Fueter Theory for Biquaternions

Introduction

This article develops Fueter theory in the quaternionic and biquaternionic setting: the Fueter operator, the regular functions it defines, and the construction that produces them from holomorphic functions of one complex variable. It follows Biquaternion Analysis, Biquaternion Regular Functions, Biquaternion Integration, Biquaternion Norm and Invertibility and Biquaternion Zero Divisors.

The physical content is the passage from a one-component complex amplitude to a four-component field. A holomorphic function of one complex variable is a scalar amplitude; applying a power of the four-dimensional Laplacian to its axial extension produces a function annihilated by the Dirac-type operator, that is, a spinor field. Fueter's theorem is the precise algebraic form of this upgrade, and it is why the algebra can host both the Klein–Gordon and the Dirac descriptions of one object —— the reason the framework uses it in The Dirac Equation in Biquaternionic Form and Spinors as Biquaternion Idempotents.

Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$, $e_1e_2=e_3$ and so on, central scalar imaginary $i$, and general element $\tilde{Q}=\sum_{\mu=0}^{3}Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$. On the quaternion subspace $\mathbb{H}_{\mathbb{B}}$ the coordinates are real, $\tilde{Q} = \sum_\mu q_\mu e_\mu$, with $\mathbf{q} = q_1 e_1 + q_2 e_2 + q_3 e_3$, $\rho = \|\mathbf{q}\|_E$, and $\partial_\mu = \partial/\partial q_\mu$. The subspaces $\mathbb{M}_-$, $\mathbb{M}_+$ and $i\mathbb{H}_{\mathbb{B}}$ are those of Biquaternion Algebra, with the physical readings of The Anti-Hermitian Subspace $\mathbb{M}_-$ as the Material Sector and The Hermitian Subspace $\mathbb{M}_+$ as the Informational Sector. The Dirac operator and the d'Alembertian are built in Biquaternion Analysis, and the Clifford identification in The Clifford Structure of the Biquaternion Algebra.

The Fueter Operator

Definition and Conjugate

On $\mathbb{H}_{\mathbb{B}}$ the Fueter operator, also called the Cauchy–Riemann–Fueter operator, is the biquaternion-valued first-order operator

$$ \tilde{\nabla} = \sum_{\mu=0}^{3} e_\mu \partial_\mu, \qquad \tilde{\nabla}\tilde{F} = \sum_{\mu=0}^{3} e_\mu \partial_\mu \tilde{F}, $$

acting on the left. Its conjugate is obtained by negating the vector part, $e_k \mapsto -e_k$:

$$ \tilde{\nabla}^{\natural} = \partial_0 - \mathbf{D}, \qquad \mathbf{D} = e_1 \partial_1 + e_2 \partial_2 + e_3 \partial_3, \qquad \tilde{\nabla} = \partial_0 + \mathbf{D}. $$

The operator $\tilde{\nabla}$ is exactly the biquaternionic gradient of Biquaternion Analysis restricted to $\mathbb{H}_{\mathbb{B}}$, where the complex coefficients $Q_\mu$ reduce to the real coordinates $q_\mu$. It is the Cauchy–Riemann operator of the theory, classically the Dirac operator: on the material slice it is the massless Dirac operator in the four-vector form the framework uses.

Factorization of the Laplacian

Proposition. On $\mathbb{H}_{\mathbb{B}}$, $\tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$, where $\Delta_4 = \sum_{\mu=0}^{3} \partial_\mu^2$.

Proof. Expand $\tilde{\nabla}\tilde{\nabla}^{\natural} = \sum_{\mu,\nu} e_\mu \bar{e}_\nu \partial_\mu \partial_\nu$, with $\bar{e}_0 = e_0$ and $\bar{e}_k = -e_k$. The diagonal contribution is $\sum_\mu e_\mu \bar{e}_\mu \partial_\mu^2 = \Delta_4 e_0$, since $e_0\bar{e}_0 = 1$ and $e_k\bar{e}_k = -e_k^2 = 1$. For $\mu < \nu$ the coefficient is $e_\mu\bar{e}_\nu + e_\nu\bar{e}_\mu$, equal to $-e_k + e_k = 0$ when $\mu = 0 < \nu = k$ and to $-e_\mu e_\nu - e_\nu e_\mu = 0$ when $1 \le \mu < \nu$ by anticommutativity. Hence the off-diagonal vanishes, and the same argument gives $\tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$.

