Biquaternion Regular Functions

Introduction

This article studies regular (equivalently monogenic) biquaternion-valued functions. It follows Biquaternion Analysis, Biquaternion Integration and Biquaternion Algebra, and assumes the algebra $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, its four conjugations, its biquaternion norm $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural} = \sum_\mu Q_\mu^2$, its Euclidean norm and its zero divisors.

The decisive fact is that $\mathbb{B}$ is not a division algebra: it has nonzero elements with no inverse, the zero divisors, forming the null quadric $N(\tilde{Q}) = 0$. Every failure of the complex analogy on the full algebra and on the indefinite subspaces is traceable to this fact; on $\mathbb{H}_{\mathbb{B}}$ the remaining failures are those of dimension and noncommutativity. A statement of regularity must therefore specify both the operator with respect to which the function is regular and the domain on which it is defined.

Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with $e_0=1$, $e_1^2=e_2^2=e_3^2=-e_0$, $e_1e_2=e_3$, $e_2e_3=e_1$, $e_3e_1=e_2$, and central scalar imaginary $i$; a general element is $\tilde{Q}=\sum_\mu Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$, written $\tilde{Q}=Q_0e_0+\mathbf{Q}$ with $\mathbf{Q}=\sum_{k=1}^{3}Q_ke_k$. The two polar forms, the four conjugations and the six subspaces are those of Biquaternion Algebra; the norm and the null quadric are Biquaternion Norm and Invertibility, and the zero divisors are Biquaternion Zero Divisors. The underlying analysis operators are built in Biquaternion Analysis; the integral theory is Biquaternion Integration; the construction that turns holomorphic data into regular functions is Fueter Theory for Biquaternions.

The Biquaternion Variable and Its Two Complex Structures

A complex structure on a real vector space is a real-linear map $J$ with $J^2 = -\mathrm{id}$. The algebra carries a pair of commuting complex structures,

$$ J_i(\tilde{Q}) = i\tilde{Q}, \qquad J_1(\tilde{Q}) = e_1\tilde{Q}. $$

Both square to $-\mathrm{id}$ and commute because $i$ is central; the same construction with $e_2$ or $e_3$ gives further ones, and every root of $-1$ in $\mathbb{B}$ defines one. The two named structures are the coefficient complex structure $J_i$, which complexifies the coefficients $Q_\mu$, and the quaternionic complex structure $J_1$, which complexifies the plane spanned by $e_0, e_1$.

The quaternionic structure turns that plane into a complex line with coordinate $z = q_0 + e_1 q_1$, and the complex combinations of $e_0, e_1$ form the commutative subalgebra

$$ \mathbb{C}[e_1] = \{a e_0 + b e_1 : a, b \in \mathbb{C}\} = \mathrm{span}_{\mathbb{R}}\{e_0, e_1, i e_0, i e_1\} \cong \mathbb{C} \times \mathbb{C}, $$

the largest commutative subalgebra in which $e_1$ is the imaginary unit; the coefficient structure similarly singles out $\mathbb{C}_{\mathbb{B}} = \mathbb{C} e_0$. These two structures give two inequivalent notions of holomorphy, distinguished in §Regular Functions: Single-Plane versus Hypercomplex: no single complex structure reduces the four real variables to one. Finally, $\mathbb{B}$ is simple with centre $\mathbb{C}_{\mathbb{B}}$, and its norm vanishes exactly on the zero divisors and the origin.

Physical reading: two complex structures, two physical roles. The coefficient structure $J_i$ is the one the physics uses as complex conjugation of amplitudes — it is the $i$ of the wave function and of quantum phases, and the one whose square gives the $ict$ time. The quaternionic structure $J_1$ is a spatial rotation of the algebra's own action, and it is the one whose complex line $z = q_0+e_1q_1$ is a two-dimensional slice of a four-position. Keeping the two apart is what prevents the common error of treating the physical $i$ and the algebra's $e_k$ as interchangeable: they are different square roots of $-1$ in different roles, and Biquaternion Algebra is where their two polar forms are separated.

The Cauchy–Riemann Operator and Its Conjugate

On a four-dimensional real subspace $V \subset \mathbb{B}$ with complex coefficients $Q_0, \dots, Q_3$, the biquaternionic gradient, called here the biquaternionic Cauchy–Riemann operator, and its quaternion conjugate are

$$ \tilde{\nabla} = \sum_{\mu=0}^{3} e_\mu \frac{\partial}{\partial Q_\mu}, \qquad \tilde{\nabla}^{\natural} = e_0 \frac{\partial}{\partial Q_0} - \sum_{k=1}^{3} e_k \frac{\partial}{\partial Q_k}. $$

This is the Cauchy–Riemann operator of the biquaternion theory, classically the Dirac operator. The pair $(\tilde{\nabla}, \tilde{\nabla}^{\natural})$ plays the role of $(\partial_{\bar{z}}, \partial_z)$ in one complex variable. On the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, where all coefficients are real and the partials are ordinary real derivatives, $\tilde{\nabla}$ is the Cauchy–Riemann operator on $\mathbb{R}^4$. The units satisfy the Clifford relation $e_\mu \bar{e}_\nu + e_\nu \bar{e}_\mu = 2\delta_{\mu\nu} e_0$, so the cross terms cancel and

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Box := \left(\partial^2_{Q_0} + \partial^2_{Q_1} + \partial^2_{Q_2} + \partial^2_{Q_3}\right) e_0. $$

