the right converges absolutely and normally on $\|\tilde q\|_E < R$, where $R^{-1} = \limsup_n\|a_n\|_E^{1/n}$, and its sum is slice-regular (Cullen-regular): holomorphic on each slice.

Proposition. The coordinate function $\tilde q$ is slice-regular but not Fueter-regular: $\nabla \tilde q = \sum_\mu e_\mu e_\mu = e_0^2+e_1^2+e_2^2+e_3^2 = 1-1+1+1 = 2e_0 \neq 0$. In general the slice-regular class is strictly larger than the Fueter-regular class, and the Fueter construction is the operation that converts the first into the second.

Theorem (Fischer decomposition). Let $\mathcal{P}_k$ be the $\mathbb{H}_{\mathrm{s}}$-valued homogeneous polynomials of degree $k$ and $\mathcal{M}_k = \{P \in \mathcal{P}_k : \nabla P = 0\}$ the monogenic homogeneous polynomials. Then

$$ \mathcal{P}_k = \bigoplus_{j=0}^{k} \tilde q^j \mathcal{M}_{k-j}, $$

so every Fueter-regular function on a ball has a normally convergent expansion in monogenic homogeneous polynomials, the analogue of the Taylor series of complex analysis.

Proof. This is the standard Fischer decomposition for a real Clifford algebra with a non-degenerate quadratic form; it depends only on the non-degeneracy of $N$ and the factorization $\Box = \nabla\bar\nabla$, both of which hold with the split signature.

Relation to the Quaternion and Biquaternion Fueter Theories

The biquaternion article Fueter Theory for Biquaternions develops the operator on the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, where $\tilde\nabla\tilde{\nabla}^{\natural} = \Delta_4 e_0$ is the definite Laplacian; there the imaginary units form the sphere $S^2$, the axial representation and the Fueter construction are classical, the Fueter–Sce theorem holds for odd $n$ with the power $(n-1)/2$, and the Cauchy kernel is singular only at the origin. On the indefinite biquaternion subspaces $\mathbb{M}_\pm$ the second-order operator becomes a wave operator and the elliptic tools disappear, while the full algebra $\mathbb{B}$ has the six-dimensional null quadric as an obstruction.

The split-quaternion theory is the case in which every direction is of that indefinite kind: the second-order operator is ultrahyperbolic on the whole algebra, the imaginary units form a two-sheeted hyperboloid rather than a sphere, and the axial construction is obstructed on the null cone and replaced by a hyperbolic structure outside the timelike region. The slice statements survive because each slice restores a definite (complex) structure, and the Fischer decomposition survives because it needs only non-degeneracy; the radial Fueter construction and the Fueter–Sce parity theorem do not transfer, because they rest on spherical symmetry of a definite form. In Clifford terms the split-quaternion Fueter theory is the monogenic function theory of $\mathrm{Cl}_{2,2}$, to be compared with the $\mathrm{Cl}_{0,3}$ theory of the quaternions and the $\mathrm{Cl}_{1,3}$ theory of the biquaternions. Nothing quaternion- or biquaternion-specific — no imaginary sphere $S^2$, no definite half, no Euclidean Cauchy kernel — is imported.

Summary

The Fueter operator $\nabla = \sum_\mu e_\mu\partial_\mu$ and its conjugate satisfy $\nabla\bar\nabla = \bar\nabla\nabla = \Box e_0$, the ultrahyperbolic operator $\partial_{q_0}^2+\partial_{q_1}^2-\partial_{q_2}^2-\partial_{q_3}^2$ of signature $(2,2)$; the Fueter operator is the Cauchy–Riemann operator of $\mathrm{Cl}_{2,2}$, non-elliptic, with the null cone as its characteristic set. Fueter-regular functions are the solutions of $\nabla F = 0$, equivalently of the split Cauchy–Riemann–Fueter system, and every such function is $\Box$-harmonic. The imaginary units are the two-sheeted hyperboloid $\{N(I)=1\}$ in $V$, not a sphere; on any slice $\mathbb{C}_I$ the operator reduces to the classical Cauchy–Riemann operator $\partial_{q_0} + I\partial_\rho$, so holomorphic functions of the slice variable are regular. The axial representation $F = A + \hat{\mathbf v}B$ is available exactly on the timelike region $N(\mathbf v)>0$, degenerates on the null cone, and is replaced by a hyperbolic structure on the spacelike region; and the radial Fueter construction does not transfer, because the level sets of $N(\mathbf v)$ are hyperboloids rather than spheres and the radial identity acquires the vector part $2v_1(-v_3e_2+v_2e_3)$. Right-coefficient series are slice-regular on $\|\tilde q\|_E < R$, the coordinate $\tilde q$ is slice-regular but not regular, and the Fischer decomposition $\mathcal{P}_k = \bigoplus_j \tilde q^j\mathcal{M}_{k-j}$ gives the monogenic Taylor expansion. The split-quaternion theory is the fully indefinite member of the family: the slice and Fischer parts survive, and the radial, spherical and Fueter–Sce parts do not.

Summary of Notation

Symbol	Meaning	Article
$\mathbb{H}_{\mathrm{s}}$	the split-quaternion algebra	Split-Quaternion Algebra
$\nabla = \partial_{q_0} + \mathbf{D}$, $\bar\nabla = \partial_{q_0} - \mathbf{D}$	the Fueter (Cauchy–Riemann–Fueter) operator and its conjugate	Split-Quaternion Regular Functions
$\mathbf{D} = \sum_k e_k\partial_k$	the vector-derivative part	this article
$\Box = \partial_{q_0}^2+\partial_{q_1}^2-\partial_{q_2}^2-\partial_{q_3}^2$	the ultrahyperbolic operator, signature $(2,2)$	this article
$\nabla F = 0$	left-Fueter-regular (monogenic); $F\nabla=0$ right-regular	this article
$I$, $\{N(I)=1\}$	an imaginary unit, and the two-sheeted hyperboloid of imaginary units	Split-Quaternion Roots of Minus One
$\mathbb{C}_I = \mathbb{R}+I\mathbb{R}$	a slice, a copy of the complex plane	this article
$\rho = \sqrt{N(\mathbf v)}$, $\hat{\mathbf v} = \mathbf v/\rho$	axial radius and direction, on the timelike region $N(\mathbf v)>0$	this article
$A, B$	axial coefficients, $F = A(q_0,\rho) + \hat{\mathbf v}B(q_0,\rho)$	this article
$\mathcal{P}_k, \mathcal{M}_k$	homogeneous and monogenic homogeneous polynomials	this article
$N(\tilde q)=q_0^2+q_1^2-q_2^2-q_3^2$	the split-quaternion norm, signature $(2,2)$	Split-Quaternion Norm and Invertibility
$\mathcal{N} = \{N=0\}$	the null cone / zero-divisor set	Split-Quaternion Zero Divisors