Physical reading: a square root of the wave operator. The factorization is the algebraic statement that the Dirac operator is a square root of the d'Alembertian: applying the first-order operator twice gives the second-order wave operator, and this is exactly why a massive field's Klein–Gordon equation is the square of its Dirac equation. The notation $\Delta_4$ here is the four-dimensional Laplacian of the Euclidean reading; on the material slice, where the zeroth coordinate is $ict$, it is the wave operator, of signature $(3,1)$. The two faces are the two subspace readings of one factorisation, treated in Biquaternion Analysis.

Relation to the Cauchy–Riemann Operator

The factorization makes $\tilde{\nabla}$ a square root of the Laplacian, hence the Cauchy–Riemann operator of $\mathbb{R}^4$ up to normalization: since $\mathbf{D}^2 = -\Delta_3 e_0$, the product $(\partial_0 + \mathbf{D})(\partial_0 - \mathbf{D})$ equals $\partial_0^2 + \Delta_3 = \Delta_4$. On a slice $\mathbb{C}_I$ (below), the part of $\tilde{\nabla}$ differentiating along the slice is the complex Cauchy–Riemann operator $\partial_{q_0} + I\partial_\rho$, so $\tilde{\nabla}\tilde{F} = 0$ is the quaternionic Cauchy–Riemann equation. In Clifford-algebra language, $\mathbb{H}$ is the even subalgebra of $\mathrm{Cl}_{0,3}$ and $\mathbb{B}$ its complexification, the even subalgebra of $\mathrm{Cl}_{1,3}$; hence the term monogenic for Fueter-regular functions.

Fueter-Regular Functions

Left and Right Regularity

Definition. Let $\tilde{F} : \Omega \to \mathbb{B}$ be differentiable on an open set $\Omega \subseteq \mathbb{H}_{\mathbb{B}}$. Then $\tilde{F}$ is left-regular if $\tilde{\nabla}\tilde{F} = 0$, and right-regular if

$$ \tilde{F}\tilde{\nabla} = 0, \qquad \tilde{F}\tilde{\nabla} = \sum_{\mu=0}^{3} \partial_\mu \tilde{F}\, e_\mu . $$

The conditions differ in general because $\mathbb{B}$ is not commutative. Left-regular functions form a right $\mathbb{B}$-module ($\tilde{\nabla}(\tilde{F}a) = (\tilde{\nabla}\tilde{F})a = 0$ for constant $a$) and right-regular functions a left module ($(a\tilde{F})\tilde{\nabla} = a(\tilde{F}\tilde{\nabla}) = 0$); neither is two-sided in general.

The Componentwise System

Write $\tilde{F} = \sum_\mu F_\mu e_\mu$ and $\mathbf{F} = F_1 e_1 + F_2 e_2 + F_3 e_3$. The scalar-vector decomposition of Biquaternion Analysis gives

$$ \tilde{\nabla}\tilde{F} = (\partial_0 F_0 - \mathrm{div}\,\mathbf{F}) + (\partial_0 \mathbf{F} + \mathrm{grad}\,F_0 + \mathrm{rot}\,\mathbf{F}), $$

with $\mathrm{div}\,\mathbf{F} = \sum_k \partial_k F_k$, $\mathrm{grad}\,F_0 = \sum_k (\partial_k F_0)e_k$ and $\mathrm{rot}\,\mathbf{F} = \sum_{j,k,l} \epsilon_{jkl} (\partial_j F_k)e_l$. Hence left-regularity is

$$ \partial_0 F_0 = \mathrm{div}\,\mathbf{F}, \qquad \partial_0 \mathbf{F} = -\mathrm{grad}\,F_0 - \mathrm{rot}\,\mathbf{F}, $$

while the right action gives the same system with $+\mathrm{rot}\,\mathbf{F}$. The two systems differ only in the sign of the curl term and coincide exactly when $\mathrm{rot}\,\mathbf{F} = 0$. This is the quaternionic Cauchy–Riemann system: one biquaternion equation, equivalently eight real equations for the eight real components of $\tilde{F}$. The reflection $q_k \mapsto -q_k$ interchanges $\tilde{\nabla}$ and $\tilde{\nabla}^{\natural}$, so the conjugate theory is the mirror image; conjugating $\tilde{\nabla}\tilde{F} = 0$ shows that $\tilde{F}^{\natural}$ is right-regular for $\tilde{\nabla}^{\natural}$.