The d'Alembertian $\Box$ is scalar and acts component-wise. On $\mathbb{H}_{\mathbb{B}}$ it is the Euclidean Laplacian of signature $(4,0)$; on $\mathbb{M}_-$ it is $-\partial^2_{q'_0} + \Delta_{\mathbb{M}_-}$ of signature $(3,1)$; on $\mathbb{M}_+$ it is $\partial^2_{q_0} - \Delta_{\mathbb{M}_+}$ of signature $(1,3)$; these are coordinate expressions of one abstract operator, the factors of $-i$ being a change of coordinates. The second-order operator $\tilde{\nabla}^2 = \tilde{\nabla}\tilde{\nabla} = (\partial^2_{Q_0} - \Delta_Q)e_0 + 2\sum_k e_k \partial^2_{Q_0 Q_k}$ also appears, with $\tilde{\nabla}^2 = 2\partial_{Q_0}\tilde{\nabla} - \Box \neq \Box$; it must not be confused with $\Box$, the natural second-order operator. For a function depending only on $q_0, q_1$,

$$ \tilde{\nabla}\tilde{F} = 2\partial_{\bar{z}}\tilde{F}, \qquad \tilde{\nabla}^{\natural}\tilde{F} = 2\partial_{z}\tilde{F}, \qquad \partial_{\bar{z}} = \tfrac{1}{2}(\partial_{q_0} + e_1\partial_{q_1}), \qquad \partial_{z} = \tfrac{1}{2}(\partial_{q_0} - e_1\partial_{q_1}). $$

Physical reading: the massless Dirac operator. With the material four-position $\tilde{Q} = ict\,e_0 + \mathbf{x}$ in $\mathbb{M}_-$, the equation $\tilde{\nabla}\tilde{F} = 0$ is the massless Dirac equation: one first-order biquaternion-valued equation for a four-component field, whose square is the wave equation. The pairing of $\tilde{\nabla}$ with $\tilde{\nabla}^{\natural}$ is the pairing of the two chiralities of a massless field, and the fact that the operator is first order rather than second is the reason a fermion's equation is first order. The combination $\tilde{\nabla}^2 = 2\partial_{Q_0}\tilde{\nabla} - \Box$ is the piece that appears when the operator is applied twice on the same side; it is not the wave operator, and confusing the two is the most common error the framework warns about. The Lorentz action the operator carries is the subject of Biquaternion Rotations and Lorentz Transformations.

Regular Functions: Single-Plane versus Hypercomplex

Definition (left-regular). Let $\Omega$ be open in a four-dimensional real subspace $V \subset \mathbb{B}$ and let $\tilde{F} : \Omega \to \mathbb{B}$ be continuously differentiable. Then $\tilde{F}$ is left-regular, or left-monogenic, if $\tilde{\nabla}\tilde{F} = 0$ on $\Omega$. It is right-regular if $\tilde{F}\tilde{\nabla} := \sum_\mu \partial_\mu\tilde{F}\, e_\mu = 0$ on $\Omega$. The two differ by noncommutativity — the units act on the left in the first and on the right in the second — and are exchanged by quaternion conjugation together with the interchange of $\tilde{\nabla}$ and $\tilde{\nabla}^{\natural}$ ($\tilde{\nabla}\tilde{F} = 0 \iff \tilde{F}^{\natural}\,\tilde{\nabla}^{\natural} = 0$); a function regular with respect to $\tilde{\nabla}^{\natural}$ is anti-regular, the analogue of an anti-holomorphic function.

Convention. In this series regular without qualification means left-regular, $\tilde{\nabla}\tilde{F} = 0$. This is fixed by the analysis and integration articles and determines which operator is inverted: the operator being inverted is the first-order operator $\tilde{\nabla}$, whose fundamental solution is the Cauchy kernel. It is not $\Box$, and it is not $\tilde{\nabla}^2$; several second-order operators can be built from $\tilde{\nabla}$, and only one has the Cauchy kernel as its fundamental solution. The opposite convention, $\tilde{\nabla}^{\natural}\tilde{F} = 0$, is also common and merely interchanges regular and anti-regular.

Single-plane holomorphy. Fix $z = q_0 + e_1 q_1$. A function independent of $q_2, q_3$ is holomorphic in $z$ in the classical sense precisely when $\tilde{\nabla}\tilde{F} = 2\partial_{\bar{z}}\tilde{F} = 0$. Hence every classical holomorphic function of $z$, extended by constancy in the orthogonal directions, is regular: the powers $z^n$, the inverse $(z - w)^{-1}$, and so on. Such functions use one complex structure and carry no information about $q_2, q_3$.

Hypercomplex regularity. The hypercomplex notion uses the full dependence on all four variables and the full Clifford structure. It is strictly larger than the single-plane class: the Cauchy kernel $\tilde{G} = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$ is regular on $\mathbb{H}_{\mathbb{B}} \setminus \{0\}$ and is not holomorphic in any single-plane variable. The two notions coincide only in two real dimensions, where the biquaternionic operator reduces to the classical Cauchy–Riemann operator in one complex variable.

The System of Regularity Equations

Writing $\tilde{F} = F_0 + \mathbf{F}$ with $\mathbf{F} = F_1 e_1 + F_2 e_2 + F_3 e_3$, Biquaternion Analysis gives

$$ \tilde{\nabla}\tilde{F} = \left(\partial_{Q_0}F_0 - \mathrm{div}\,\mathbf{F}\right) + \left(\partial_{Q_0}\mathbf{F} + \mathrm{grad}\,F_0 + \mathrm{rot}\,\mathbf{F}\right), $$

with $\mathrm{div}\,\mathbf{F} = \sum_k \partial_{Q_k}F_k$, $\mathrm{grad}\,F_0 = \sum_k (\partial_{Q_k}F_0)e_k$ and $\mathrm{rot}\,\mathbf{F} = \sum_{j,k,l}\epsilon_{jkl}(\partial_{Q_j}F_k)e_l$. Therefore $\tilde{F}$ is regular if and only if

$$ \partial_{Q_0}F_0 = \mathrm{div}\,\mathbf{F}, \qquad \partial_{Q_0}\mathbf{F} + \mathrm{grad}\,F_0 + \mathrm{rot}\,\mathbf{F} = 0. $$

This is a system of four equations for the four coefficients, the biquaternionic Cauchy–Riemann–Fueter system. It is the precise sense in which regularity is expressed by a system of equations rather than by one complex equation, and it is the local expression of the regularity notion inherited from the Clifford algebra, coupling $F_0$ to $\mathbf{F}$ through $\mathrm{div}$, $\mathrm{grad}$ and $\mathrm{rot}$.

