Split-Biquaternion Analysis

Introduction

This article introduces the analysis of split-biquaternion-valued functions. It follows the article on the split biquaternion exponential, which defined the exponential and the polar decomposition of the algebra, and it uses the zero divisors article, which characterized the zero divisors as the union of two four-dimensional linear subspaces.

The treatment is purely mathematical. The independent variables are four real variables. They are the coordinates of $\mathbb{R}^4$, and they are independent of any physical interpretation. The split complex structure of the coefficients and the non-commutative structure of the quaternion units are the only algebraic ingredients.

Every claim is either proved or stated as a definition. Where a computation is long, all steps are shown.

The integral theory of split-biquaternion-valued functions is the subject of the companion article on split biquaternion integration. The specialization to the four fixed-point subspaces is treated in the four subspace articles of the category, from Split-Biquaternion Split-Complex Subspace to Split-Biquaternion Anti-Hermitian Subspace, each of which carries the analysis on its own subspace.

Throughout, a split biquaternion is written

$$ \tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3, \qquad Q_\mu = q_\mu + j q'_\mu, \quad q_\mu, q'_\mu \in \mathbb{R}. $$

The split complex unit is $j$, with $j^2 = +1$, and it commutes with the quaternion units. The quaternion conjugate is $\tilde{Q}^{\natural} = Q_0 e_0 - \mathbf{Q}$, where $\mathbf{Q} = Q_1 e_1 + Q_2 e_2 + Q_3 e_3$. The split complex conjugate is $\bar{\tilde{Q}} = \bar{Q_0} e_0 + \mathbf{Q}^*$, with $Q_{\bar{\mu}} = q_\mu - j q'_\mu$. The Hermitian conjugate is $\tilde{Q}^{*} = \overline{\tilde{Q}^{\natural}}$, and the anti-Hermitian conjugate is $\tilde{Q}^\flat = -\tilde{Q}^{*}$. The split-biquaternion norm is $N(\tilde{Q}) = \tilde{Q} \tilde{Q}^{\natural}$.

The idempotents of the split complex algebra are $\tilde\Pi_+ = \tfrac{1}{2}(1 + j)$ and $\tilde\Pi_- = \tfrac{1}{2}(1 - j)$. The idempotent decomposition of a split biquaternion is

$$ \tilde{Q} = \tilde{Q}_+ \tilde\Pi_+ + \tilde{Q}_- \tilde\Pi_-, $$

with $\tilde{Q}_\pm = \tilde{Q} \tilde\Pi_\pm \in \mathbb{H}$ ordinary quaternions.

The Metric Structure of $\mathbb{H}_{\mathbb{D}}$

The Euclidean Norm

The split biquaternion algebra $\mathbb{H}_{\mathbb{D}}$ is a real vector space of dimension $8$. The Euclidean norm is

$$ \|\tilde{Q}\|_E = \sqrt{\sum_{\mu=0}^{3} (q_\mu^2 + q'^2_\mu)}. $$

It is a genuine norm on the real vector space $\mathbb{H}_{\mathbb{D}} \cong \mathbb{R}^8$: positive-definite, subadditive, and homogeneous of degree one. It satisfies

$$ \|\tilde{Q} + \tilde{R}\|_E \leq \|\tilde{Q}\|_E + \|\tilde{R}\|_E, \qquad \|\lambda \tilde{Q}\|_E = |\lambda| \|\tilde{Q}\|_E, \qquad \lambda \in \mathbb{R}. $$

The Euclidean norm is not the square root of the split-biquaternion norm or of the Hermitian form. The split-biquaternion norm is split complex-valued and is not positive-definite; the Hermitian form is real but indefinite of signature $(4, 4)$. The Euclidean norm is defined separately, and it is the split-biquaternion norm that gives $\mathbb{H}_{\mathbb{D}}$ its structure as a metric space.