Harmonicity and the Mean Value Property

Proposition. Every left- or right-regular function is harmonic: $\Delta_4 \tilde{F} = 0$ componentwise.

Proof. If $\tilde{\nabla}\tilde{F} = 0$, then $\Delta_4 \tilde{F} = \tilde{\nabla}^{\natural}(\tilde{\nabla}\tilde{F}) = 0$; if $\tilde{F}\tilde{\nabla} = 0$, then $\Delta_4 \tilde{F} = (\tilde{F}\tilde{\nabla})\tilde{\nabla}^{\natural} = 0$, with the operators acting on the right.

Consequently every regular function is real-analytic and enjoys the maximum principle, Liouville's theorem, the identity theorem and the Cauchy estimates (see Biquaternion Regular Functions). Since each component is harmonic, the mean value property holds: for $\tilde{F}$ regular near the closed ball $\bar{B}(\tilde{Q}_0, r)$,

$$ \tilde{F}(\tilde{Q}_0) = \frac{1}{|B(\tilde{Q}_0,r)|}\int_{B(\tilde{Q}_0,r)} \tilde{F}\, dV = \frac{1}{|\partial B(\tilde{Q}_0,r)|}\int_{\partial B(\tilde{Q}_0,r)} \tilde{F}\, dS, $$

with the ordinary measures on $\mathbb{R}^4$; the property follows equally from the Cauchy formula, the proof above showing that it is already a consequence of the factorization.

The Fueter Construction

The Axial Extension of a Holomorphic Function

Let $f_0$ be holomorphic on the disc $D(0,R) \subseteq \mathbb{C}$, written $f_0(z) = u(x,y) + i\,v(x,y)$ with $z = x + iy$ and $u, v$ real-valued; then $u, v$ are harmonic and satisfy the Cauchy–Riemann equations $u_x = v_y$, $u_y = -v_x$. The axial extension of $f_0$ is

$$ \tilde{f}_0(\tilde{Q}) = u(q_0, \rho) + \hat{\mathbf{q}}\, v(q_0, \rho), \qquad \hat{\mathbf{q}} = \mathbf{q}/\rho, $$

defined for $\rho > 0$ by replacing $x$ by $q_0$, $y$ by $\rho$, and the complex imaginary unit $i$ by $\hat{\mathbf{q}}$; the replacement is legitimate because $\hat{\mathbf{q}}^2 = -1$. When $f_0$ has real Taylor coefficients, $v(q_0,0) = 0$, the extension is continuous at $\rho = 0$ with value $f_0(q_0)$, and $\tilde{f}_0(\tilde{Q}) = \sum_{n \ge 0} a_n \tilde{Q}^n$ on $B(0,R)$. Complex coefficients are handled by $\mathbb{C}$-linearity: writing $f_0 = f_{0,1} + i f_{0,2}$ with real-coefficient parts, one sets $\tilde{f}_0 = \tilde{f}_{0,1} + i\,\tilde{f}_{0,2}$.

Fueter's Theorem

Theorem (Fueter's construction). Let $f_0$ be holomorphic on $D(0,R)$, with axial extension $\tilde{f}_0$. Then

$$ \tilde{F}(\tilde{Q}) = \Delta_4\, \tilde{f}_0(\tilde{Q}) = \frac{2\,\partial_\rho u(q_0,\rho)}{\rho} + \hat{\mathbf{q}}\left(\frac{2\,\partial_\rho v(q_0,\rho)}{\rho} - \frac{2\,v(q_0,\rho)}{\rho^2}\right) $$

is defined and real-analytic on $B(0,R)$, by continuity at $\rho = 0$, and is both left- and right-Fueter-regular there.