On $\mathbb{H}_{\mathbb{B}}$ the principal symbol $s(\xi) = \sum_\mu \xi_\mu e_\mu$ satisfies $s(\xi)\bar{s}(\xi) = |\xi|^2 e_0$, so the system is elliptic and its solutions are smooth (indeed real-analytic). On $\mathbb{M}_-$, with $s(\xi) = -i\xi_0 e_0 + \sum_k \xi_k e_k$, one finds

$$ s(\xi)\bar{s}(\xi) = \left(-\xi_0^2 + \xi_1^2 + \xi_2^2 + \xi_3^2\right)e_0, $$

which vanishes when $\xi_0^2 = \xi_1^2 + \xi_2^2 + \xi_3^2$. Thus on the indefinite subspaces $\tilde{\nabla}$ is not elliptic; it is a Cauchy–Riemann-type operator factoring the wave operator, and the null cone is its characteristic set. The elliptic tools of the complex theory — the maximum principle, the mean value property and Liouville's theorem — are therefore not available on $\mathbb{M}_\pm$.

Physical reading: elliptic versus hyperbolic. Ellipticity on $\mathbb{H}_{\mathbb{B}}$ is why there the theory is a complete analogue of complex analysis, with harmonic functions smoothing out and obeying a maximum principle. On the material slice the operator is hyperbolic instead, its characteristic set is the light cone, and the failure of the elliptic tools is exactly the statement that a massless field propagates along characteristics rather than smoothing. What looks like a loss of analytic power is the physical statement that the operator describes a wave.

Examples of Regular Functions

Constants. If $\tilde{F}(\tilde{Q}) = \tilde{C}$ is constant, then $\tilde{\nabla}\tilde{F} = 0$ on any subspace; the constants form an eight-real-dimensional space of regular functions.

Powers of a single-plane variable. On $\mathbb{H}_{\mathbb{B}}$, with $z = q_0 + e_1 q_1$, every $z^n$, $n \geq 0$, is regular, being holomorphic in $z$ and independent of $q_2, q_3$; for $n = 1$ directly, $\tilde{\nabla} z = e_0 \cdot e_0 + e_1 \cdot e_1 = e_0 - e_0 = 0$. The negative power $(z - w)^{-1}$ is regular away from $z = w$, and more generally every classical holomorphic function of $z$, extended by constancy in the orthogonal directions, is regular. These are the regular linear and power functions.

The coordinate function is not regular. On $\mathbb{H}_{\mathbb{B}}$, $\tilde{\nabla}\tilde{Q} = \sum_{\mu} e_\mu e_\mu = e_0 - 3e_0 = -2e_0 \neq 0$. So the identity function is not regular, in sharp contrast with the complex case, where $z$ is holomorphic; the regular object that replaces it is the Cauchy kernel of §The Cauchy Integral Formula Where It Holds.

The Cauchy kernel. On $\mathbb{H}_{\mathbb{B}}$, $\tilde{G}(\tilde{Q}) = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$ satisfies $\tilde{\nabla}\tilde{G} = 0$ for $\tilde{Q} \neq 0$; it is the fundamental solution of $\tilde{\nabla}$ and is singular only at the origin, since $\mathbb{H}_{\mathbb{B}}$ has no zero divisors.

Closure properties. Regular functions are closed under addition, under right multiplication by constants, $\tilde{\nabla}(\tilde{F}\tilde{C}) = (\tilde{\nabla}\tilde{F})\tilde{C} = 0$, and under left multiplication by complex scalars, since $\lambda$ is central. Left multiplication by a general constant does not preserve regularity: on $\mathbb{H}_{\mathbb{B}}$, $\tilde{\nabla}(e_2 z) = e_0 e_2 + e_1 e_2 e_1 = e_2 + e_3 e_1 = 2e_2 \neq 0$. Thus the regular functions form a right $\mathbb{B}$-module under pointwise right multiplication by constants, but not a left module. Similarly $\tilde{\nabla}(cz) = c + e_1 c e_1 = 2(c_2 e_2 + c_3 e_3)$, so $cz$ is regular exactly when $c \in \mathbb{C}[e_1]$.

Physical reading: linear superpositions of fields. That regular functions form a right module but not a left module over the algebra is the algebraic statement that linear superposition of fields is with scalars on the right, and that a general biquaternion multiplying a field from the left is not a symmetry of the free-field equation. This is why the framework's internal symmetries are built from the multiplicative group of the algebra acting by the sandwich $\tilde{F}\mapsto \tilde{R}\tilde{F}\tilde{R}^{-1}$ rather than by one-sided multiplication: the sandwich is the action that preserves regularity, and one-sided multiplication does not.