The Distance

The distance between two split biquaternions is

$$ d(\tilde{Q}, \tilde{R}) = \|\tilde{Q} - \tilde{R}\|_E. $$

It is a genuine metric on $\mathbb{H}_{\mathbb{D}}$: positive-definite, symmetric, and satisfying the triangle inequality. The pair $(\mathbb{H}_{\mathbb{D}}, d)$ is a metric space isometric to $\mathbb{R}^8$.

Balls and Neighborhoods

The open ball of radius $r > 0$ centered at $\tilde{Q}_0$ is

$$ B(\tilde{Q}_0, r) = \{\tilde{Q} \in \mathbb{H}_{\mathbb{D}} : \|\tilde{Q} - \tilde{Q}_0\|_E < r\}. $$

A neighborhood of $\tilde{Q}_0$ is any set containing some $B(\tilde{Q}_0, r)$. The topology generated by the open balls is the ordinary Euclidean topology of $\mathbb{R}^8$.

Limits and Continuity

Limits of Functions

Definition (limit). Let $\tilde{F} : \mathbb{H}_{\mathbb{D}} \to \mathbb{H}_{\mathbb{D}}$ be a function, $\tilde{Q}_0 \in \mathbb{H}_{\mathbb{D}}$ a point, and $\tilde{L} \in \mathbb{H}_{\mathbb{D}}$ a split biquaternion. We say that $\tilde{F}(\tilde{Q})$ tends to $\tilde{L}$ as $\tilde{Q}$ tends to $\tilde{Q}_0$, and write

$$ \lim_{\tilde{Q} \to \tilde{Q}_0} \tilde{F}(\tilde{Q}) = \tilde{L}, $$

if for every $\varepsilon > 0$ there exists $\delta > 0$ such that

$$ 0 < \|\tilde{Q} - \tilde{Q}_0\|_E < \delta \implies \|\tilde{F}(\tilde{Q}) - \tilde{L}\|_E < \varepsilon. $$

The punctured condition $0 < \|\tilde{Q} - \tilde{Q}_0\|_E$ ensures that the value of $\tilde{F}$ at $\tilde{Q}_0$ itself does not affect the limit.

Theorem (uniqueness). If the limit exists, it is unique.

Proof. Suppose $\tilde{F}(\tilde{Q}) \to \tilde{L}$ and $\tilde{F}(\tilde{Q}) \to \tilde{L}'$ as $\tilde{Q} \to \tilde{Q}_0$, with $\tilde{L} \neq \tilde{L}'$. Let $\varepsilon = \|\tilde{L} - \tilde{L}'\|_E/2 > 0$. By the definition, there exist $\delta_1, \delta_2 > 0$ such that $\|\tilde{F}(\tilde{Q}) - \tilde{L}\|_E < \varepsilon$ when $0 < \|\tilde{Q} - \tilde{Q}_0\|_E < \delta_1$, and $\|\tilde{F}(\tilde{Q}) - \tilde{L}'\|_E < \varepsilon$ when $0 < \|\tilde{Q} - \tilde{Q}_0\|_E < \delta_2$. Choose $\tilde{Q}$ with $0 < \|\tilde{Q} - \tilde{Q}_0\|_E < \min(\delta_1, \delta_2)$. Then

$$ \|\tilde{L} - \tilde{L}'\|_E \leq \|\tilde{L} - \tilde{F}(\tilde{Q})\|_E + \|\tilde{F}(\tilde{Q}) - \tilde{L}'\|_E < 2\varepsilon = \|\tilde{L} - \tilde{L}'\|_E, $$

a contradiction.

Theorem (sequential criterion). $\lim_{\tilde{Q} \to \tilde{Q}_0} \tilde{F}(\tilde{Q}) = \tilde{L}$ if and only if for every sequence $(\tilde{Q}_n)$ in $\mathbb{H}_{\mathbb{D}} \setminus \{\tilde{Q}_0\}$ with $\tilde{Q}_n \to \tilde{Q}_0$, we have $\tilde{F}(\tilde{Q}_n) \to \tilde{L}$.