Proof. For $g = A(q_0,\rho) + \hat{\mathbf{q}}B(q_0,\rho)$ with central $A, B$, one has $\partial_{q_k}A = (\partial_\rho A)\hat{q}_k$, $\sum_k e_k \hat{q}_k = \hat{\mathbf{q}}$, and $\sum_{j,k}(\partial_{q_k}\hat{q}_j)e_k e_j = (-3 - \hat{\mathbf{q}}^2)/\rho = -2/\rho$, because $\hat{\mathbf{q}}^2 = -1$. These give

$$ \tilde{\nabla} g = g\tilde{\nabla} = \left(\partial_0 A - \partial_\rho B - \frac{2B}{\rho}\right) + \hat{\mathbf{q}}\,(\partial_0 B + \partial_\rho A), $$

so $g$ is left-regular if and only if it is right-regular, and this holds exactly when $\partial_0 A = \partial_\rho B + 2B/\rho$ and $\partial_0 B = -\partial_\rho A$. Using the radial form of $\Delta_{\mathbb{R}^3}$, the identity $\Delta_{\mathbb{R}^3}(B\hat{\mathbf{q}}) = \hat{\mathbf{q}}(\Delta_{\mathbb{R}^3}B - 2B/\rho^2)$, and the harmonicity of $u, v$, one gets $\Delta_4 \tilde{f}_0 = P + \hat{\mathbf{q}}Q$ with $P = 2u_\rho/\rho$, $Q = 2(\rho v_\rho - v)/\rho^2$; the Cauchy–Riemann equations then give $Q_\rho + 2Q/\rho = \partial_0 P$ and $-P_\rho = \partial_0 Q$, and $P, Q$ extend continuously to $\rho = 0$.

Physical reading: from amplitude to field. A holomorphic function of one complex variable is a single-component amplitude satisfying the two-dimensional Cauchy–Riemann equations. The axial extension replaces its imaginary unit by the radial direction $\hat{\mathbf{q}}$ of the four-dimensional space, and the Laplacian then produces a field annihilated by the four-dimensional Dirac operator. The construction is thus the algebraic content of the step from a scalar wave function to a spinor field, and it is why the framework can present the Dirac equation as the "square-rooted" Klein–Gordon equation: the same holomorphic datum, read at two different orders, gives the two descriptions. The spinor nature of the result is the subject of Spinors as Biquaternion Idempotents.

The Kernel and Injectivity

Proposition. The map $\tau(f_0) = \Delta_4 \tilde{f}_0$ vanishes if and only if $f_0$ is affine, $f_0(z) = az + b$ with $a, b \in \mathbb{C}$.

Proof. $\tau(f_0) = 0$ iff $u_\rho = 0$ and $\rho v_\rho = v$, so $u = u(q_0)$, $v = c(q_0)\rho$; then $c' = 0$ and $u = cq_0 + d$ with $c, d \in \mathbb{R}$, and complex linearity gives all affine functions. Conversely, $\Delta_4$ annihilates constants and linear functions.

Thus $\tau$ is injective exactly on the holomorphic functions whose Taylor coefficients vanish to order two, $a_0 = a_1 = 0$, its kernel being the affine functions.

Physical reading. The kernel being the affine functions is the statement that a constant field and a uniform gradient carry no propagating content: the construction cannot distinguish two amplitudes that differ only by a constant and a linear term, because those are annihilated by the wave operator. The framework's counting of propagating degrees of freedom therefore works modulo affine data, and this is the algebraic version of the statement that a constant field is not a wave.

The Axial and Slice Approach

Imaginary Units and Slices

An imaginary unit is an element $I$ with $I^2 = -1$. On $\mathbb{H}_{\mathbb{B}}$ the imaginary units form the two-sphere $S^2$ of unit pure imaginary quaternions, and every $\tilde{Q}$ with $\mathbf{q} \neq 0$ is uniquely $\tilde{Q} = q_0 + I\rho$ with $I \in S^2$, $\rho > 0$. The slice $\mathbb{C}_I = \mathbb{R} + I\mathbb{R}$ is a copy of the complex plane, and two slices meet only in $\mathbb{R}$ unless $I = \pm J$. On the full algebra the roots of $-1$ are, by Biquaternion Square Roots of Minus One, Zero and Plus One, the family $\pm i$, $\pm\mu$ and $b\mu + d\nu i$, with $\mu, \nu$ perpendicular unit pure real quaternions and $b^2 - d^2 = 1$, $b, d > 0$. Each spans a slice $\mathbb{C}_\xi \cong \mathbb{C}$, but this set of roots is no longer a sphere, so there is no single imaginary sphere on which to base the theory: the axial construction must select one root, hence one slice, at a time.