Harmonicity and the Factorization of the Laplacian

Since $\Box = \tilde{\nabla}^{\natural}\tilde{\nabla}$, every left-regular function is harmonic: $\Box\tilde{F} = \tilde{\nabla}^{\natural}(\tilde{\nabla}\tilde{F}) = 0$, that is, $(\sum_\mu \partial^2_{Q_\mu})\tilde{F} = 0$. On $\mathbb{H}_{\mathbb{B}}$ this is the ordinary Laplace equation for each coefficient $F_\nu$; on $\mathbb{M}_\pm$ it is the wave equation. The converse fails: $q_0$ on $\mathbb{H}_{\mathbb{B}}$ is harmonic but $\tilde{\nabla} q_0 = e_0 \neq 0$. The factorization is the analogue of $\partial_z \partial_{\bar{z}} = \tfrac{1}{4}\Delta$ in one complex variable, and it explains why the harmonic functions form a strictly larger class. Right-regular functions are likewise harmonic: if $\tilde{F}\tilde{\nabla} = 0$, then

$$ 0 = \left(\tilde{F}\tilde{\nabla}\right)\tilde{\nabla}^{\natural} = \sum_{\mu,\nu} \partial^2_{Q_\mu Q_\nu}\tilde{F}\, e_\mu \bar{e}_\nu = \sum_{\mu=0}^{3} \partial^2_{Q_\mu}\tilde{F} = \Box\tilde{F}, $$

the mixed terms cancelling by the Clifford relation. Every regular function also satisfies $\tilde{\nabla}^2\tilde{F} = 0$, but $\tilde{\nabla}^2$ is not the natural second-order operator.

Physical reading. That regularity implies harmonicity is the statement that a massless field satisfies the wave equation; the converse fails, and this is the statement that not every solution of the wave equation is a massless spinor field. The distinction matters for the framework's counting of degrees of freedom: the wave equation governs a scalar amplitude, while the regularity equation governs a four-component field and imposes the further first-order constraints.

The Cauchy Integral Formula Where It Holds

The integral theory is developed on $\mathbb{H}_{\mathbb{B}}$, where it is the standard Clifford analysis of $\mathbb{R}^4$ and is complete because $\mathbb{H}_{\mathbb{B}}$ is a division algebra. The following results are established in Biquaternion Integration.

Theorem (fundamental solution). On $\mathbb{H}_{\mathbb{B}}$, the function $\tilde{G}(\tilde{Q}) = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$ satisfies $\tilde{\nabla}\tilde{G} = 0$ for $\tilde{Q} \neq 0$ and, in the sense of distributions, $\tilde{\nabla}\tilde{G} = -2\pi^2 \delta_0\, e_0$.

Theorem (Cauchy integral formula). Let $\tilde{F}$ be continuously differentiable on a domain $\Omega \subset \mathbb{H}_{\mathbb{B}}$ with piecewise smooth boundary $\partial\Omega$, and let $\tilde{Q}_0$ be an interior point. Then

$$ \tilde{F}(\tilde{Q}_0) = \frac{1}{2\pi^2}\int_{\partial\Omega}\tilde{G}(\tilde{Q} - \tilde{Q}_0)\tilde{n}\tilde{F}(\tilde{Q})\,dS - \frac{1}{2\pi^2}\int_{\Omega}\tilde{G}(\tilde{Q} - \tilde{Q}_0)(\tilde{\nabla}\tilde{F})(\tilde{Q})\,dV, $$

where $\tilde{n}$ is the biquaternion-valued outward unit normal. If $\tilde{F}$ is regular, the volume term vanishes and

$$ \tilde{F}(\tilde{Q}_0) = \frac{1}{2\pi^2}\int_{\partial\Omega}\tilde{G}(\tilde{Q} - \tilde{Q}_0)\tilde{n}\tilde{F}(\tilde{Q})\,dS. $$

Three hypotheses must be emphasised: the formula holds on the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, where all four coefficients of $\tilde{Q}$ are real, and is not asserted on $\mathbb{M}_+$, on $\mathbb{M}_-$ or on the full algebra $\mathbb{B}$ (§Where the Complex Analogy Fails: Zero Divisors and the Null Cone); regularity is required on all of $\Omega$, not merely on the boundary; and the operator inverted is the first-order operator $\tilde{\nabla}$, normalised by $\tilde{\nabla}\tilde{G} = -2\pi^2\delta_0 e_0$. The mean value property, the maximum principle, Liouville's theorem, the identity theorem, the Cauchy estimates and the residue theory for isolated singularities then follow, as in Biquaternion Integration, on $\mathbb{H}_{\mathbb{B}}$ only.

Physical reading: a field from its boundary values. The Cauchy formula says that a regular field inside a region is fixed by its values on the boundary. That is the field-theoretic statement that the free field has no independent bulk degrees of freedom: the initial or boundary data determine the interior, which is why a massless field can be quantised from boundary modes. The formula holds on $\mathbb{H}_{\mathbb{B}}$ and not on the material slice, and the reason is physical as much as analytic: on the material slice the operator is hyperbolic, and a hyperbolic equation is fixed by initial data on a spacelike surface rather than by data on an enclosing boundary of arbitrary shape.

The Shifted Operator and a Biquaternionic Parameter

The theory above is the theory of $\tilde{\nabla}$, the operator obtained by letting the units differentiate. It has a shift, and the shift is the theory that problems with a boundary condition need. The definition is

$$ D_\alpha = D + M_\alpha , $$

where $D$ is the spatial Moisil–Teodoresco operator in the normalisation in which it is a square root of the Laplacian,

$$ D = i\sum_{k=1}^{3} e_k\partial_k, \qquad D^2 = \Delta , $$

and $M_\alpha$ is right multiplication by $\alpha\in\mathbb{B}$. A function $f$ with $D_\alpha f = 0$ is $\alpha$-hyperholomorphic. The essential difference from the central shift $\tilde{\nabla}_\kappa = \tilde{\nabla} + \kappa$ is that $\alpha$ is an arbitrary element of the algebra: it may be a zero divisor, its square need not be a scalar, and it need not commute with the units. The shifted theory is therefore a genuinely larger object than a Helmholtz shift, and it is the general case the source develops.