Proof. The forward direction is immediate from the definition. For the converse, suppose the limit is not $\tilde{L}$. Then there exists $\varepsilon > 0$ such that for every $\delta > 0$ there exists $\tilde{Q}$ with $0 < \|\tilde{Q} - \tilde{Q}_0\|_E < \delta$ and $\|\tilde{F}(\tilde{Q}) - \tilde{L}\|_E \geq \varepsilon$. Taking $\delta = 1/n$ gives a sequence $\tilde{Q}_n \to \tilde{Q}_0$ with $\tilde{F}(\tilde{Q}_n) \not\to \tilde{L}$, contradicting the hypothesis.

Theorem (algebra of limits). If $\tilde{F}(\tilde{Q}) \to \tilde{L}$ and $\tilde{G}(\tilde{Q}) \to \tilde{M}$ as $\tilde{Q} \to \tilde{Q}_0$, then

$$ \tilde{F}(\tilde{Q}) + \tilde{G}(\tilde{Q}) \to \tilde{L} + \tilde{M}, \qquad \tilde{F}(\tilde{Q}) \circ \tilde{G}(\tilde{Q}) \to \tilde{L} \circ \tilde{M}. $$

Proof. For the sum, use the triangle inequality:

$$ \|(\tilde{F} + \tilde{G}) - (\tilde{L} + \tilde{M})\|_E \leq \|\tilde{F} - \tilde{L}\|_E + \|\tilde{G} - \tilde{M}\|_E. $$

For the product, use the identity

$$ \tilde{F} \tilde{G} - \tilde{L} \tilde{M} = (\tilde{F} - \tilde{L}) \tilde{G} + \tilde{L} (\tilde{G} - \tilde{M}), $$

and the fact that $\tilde{G}$ is bounded near $\tilde{Q}_0$ (because it has a limit).

Continuity

Definition (continuity). A function $\tilde{F} : \mathbb{H}_{\mathbb{D}} \to \mathbb{H}_{\mathbb{D}}$ is continuous at a point $\tilde{Q}_0 \in \mathbb{H}_{\mathbb{D}}$ if the limit exists and equals the value of the function at that point:

$$ \lim_{\tilde{Q} \to \tilde{Q}_0} \tilde{F}(\tilde{Q}) = \tilde{F}(\tilde{Q}_0). $$

Equivalently, for every $\varepsilon > 0$ there exists $\delta > 0$ such that

$$ \|\tilde{Q} - \tilde{Q}_0\|_E < \delta \implies \|\tilde{F}(\tilde{Q}) - \tilde{F}(\tilde{Q}_0)\|_E < \varepsilon. $$

The function is continuous on an open set if it is continuous at every point of the set.

Theorem (basic properties). If $\tilde{F}$ and $\tilde{G}$ are continuous at $\tilde{Q}_0$, then so are $\tilde{F} + \tilde{G}$ and $\tilde{F} \circ \tilde{G}$. If $\tilde{F}$ is continuous at $\tilde{Q}_0$ and $\tilde{G}$ is continuous at $\tilde{F}(\tilde{Q}_0)$, then $\tilde{G} \circ \tilde{F}$ is continuous at $\tilde{Q}_0$.

Proof. The first two claims follow from the algebra of limits. The third follows from the definition: given $\varepsilon > 0$, choose $\eta > 0$ with $\|\tilde{G}(\tilde{R}) - \tilde{G}(\tilde{F}(\tilde{Q}_0))\|_E < \varepsilon$ when $\|\tilde{R} - \tilde{F}(\tilde{Q}_0)\|_E < \eta$, and then choose $\delta > 0$ with $\|\tilde{F}(\tilde{Q}) - \tilde{F}(\tilde{Q}_0)\|_E < \eta$ when $\|\tilde{Q} - \tilde{Q}_0\|_E < \delta$.