Physical reading: the imaginary unit of a rest frame. The sphere $S^2$ of imaginary units of the quaternion subspace is the sphere of directions in which "the" complex structure of a rest frame can point; choosing a slice $\mathbb{C}_I$ is choosing a timelike direction and hence a rest frame. This is the algebraic statement that the decomposition of a field into "space" and "time" is frame-dependent, and it is the same sphere that appears in Biquaternion Rotations and Lorentz Transformations as the ambiguity of the rotation axis. That the full algebra's roots of $-1$ no longer form a sphere is why the framework fixes one of them — the central $i$ — and reads the physical complex structure off it.

Axially Symmetric Functions and the Harmonic Coefficients

A function is axially symmetric if $\tilde{F}(q_0 + \hat{\mathbf{q}}\rho)$ depends on the direction $\hat{\mathbf{q}}$ only through $\hat{\mathbf{q}}$. Such a function has the axial representation $\tilde{F}(\tilde{Q}) = A(q_0,\rho) + \hat{\mathbf{q}}B(q_0,\rho)$, a sum over the two-element basis $\{1,\hat{\mathbf{q}}\}$ of the slice $\mathbb{C}_{\hat{\mathbf{q}}}$; $A$ and $B$ are the axial coefficients. For such a function the left- and right-regularity conditions coincide, by the computation in the proof of Fueter's theorem, and reduce to $\partial_0 A = \partial_\rho B + 2B/\rho$ and $\partial_0 B = -\partial_\rho A$. Eliminating $B$ gives $\Delta_4 A = 0$, so the scalar axial coefficient is harmonic, while $\Delta_4 B = 2B/\rho^2$. The Fueter construction realizes this class with $P = 2u_\rho/\rho$, $Q = 2v_\rho/\rho - 2v/\rho^2$, so the axial coefficients come from the harmonic conjugate pair $(u,v)$; conversely, in the classical Fueter correspondence every axially symmetric regular function with central axial coefficients arises this way, uniquely modulo the affine kernel.

The Fueter–Sce Theorem and Its Hypotheses

Theorem (Fueter–Sce). Let $n \ge 1$ be odd, let $\mathrm{Cl}_{0,n}$ have generators $e_1, \dots, e_n$, and let $f_0$ be holomorphic on a disc, with axial extension $\tilde{f}_0$ to $\mathbb{R}^{n+1}$ (complex unit replaced by a unit vector of $\mathbb{R}^n$, $y$ by the radius). Then $\tilde{F} = \Delta^{(n-1)/2} \tilde{f}_0$ is monogenic, annihilated on the left and on the right by the Cauchy–Riemann operator $\partial_{x_0} + \sum_{j=1}^{n} e_j \partial_{x_j}$ on the ball where the extension is defined. For $n = 3$ the exponent is $1$, recovering Fueter's construction.

Three hypotheses are needed, and none can be dropped. First, $f_0$ must be holomorphic on $D(0,R)$, so its Taylor series converges there; the induced series then converges on $B(0,R)$, since $\Delta_4(\tilde{Q}^n)$ is a homogeneous polynomial of degree $n-2$ with at most polynomial growth in $n$, while $|a_n| r^n = O(1)$ for $r < R$ by Cauchy's estimates. Second, the construction has the affine kernel $az + b$, so a one-to-one correspondence requires the Taylor coefficients to vanish to order two, $a_0 = a_1 = 0$, equivalently one works modulo affine functions. Third, parity: $(n-1)/2$ must be a non-negative integer, so $n$ must be odd, as in the quaternion and biquaternion case $n = 3$.

Physical reading: why spacetime dimension is built in. The parity hypothesis, $n$ odd, is the algebraic reason the construction works in the dimensions it does: a Dirac-type square root of the Laplacian and a Fueter construction from holomorphic data exist in the dimensions where the Clifford algebra has the required even structure. The framework sits at $n=3$, that is, four spacetime dimensions, and the theorem is therefore also a statement about why that dimension is special for the construction.