The scalar case is the Helmholtz shift. For $\alpha$ a scalar the operator is the shift already used in Electromagnetism in Media: The Local Complex Structure at Work, whose $D_{3\alpha} = D_3 + \alpha$ with $D_3 = \sum_k e_k\partial_k$ has $\Delta+\alpha^2$ as its scalar second-order companion and the kernel $\Theta_\alpha = -e^{i\alpha|x|}/(4\pi|x|)$. The two normalisations differ by the factor $i$, $D = iD_3$, so $D_\alpha = i(D_3 - i\alpha)$; the identification of the two parameters is a matter of convention and should be checked against the source before it is used numerically. The scalar case is the classical Helmholtz operator; the non-scalar case is what the algebra adds.

The vector case: a non-scalar parameter with a scalar companion. A parameter need not be scalar to give a scalar second-order companion. For a pure vector $\mathbf k = k_1e_1+k_2e_2+k_3e_3$ one has $\mathbf k^2 = -\mathbf k\cdot\mathbf k\, e_0$, central, so $M_{\mathbf k}^2 = -\lvert\mathbf k\rvert^2 I$ and $M_{i\mathbf k}^2 = +\lvert\mathbf k\rvert^2 I$; and because $\mathbf k$ is constant, right multiplication by it commutes with $D_3$. Hence

$$ (D_3+M_{\mathbf k})(D_3-M_{\mathbf k})=(D_3-M_{\mathbf k})(D_3+M_{\mathbf k})=-\Delta+\lvert\mathbf k\rvert^2, $$

$$ (D_3+M_{i\mathbf k})(D_3-M_{i\mathbf k})=(D_3-M_{i\mathbf k})(D_3+M_{i\mathbf k})=-\Delta-\lvert\mathbf k\rvert^2, $$

the second being the first with $\mathbf k\mapsto i\mathbf k$. So a shift by a real vector has the Klein–Gordon-type companion $-\Delta+\lvert\mathbf k\rvert^2$ — the sign inside opposite to the scalar branch's $\Delta+\alpha^2$ — whereas a shift by an imaginary vector reproduces that branch's Helmholtz companion $\Delta+\lvert\mathbf k\rvert^2$ with $\alpha=\lvert\mathbf k\rvert$. In both lines the two orderings agree, but not because the parameter is a vector: they agree because it is constant, so that $D_3M_\alpha = M_\alpha D_3$ for any constant $\alpha$. The asymmetry of the two one-sided multiplications appears instead under the plane-wave substitution $u = e^{i\mathbf k\cdot x}v$, where, with $M_{\mathbf k}$ right multiplication and $M^{\mathbf k}$ left multiplication,

$$ (D_3+M_{i\mathbf k})\bigl(e^{i\mathbf k\cdot x}v\bigr)=e^{i\mathbf k\cdot x}\bigl(D_3v+i(\mathbf k v+v\mathbf k)\bigr), $$

and $\mathbf k v+v\mathbf k$ collapses to the pure scalar $-2\,\mathbf k\cdot v$ only when $v$ is a vector; for a general quaternion-valued $v$ the two products do not combine, so the substitution is not the plain transport $D_3v$ of the vector case. This is why a scattering problem is posed on the vector shift and why the quaternionic and the vector readings of one operator differ; the scattering problem itself is Scattering for the Quaternionic Dirac Operator. Dictionary. In the source (S. Bernstein, Seeing the Invisible and Maxwell's Equations) the operator is written $D = \sum_k e_k\partial_k$, which is this article's $D_3$ and not the corpus's $D = iD_3$; its $M_k$ is the right multiplication $M_{\mathbf k}$, its $M^k$ the left one, and its $k$ a vector biquaternion, a zero divisor exactly when $\mathbf k$ is isotropic. The factor $i$ must be restored before the source's companions are compared with the ones above, as in the scalar branch.

Three branches, and why. The integral operators of the theory — the Teodorescu transform $T_\alpha$, the Cauchy-type operator $K_\alpha$ and the operator of singular integration $S_\alpha$ — are defined for every $\alpha\in\mathbb{B}$, but their formulas differ in three cases: $\alpha$ not a zero divisor; $\alpha$ a zero divisor with $\alpha_0\neq0$; and $\alpha$ a zero divisor with $\alpha_0 = 0$. The reason is the degeneracy of the previous section: the kernel is built by inverting the symbol, and on the zero-divisor set the symbol has no inverse. The parameter of a shifted theory is thus itself an object on which the algebra's degeneracy acts — the parameter can be singular, not only the variable.

The four theorems for $D_\alpha$. With $T_\alpha$, $K_\alpha$ and $S_\alpha$ so defined, the Borel–Pompeiu formula, the Cauchy integral formula, the Plemelj–Sokhotski formulas, the involutiveness $S_\alpha^2 = I$, the Cauchy integral theorem and the Morera theorem hold in the same shapes as in the unshifted theory, with the parameter-dependent kernels in the statements; the domain of validity is a domain with Liapunov boundary and Hölder data rather than the quaternion subspace $\mathbb{H}_{\mathbb{B}}$.