Theorem (component-wise continuity). A function $\tilde{F} = \sum_\mu F_\mu e_\mu$ is continuous at $\tilde{Q}_0$ if and only if each split complex-valued coefficient $F_\mu$ is continuous at $\tilde{Q}_0$, which in turn holds if and only if each of the eight real-valued functions $\Re(F_\mu)$, $\Im(F_\mu)$ (with respect to $j$) is continuous at $\tilde{Q}_0$.

Proof. The Euclidean norm is equivalent to the maximum of the moduli of the coefficients, and the modulus of a split complex number is equivalent to the maximum of the absolute values of its real and split parts. So convergence in $\mathbb{H}_{\mathbb{D}}$ is equivalent to convergence of each of the eight real components.

Theorem (uniform continuity). A continuous function on a compact subset of $\mathbb{H}_{\mathbb{D}}$ is uniformly continuous.

Proof. This is the standard argument: if $\tilde{F}$ is not uniformly continuous, there exist $\varepsilon > 0$ and sequences $(\tilde{Q}_n)$, $(\tilde{R}_n)$ in the compact set with $\|\tilde{Q}_n - \tilde{R}_n\|_E \to 0$ but $\|\tilde{F}(\tilde{Q}_n) - \tilde{F}(\tilde{R}_n)\|_E \geq \varepsilon$. By compactness, pass to a subsequence with $\tilde{Q}_n \to \tilde{Q}$. Then $\tilde{R}_n \to \tilde{Q}$, and by continuity $\tilde{F}(\tilde{Q}_n) \to \tilde{F}(\tilde{Q})$ and $\tilde{F}(\tilde{R}_n) \to \tilde{F}(\tilde{Q})$, contradicting the inequality.

The Problem of Differentiability

The Naive Definition

The derivative of a function $\tilde{F} : \mathbb{H}_{\mathbb{D}} \to \mathbb{H}_{\mathbb{D}}$ at a point $\tilde{Q}_0$ would be defined, in analogy with ordinary calculus, as the limit

$$ \tilde{F}'(\tilde{Q}_0) = \lim_{\tilde{H} \to 0} \frac{\tilde{F}(\tilde{Q}_0 + \tilde{H}) - \tilde{F}(\tilde{Q}_0)}{\tilde{H}}, $$

where $\tilde{H}$ is a split biquaternion and the quotient is a split biquaternion quotient. This definition runs into two difficulties, both specific to the non-commutative and non-division structure of $\mathbb{H}_{\mathbb{D}}$.

The First Obstruction: Left and Right Division

In a non-commutative algebra, the quotient $\tilde{A} / \tilde{H}$ is ambiguous. There are two natural definitions:

Left division: $\tilde{A} / \tilde{H}$ means $\tilde{H}^{-1} \tilde{A}$, multiplying by $\tilde{H}^{-1}$ on the left.
Right division: $\tilde{A} / \tilde{H}$ means $\tilde{A} \tilde{H}^{-1}$, multiplying by $\tilde{H}^{-1}$ on the right.

These two conventions give different results in general. So the derivative would depend on the convention, and there would be two distinct notions: the left derivative and the right derivative. Neither is privileged over the other.

The Second Obstruction: Zero Divisors

The quotient $\tilde{A} / \tilde{H}$ requires $\tilde{H}^{-1}$ to exist. In a division algebra, every nonzero element has an inverse, so this is automatic. In $\mathbb{H}_{\mathbb{D}}$, however, the zero divisor set is the union of two four-dimensional linear subspaces $Z_+$ and $Z_-$, and for $\tilde{H}$ in this set (with $\tilde{H} \neq 0$), the inverse does not exist.