Power Series Representations

A series with biquaternionic coefficients placed on the right, $f(\tilde{Q}) = \sum_{n \ge 0} \tilde{Q}^n a_n$, converges absolutely on $\|\tilde{Q}\|_E < R$ with $R^{-1} = \limsup_n \|a_n\|_E^{1/n}$, because $\|\tilde{Q}^n\|_E = \|\tilde{Q}\|_E^n$. Its sum is slice-regular, or Cullen-regular: holomorphic on each slice. Slice-regular functions form a different class from the Fueter-regular ones; for example, $\tilde{Q} \mapsto \tilde{Q}$ is slice-regular, but

$$ \tilde{\nabla}\tilde{Q} = \sum_{\mu=0}^{3} e_\mu e_\mu = e_0 - e_1^2 - e_2^2 - e_3^2 = -2e_0 \neq 0. $$

The Fueter construction is precisely the operation converting slice-regular, or holomorphic, data into Fueter-regular functions.

For real Taylor coefficients, $\tilde{f}_0(\tilde{Q}) = \sum_{n \ge 0} a_n \tilde{Q}^n$, and term-by-term application of $\Delta_4$ gives the induced series $\tau(f_0) = \sum_{n \ge 0} a_n \Delta_4(\tilde{Q}^n)$, which begins at $n = 2$ and converges normally on $B(0,R)$ because $\Delta_4(\tilde{Q}^n)$ has degree $n-2$ and at most polynomial growth; term-by-term differentiation is therefore justified. In general, with $\mathcal{P}_k$ the biquaternion-valued homogeneous polynomials of degree $k$ and $\mathcal{M}_k = \{P \in \mathcal{P}_k : \tilde{\nabla}P = 0\}$ the monogenic homogeneous polynomials, the Fischer decomposition $\mathcal{P}_k = \bigoplus_{j=0}^{k} \tilde{Q}^j \mathcal{M}_{k-j}$ gives every Fueter-regular function on $B(0,R)$ a normally convergent expansion $\tilde{F} = \sum_{k \ge 0} \tilde{F}_k$ with $\tilde{F}_k \in \mathcal{M}_k$, the analogue of the Taylor series of complex analysis.

Physical reading. The monogenic Taylor expansion is the multipole expansion of a regular field: each homogeneous monogenic polynomial is one multipole order, and every regular field is a sum of them. The distinction from slice regularity is the statement that a field which is merely holomorphic on each slice is not automatically a solution of the Dirac equation; the construction in the other direction — Fourier-like synthesis from monogenic polynomials — is what makes the expansion useful, and it is the four-dimensional analogue of the plane-wave expansion of a free field.

The Biquaternionic Case

On $\mathbb{H}_{\mathbb{B}}$ the theory is the classical one with biquaternion coefficients, since the quaternion subspace is a division algebra: $N(\tilde{Q}) = \sum_\mu q_\mu^2$ is positive-definite, so every nonzero element is invertible. The fundamental solution $\tilde{G}(\tilde{Q}) = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$ satisfies $\tilde{\nabla}\tilde{G} = 0$ off the origin and $\tilde{\nabla}\tilde{G} = -2\pi^2 \delta_0 e_0$ distributionally, so the only singularity of the Cauchy theory is the point $\tilde{Q} = 0$.

On $i\mathbb{H}_{\mathbb{B}}$ the theory is the same, up to the constant central factor by which the two gradients differ. Every coefficient there is purely imaginary, so the gradient is $\tilde{\nabla}_{i\mathbb{H}} = -i\,\tilde{\nabla}_{\mathbb{H}}$ in the same four coordinates; since $-i$ is an invertible central constant, $\tilde{\nabla}_{i\mathbb{H}}\tilde{F} = 0$ if and only if $\tilde{\nabla}_{\mathbb{H}}\tilde{F} = 0$. The left- and right-regular functions are therefore the same class as on $\mathbb{H}_{\mathbb{B}}$, and the Cauchy kernel is the kernel of $\mathbb{H}_{\mathbb{B}}$ multiplied by $i$. Only the second-order operator changes sign, $\tilde{\nabla}\tilde{\nabla}^{\natural} = -\Delta_4 e_0$; its solutions are again the ordinary harmonic functions, so the harmonicity, the mean value property and the monogenic Taylor expansion carry over unchanged. In this sense $i\mathbb{H}_{\mathbb{B}}$ adds no new case to Fueter theory: it is the definite counterpart of $\mathbb{H}_{\mathbb{B}}$, as $\mathbb{M}_+$ is the indefinite counterpart of $\mathbb{M}_-$.