Boundary-value criterion. The sharpest of the statements characterises the boundary values themselves: a Hölder function $f$ on the boundary $\Gamma$ of a domain is the boundary value of a solution of $D_\alpha g = 0$ in that domain if and only if

$$ P_\alpha f = f \ \text{ on } \Gamma, \qquad P_\alpha = \tfrac12 (I + S_\alpha), $$

and in that case the solution is $g = K_\alpha f$. This is the homogeneous companion of the inhomogeneous solvability criterion $Q_\alpha v = T_\alpha g$ for $D_\alpha f = g$, $f|_\Gamma = v$ quoted in Electromagnetism in Media: The Local Complex Structure at Work. Where the inhomogeneous criterion decides whether prescribed data can be matched by some solution with a source, this one decides whether prescribed data are the trace of a solution of the homogeneous equation — which is exactly what a boundary condition alone gives. The criterion is therefore what turns an operator boundary condition into a solvability condition, and the instance worked out in the corpus is the bag model (Confinement and the Loss of Partonic Information in Biquaternionic Form), whose driver, the time-harmonic massive Dirac field, is the subject of The Dirac Equation in Biquaternionic Form.

Physical reading: the mass shell as a nilpotent parameter. The parameter of a shifted equation can be pure, $\alpha_0 = 0$, and then the criterion of Biquaternion Zero Divisors applies verbatim: a pure element is a zero divisor if and only if its norm vanishes, and then its square is zero. The instance is the bag. There the parameter is $\alpha = -(i\omega e_1 + m e_2)$, whose norm is

$$ N(\alpha) = m^2 - \omega^2 , $$

so $\alpha$ is a zero divisor, indeed a nilpotent with $\alpha^2 = 0$, exactly on the mass shell $\omega^2 = m^2$. The physical on-shell condition and the algebraic degeneration of the parameter are one equation, not two: a massive field's frequency lies on the mass shell exactly when the shift that carries its mass is a nilpotent of the algebra. This is the parameter-space counterpart of the field statements of Zero Divisors as a Physical Locus in Biquaternionic Form — there the field's momentum is null on the light cone, here the operator's parameter is nilpotent on the mass shell — and it also explains why the bag's reduction lands in the third branch above, $\alpha$ a zero divisor with $\alpha_0 = 0$.

Where the Complex Analogy Fails: Zero Divisors and the Null Cone

The complex theory rests on $\mathbb{C}$ being a field. In the biquaternion algebra this fails, and every consequence below is traceable to the zero divisors.

The zero divisors are exactly the nonzero elements with $N(\tilde{Q}) = Q_0^2 + Q_1^2 + Q_2^2 + Q_3^2 = 0$. The zero divisor set $\mathcal{Z}$ is a complex cone of complex dimension $3$ (real dimension $6$); on the indefinite subspaces it cuts out the double cones $q_0^2 = (q'_1)^2 + (q'_2)^2 + (q'_3)^2$ in $\mathbb{M}_+$ and $(q'_0)^2 = q_1^2 + q_2^2 + q_3^2$ in $\mathbb{M}_-$, each three-dimensional with apex at the origin; in $\mathrm{Vect}(\mathbb{B})$ the norm is complex and vanishes on the nilpotent cone, of real dimension $4$; while on $\mathbb{C}_{\mathbb{B}}$, $\mathbb{H}_{\mathbb{B}}$ and $i\mathbb{H}_{\mathbb{B}}$ the norm is definite and there are none. On the null cone there is no inverse, so no quotient $\tilde{A}/\tilde{Q}$ is defined, and the naive difference quotient of Biquaternion Analysis requires $\tilde{H}^{-1}$, which may not exist.

The proof that $\tilde{\nabla}\tilde{G} = 0$ uses $\tilde{Q}^{\natural}\tilde{Q} = \|\tilde{Q}\|_E^2 e_0$, which requires real coefficients; on $\mathbb{M}_\pm$ this identity fails, so $\tilde{G}$ is not a fundamental solution and no Cauchy formula of the stated form holds, and whether a modified kernel exists is open (see Biquaternion Integration). By §The System of Regularity Equations the principal symbol degenerates on the null cone on $\mathbb{M}_\pm$, so the system is not elliptic and the maximum principle, the mean value property and Liouville's theorem do not generalise; a regular function there, if defined, solves a hyperbolic rather than an elliptic system. The natural singular set is the six-real-dimensional null quadric, not a point, so there is no punctured-disk model and the residue theory is correspondingly delicate.

On the indefinite subspaces the null cone is thus simultaneously the zero divisor set, the characteristic set of $\tilde{\nabla}$, and the set where the Cauchy kernel ceases to be a fundamental solution. A theorem about regular functions must therefore either restrict to a domain avoiding the cone — or, better, to a subspace such as $\mathbb{H}_{\mathbb{B}}$ — or state explicitly which weakened conclusion replaces the classical one. No Cauchy formula may be asserted beyond the quaternion subspace.

Physical reading: the light cone. On the material slice the null cone is the light cone, and the three roles it plays are three faces of one physical fact. As zero divisor set it is where a biquaternion has no inverse, which is the algebraic statement that a lightlike direction cannot be normalised; as characteristic set of $\tilde{\nabla}$ it is where the massless equation degenerates and disturbances propagate; and as the failure set of the Cauchy kernel it is where the free field's Green's function is singular. That the same cone does all three jobs is why the framework can speak of "the light cone" in a purely algebraic setting, and it is the content of Zero Divisors as a Physical Locus in Biquaternionic Form, which reads the null quadric physically, and of the norm article's identification of the null cone with the causal boundary.

The Naive Inverse Function and the Role of the Null Cone

The naive transcription of $1/(z - w)$ is $\tilde{F}(\tilde{Q}) = \tilde{Q}^{-1} = \tilde{Q}^{\natural}/N(\tilde{Q})$, defined exactly on the group of units, that is, off the null quadric with the origin removed; it does not exist on the null cone. Its singular set is six-dimensional, not isolated, so there is no Laurent expansion about an isolated pole.

On $\mathbb{H}_{\mathbb{B}}$, where $N(\tilde{Q}) = \|\tilde{Q}\|_E^2$ is real and positive definite, the inverse exists for all $\tilde{Q} \neq 0$, but it is not regular: a direct computation gives

$$ \tilde{\nabla}\left(\frac{\tilde{Q}^{\natural}}{\|\tilde{Q}\|_E^2}\right) = \frac{4e_0}{\|\tilde{Q}\|_E^2} - \frac{2e_0}{\|\tilde{Q}\|_E^2} = \frac{2e_0}{\|\tilde{Q}\|_E^2} \neq 0. $$

The genuine regular radial function is the Cauchy kernel $\tilde{G} = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$. In four variables the fundamental solution of $\tilde{\nabla}$ has homogeneity $1 - 4 = -3$, and the exponent $4$ is exactly this homogeneity; the naive inverse carries the two-dimensional exponent and is the fundamental solution of the Cauchy–Riemann operator only in the complex plane. The single-plane inverse $(z - w)^{-1}$ is regular, so this failure is a genuinely hypercomplex phenomenon — the change of dimension from two to four — not merely a consequence of noncommutativity. The null cone thus enters in two ways: on the full algebra it makes the inverse undefined on a positive-dimensional set, and on $\mathbb{H}_{\mathbb{B}}$, where the cone is absent, the naive inverse is still the wrong function, the correct kernel being selected by the homogeneity required in four variables.

Physical reading. The homogeneity $1-4=-3$ is the statement that the Green's function of the massless field in four spacetime dimensions falls off like $1/r^3$ — the exponent that controls why the Coulomb potential in four dimensions goes as $1/r$ after one integration, and why the massless two-point function has the scaling it does. That the naive inverse carries the two-dimensional exponent is the algebraic trace of the plane analogy: only the four-dimensional kernel is the physical one. The singular set being the cone rather than a point is the statement that the massless propagator is singular on the light cone, not at the origin alone.

Relation to Fueter and Quaternionic Theories

Restricting to real quaternion-valued functions on $\mathbb{H}_{\mathbb{B}}$ recovers Fueter's quaternionic analysis: $\tilde{\nabla}$ is the Cauchy–Riemann operator $D$, the equation $DF = 0$ is the Cauchy–Riemann–Fueter equation, and the Cauchy formula, mean value property, maximum principle, Liouville theorem, identity theorem, Taylor and Laurent expansions and residue theorem all hold (see Fueter Theory for Biquaternions). Since $\mathbb{H}$ is a division algebra, this case has no zero divisors and is a complete analogue of the complex theory in the sense appropriate to four real variables; the biquaternion theory is its complexification, the variable remaining quaternionic in structure while the coefficients become complex.

The general framework is Clifford analysis. Since $\mathbb{B} \cong \mathrm{Cl}_{1,3}^{+}$ (see The Clifford Structure of the Biquaternion Algebra), the regular functions of this article are the monogenic functions of the even Clifford algebra in four dimensions; relative to the quaternionic case, the additional structure is the complex coefficients, the four conjugations, the two complex structures and the zero divisors. The bridge between the single-plane and hypercomplex notions is the Fueter–Sce construction: a slice-regular function is generally not monogenic, but applying the appropriate power of the Laplacian to a slice-regular function produces a monogenic one. Due to Fueter and completed by Sce, this is the mechanism converting holomorphic data of a single complex variable into regular functions of four real variables, treated in Fueter Theory for Biquaternions.

Physical reading. The Fueter–Sce construction is the algebraic form of the passage from a one-component analytic amplitude to a four-component field: take a complex amplitude, apply the appropriate power of the wave operator, and the result is a field satisfying the Dirac-type equation. It is the framework's answer to how a scalar wave function is upgraded to a spinor field, and it is why the algebra can host both the Klein–Gordon and the Dirac descriptions of one object.

Summary

Regularity for biquaternion-valued functions must specify both the operator and the domain. A function is left-regular (left-monogenic) if $\tilde{\nabla}\tilde{F} = 0$ and right-regular if $\tilde{F}\tilde{\nabla} = 0$, where $\tilde{\nabla} = \sum_\mu e_\mu \partial_{Q_\mu}$ is the biquaternionic Cauchy–Riemann operator; the two notions are exchanged by quaternion conjugation, and a function regular for $\tilde{\nabla}^{\natural}$ is anti-regular. In this series regular unqualified means left-regular for the first-order operator $\tilde{\nabla}$, the operator whose fundamental solution is the Cauchy kernel; it is not $\Box$ and not $\tilde{\nabla}^2$.

Two notions of regularity must be kept apart. Single-plane holomorphy takes one complex variable $z = q_0 + e_1 q_1$, with $q_2, q_3$ entering only as parameters; every classical holomorphic function, extended by constancy in the orthogonal directions, is regular. Hypercomplex regularity uses the full dependence on all four variables and the full Clifford structure, is strictly larger, and contains the Cauchy kernel $\tilde{G} = \tilde{Q}^{\natural}/\|\tilde{Q}\|_E^4$, which is regular on $\mathbb{H}_{\mathbb{B}}\setminus\{0\}$ and holomorphic in no single-plane variable. Since $\Box = \tilde{\nabla}^{\natural}\tilde{\nabla}$ is scalar, every regular function is harmonic, but the converse fails, and $\tilde{\nabla}^2$ annihilates regular functions without being the natural second-order operator.