On $\mathbb{M}_-$ and $\mathbb{M}_+$ the second-order operator is the wave operator, $\Box_{\mathbb{M}_-} = (-\partial_{q'_0}^2 + \Delta_{\mathbb{R}^3})e_0$ and $\Box_{\mathbb{M}_+} = (\partial_{q_0}^2 - \Delta_{\mathbb{R}^3})e_0$, of signatures $(3,1)$ and $(1,3)$: hyperbolic, hence without elliptic regularity, and the axial coefficients then satisfy a wave-type system rather than a Laplace system. The zero divisors form the null cone $(q'_0)^2 = q_1^2 + q_2^2 + q_3^2$ on $\mathbb{M}_-$ (and $q_0^2 = (q'_1)^2 + (q'_2)^2 + (q'_3)^2$ on $\mathbb{M}_+$), exactly where the biquaternion norm vanishes. Its nonzero elements are not invertible, so the pointwise inversion of $\tilde{Q}$ underlying the Cauchy kernel fails there, and the Euclidean kernel is not the fundamental solution of the gradient on these subspaces, as Biquaternion Integration, Biquaternion Regular Functions and Biquaternion Analysis on Subspaces record. Domains for the Cauchy theory must avoid the cone, not merely the origin.

On the full algebra $\mathbb{B}$ the biquaternion norm $N(\tilde{Q}) = \sum_\mu Q_\mu^2$ is complex-valued, and its zero set is the null quadric, whose nonzero points are exactly the zero divisors and which has real dimension $6$. Every construction dividing by $\tilde{Q}$, and every kernel built from $\tilde{Q}^{-1}$, is therefore unavailable there. The Fueter operator is also not elliptic over $\mathbb{C}$: its symbol $\sum_\mu e_\mu \xi_\mu$ has norm $\sum_\mu \xi_\mu^2$, vanishing for nonzero complex covectors such as $(1, i, 0, 0)$, so ellipticity holds for real covectors but fails over $\mathbb{C}$ and the second-order operator is not the real Laplacian. Finally, the imaginary units are the larger family of the preceding section, so the slice structure must be built one root at a time. The Fueter theory is therefore developed on the four-dimensional subspaces, principally $\mathbb{H}_{\mathbb{B}}$, and extended to the indefinite subspaces only off the null cone; the full-algebra case remains the open problem noted in Biquaternion Analysis.

Physical reading: the material slice versus the informational slice. The two indefinite subspaces are the two real signatures the framework uses: $\mathbb{M}_-$ with signature $(3,1)$ is the material slice whose null cone is the light cone, and $\mathbb{M}_+$ with signature $(1,3)$ is the informational slice. On them the operator is hyperbolic and the Cauchy theory must be replaced by a characteristic-initial-value theory, which is the field-theoretic content of The Wave Equation on $\mathbb{M}_-$ and of Biquaternion Analysis on Subspaces. The vanishing of the norm on the cone is the algebraic statement that lightlike directions are not normalisable, and the six-real-dimensional null quadric of the full algebra is the obstruction that keeps Fueter theory on subspaces rather than on the whole algebra.

Summary

The Fueter operator $\tilde{\nabla} = \sum_\mu e_\mu \partial_\mu$ and its conjugate factor the four-dimensional Laplacian on the quaternion subspace, $\tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$. On the anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$ the gradient differs from this one by the constant central factor $-i$, so the regular functions and the Cauchy kernel are those of $\mathbb{H}_{\mathbb{B}}$ up to $i$, and only the sign of the second-order operator changes. Left-regularity $\tilde{\nabla}\tilde{F} = 0$ and right-regularity $\tilde{F}\tilde{\nabla} = 0$ differ only in the sign of the curl term of the componentwise Cauchy–Riemann–Fueter system and coincide for axially symmetric functions; every regular function is harmonic, hence real-analytic, and satisfies the mean value property.