The integral theory is complete on the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, where the algebra is a division ring: the Cauchy integral formula holds there with fundamental solution $\tilde{\nabla}\tilde{G} = -2\pi^2\delta_0 e_0$, and the mean value property, maximum principle, Liouville theorem, identity theorem, Cauchy estimates and residue theory follow. Everything that fails elsewhere fails through the zero divisors: on $\mathbb{M}_\pm$ and on the full algebra the identity $\tilde{Q}^{\natural}\tilde{Q} = \|\tilde{Q}\|_E^2 e_0$ fails, the principal symbol degenerates on the null cone — the light cone — the system is hyperbolic rather than elliptic, and no Cauchy formula may be asserted. The naive inverse $\tilde{Q}^{-1} = \tilde{Q}^{\natural}/N(\tilde{Q})$ fails in two ways: on the full algebra it is undefined on the positive-dimensional null cone, and even on $\mathbb{H}_{\mathbb{B}}$, where it exists, it is not regular, the correct kernel in four variables being fixed by the homogeneity $1 - 4 = -3$ rather than the two-dimensional exponent.

The shifted operator $D_\alpha = D + M_\alpha$, with $D = i\sum_k e_k\partial_k$ and $M_\alpha$ right multiplication by an arbitrary $\alpha\in\mathbb{B}$, extends the theory to the equations that carry a mass or a wave number. For a scalar parameter it is the Helmholtz shift $D_{3\alpha} = D_3+\alpha$ with $D_3 = \sum_k e_k\partial_k$ and the kernel $\Theta_\alpha = -e^{i\alpha|x|}/(4\pi|x|)$, already used in the chiral-media article. A vector parameter is the intermediate case: being constant it commutes with $D_3$, so both orderings agree and the shift by a real vector has the scalar companion $-\Delta+\lvert\mathbf k\rvert^2$ while the shift by an imaginary vector has the Helmholtz companion $-\Delta-\lvert\mathbf k\rvert^2$; the asymmetry of the two one-sided multiplications appears instead in the plane-wave substitution, where $\mathbf k v+v\mathbf k$ collapses only for vector-valued $v$. For a general parameter the shift is genuinely non-scalar and the theory is the source's. Its integral operators $T_\alpha$, $K_\alpha$ and $S_\alpha$ obey the same four theorems as the unshifted theory — Borel–Pompeiu, Cauchy, Plemelj–Sokhotski, involutiveness — but their formulas branch according to whether $\alpha$ is a unit, a zero divisor with nonzero scalar part, or a zero divisor with vanishing scalar part, because the kernel is built by inverting the symbol. The sharpest statement is the boundary-value criterion: a Hölder function $f$ on $\Gamma$ is the trace of a solution of $D_\alpha g = 0$ in the domain if and only if $P_\alpha f = f$ on $\Gamma$, with $P_\alpha = \tfrac12(I+S_\alpha)$, and then the solution is $K_\alpha f$. Because the bag model's parameter $\alpha = -(i\omega e_1+me_2)$ is pure with $N(\alpha) = m^2-\omega^2$, the parameter of that shifted equation is a nilpotent, $\alpha^2 = 0$, exactly on the mass shell.

Restricting to real quaternion-valued functions recovers Fueter's quaternionic analysis, where the analogous theory is complete because $\mathbb{H}$ is a division algebra, and the general framework is Clifford analysis on $\mathrm{Cl}_{1,3}^{+}$; the Fueter–Sce construction converts slice-regular data of one complex variable into monogenic functions of four real variables.

Summary of Notation

Further Reading

• R. Fueter, "Die Funktionentheorie der Differentialgleichungen $\Delta u = 0$ und $\Delta\Delta u = 0$ mit vier reellen Variablen", Commentarii Mathematici Helvetici 7 (1934–35) 307–330, for quaternionic analysis.
• A. Sudbery, "Quaternionic analysis", Mathematical Proceedings of the Cambridge Philosophical Society 85 (1979) 199–225, for the limits of the complex analogy in four real dimensions.
• F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Research Notes in Mathematics 76 (Pitman, 1982), for monogenic functions and the Cauchy kernel.
• R. Delanghe, F. Sommen and V. Souček, Clifford Algebra and Spinor-Valued Functions (Kluwer, 1992), for monogenic functions and harmonicity.
• G. Gentili, C. Stoppato and D. C. Struppa, Regular Functions of a Quaternionic Variable (Springer, 2013), for slice regularity and the Fueter–Sce construction.
• V. V. Kravchenko and M. V. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, Pitman Research Notes in Mathematics 351 (Addison-Wesley Longman, 1996), for the Teodorescu transform, the Cauchy-type operator and the singular integral operator with a biquaternionic parameter, for the Borel–Pompeiu and Plemelj–Sokhotski formulas in that setting, and for the boundary-value criterion $P_\alpha f = f$.
• V. V. Kravchenko, "On a Biquaternionic Bag Model," Zeitschrift für Analysis und ihre Anwendungen 14 (1995), no. 1, 3–14, DOI 10.4171/ZAA/658, for an application of the shifted operator: the linear bag model reduced to the boundary equation $P_\alpha\tilde{p} = S^+\tilde{p}$, with the parameter $\alpha = -(i\omega e_1+me_2)$ that becomes a nilpotent on the mass shell.
• Swanhild Bernstein, "Seeing the Invisible and Maxwell's Equations", chapter, DOI 10.1007/978-3-0348-0603-9_13 (2013), for the two one-sided multiplications $M_k$ (right) and $M^k$ (left) by a vector parameter, the two factorisations with a vector and an imaginary vector parameter and their scalar companions $-\Delta+\lvert\mathbf k\rvert^2$ and $-\Delta-\lvert\mathbf k\rvert^2$, and the plane-wave substitution $u=e^{i\mathbf k\cdot x}v$, whose behaviour distinguishes vector-valued from general quaternion-valued $v$. This is a scattering paper; it is not the "Bernstein (1996)" or the "Bernstein–Gürlebeck (1999)" of the electric-media article, which are separate papers by the same author on the Riccati form of the factorisation. Its scattering theory is the subject of Scattering for the Quaternionic Dirac Operator.