Physically the factorization is the statement that the Dirac operator is a square root of the wave operator, and the Fueter construction is the passage from a one-component holomorphic amplitude to a four-component field: it sends $f_0$ to the induced function $\tilde{F} = \Delta_4 \tilde{f}_0 = 2u_\rho/\rho + \hat{\mathbf{q}}(2v_\rho/\rho - 2v/\rho^2)$, which is both left- and right-regular. It is $\mathbb{C}$-linear with the affine functions as kernel, hence injective exactly for holomorphic functions whose Taylor coefficients vanish to order two — the statement that a constant field and a uniform gradient carry no propagating content. The Fueter–Sce theorem extends it to odd-dimensional Clifford algebras with the power $(n-1)/2$, under the hypotheses of holomorphic convergence, order-two vanishing and odd dimension; the parity hypothesis is why the construction lives in four spacetime dimensions. Regular functions have the induced series and the monogenic Taylor expansion, while slice-regular series form the distinct class that the construction converts into regular functions.

On $\mathbb{H}_{\mathbb{B}}$ the origin is the only singularity, but on the full algebra the six-dimensional null quadric obstructs inversion, the Fueter operator is not elliptic over $\mathbb{C}$, and the imaginary units are no longer the sphere $S^2$; the full-algebra case remains open.

Summary of Notation

Symbol	Meaning
$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$	Biquaternion algebra
$\mathbb{H}_{\mathbb{B}}$	Quaternion subspace, with real coordinates $q_\mu$; the domain of Fueter theory
$i\mathbb{H}_{\mathbb{B}}$	Anti-quaternion subspace, all coefficients purely imaginary; regularity is the same class as on $\mathbb{H}_{\mathbb{B}}$, with $\Box = -\Delta_4 e_0$
$\tilde{Q} = \sum_\mu Q_\mu e_\mu$	Biquaternion; $\tilde{Q} = \sum_\mu q_\mu e_\mu$ on $\mathbb{H}_{\mathbb{B}}$
$\mathbf{q} = \sum_k q_k e_k$, $\rho = \\|\mathbf{q}\\|_E$, $\hat{\mathbf{q}} = \mathbf{q}/\rho$	Vector part, its modulus, and its direction
$\tilde{\nabla} = \partial_0 + \mathbf{D}$, $\mathbf{D} = \sum_k e_k \partial_k$	Fueter (Cauchy–Riemann–Fueter) operator, acting on the left; the massless Dirac operator on the material slice
$\tilde{\nabla}^{\natural} = \partial_0 - \mathbf{D}$	Conjugate Fueter operator
$\Delta_4 = \sum_\mu \partial_\mu^2$	Four-dimensional Laplacian; $\tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta_4 e_0$
$\tilde{F}$ left-regular	$\tilde{\nabla}\tilde{F} = 0$; monogenic in Clifford language
$\tilde{F}$ right-regular	$\tilde{F}\tilde{\nabla} = 0$
$I$, $S^2$	Imaginary unit, $I^2 = -1$; on $\mathbb{H}_{\mathbb{B}}$ these form the two-sphere, the sphere of rest-frame directions
$\mathbb{C}_I = \mathbb{R} + I\mathbb{R}$	Slice, a copy of the complex plane; slice-regular functions live on unions of slices
$A, B$	Axial coefficients of an axially symmetric function, $\tilde{F} = A(q_0,\rho) + \hat{\mathbf{q}}B(q_0,\rho)$
$\tilde{f}_0$, $\Delta_4\tilde{f}_0$	Axial extension of a holomorphic $f_0$ and its Fueter-induced function; the passage from amplitude to field
$\mathrm{Cl}_{0,n}$, $\mathrm{Cl}_{1,3}^{+}$	Clifford algebras; $\mathbb{H}$ is $\mathrm{Cl}_{0,3}^{+}$ and $\mathbb{B}$ is $\mathrm{Cl}_{1,3}^{+}$