So the limit defining the derivative cannot be evaluated along directions in which $\tilde{H}$ is a zero divisor. The zero divisor set in the split biquaternion case is a union of two linear subspaces, which is a different geometry from the complex cone of the biquaternion case. In particular, the zero divisor set has real codimension 4, while the biquaternion zero divisor set, being the zero set of the complex norm, has real codimension 2 (complex dimension 3, real dimension 6). Neither set has interior points, so a generic direction in the ambient algebra is a zero divisor in neither case; what distinguishes the split biquaternion case is that its zero divisor set is a finite union of linear subspaces rather than a cone cut out by a single complex equation.

The Standard Approach

Together, these two difficulties make a direct definition of differentiability with respect to the split biquaternion variable problematic. The standard approach in quaternionic analysis (Fueter and his descendants) avoids the problem by working with four real variables rather than with a quaternion variable. The function is defined on $\mathbb{R}^4$, and the quaternionic structure is used to define a special class of "regular" functions, the quaternionic analogue of holomorphic functions.

We adopt this approach for the split biquaternion algebra. The function is defined on a four-dimensional real subspace of $\mathbb{H}_{\mathbb{D}}$, the coordinates are four real variables, and the split biquaternion structure is used to define the differential operators.

The General Four-Dimensional Subspace

Definition

Let $V$ be a four-dimensional real subspace of $\mathbb{H}_{\mathbb{D}}$. A general element of $V$ is written

$$ \tilde{Q} = Q_0 f_0 + Q_1 f_1 + Q_2 f_2 + Q_3 f_3, $$

where $\{f_0, f_1, f_2, f_3\}$ is a basis of $V$ and $Q_0, Q_1, Q_2, Q_3 \in \mathbb{R}$ are the coordinates. The four real numbers $Q_\mu$ are the independent variables.

In the applications below, the basis $\{f_\mu\}$ is one of: - $\{e_0, e_1, e_2, e_3\}$ (the quaternion subspace $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$). - $\{j e_0, e_1, e_2, e_3\}$ (the anti-Hermitian subspace $\mathbb{M}_-$). - $\{e_0, j e_1, j e_2, j e_3\}$ (the Hermitian subspace $\mathbb{M}_+$). - $\{e_0, j e_0, e_1, e_2\}$ or similar (subspaces involving the split complex subspace).

The general theory is independent of the choice of basis; only the interpretation of the coordinates and the algebraic properties of the subspace depend on it.

Functions on $V$

A split-biquaternion-valued function on $V$ is a map

$$ \tilde{F} : V \to \mathbb{H}_{\mathbb{D}}, \qquad \tilde{Q} \mapsto \tilde{F}(\tilde{Q}). $$

Writing $\tilde{Q} = \sum_\mu Q_\mu f_\mu$, the function $\tilde{F}$ is determined by four split-complex-valued functions $F_\mu$ of the four real variables $Q_0, Q_1, Q_2, Q_3$:

$$ \tilde{F}(\tilde{Q}) = F_0(Q_0, Q_1, Q_2, Q_3) e_0 + F_1(Q_0, Q_1, Q_2, Q_3) e_1 + F_2(Q_0, Q_1, Q_2, Q_3) e_2 + F_3(Q_0, Q_1, Q_2, Q_3) e_3. $$

The restriction to $V$ reduces the number of independent real variables from eight to four.

Partial Derivatives

For each $\mu = 0, 1, 2, 3$, we define the partial derivative of $\tilde{F}$ with respect to $Q_\mu$ by

$$ \frac{\partial \tilde{F}}{\partial Q_\mu} = \sum_{\nu=0}^{3} \frac{\partial F_\nu}{\partial Q_\mu} e_\nu. $$

The partial derivative acts component-wise on the coefficients. Since the quaternion units $e_\nu$ are constants, the rules of ordinary differential calculus apply to each coefficient separately.

We write $\partial_\mu$ for $\partial/\partial Q_\mu$. The partial derivatives commute: $\partial_\mu \partial_\nu \tilde{F} = \partial_\nu \partial_\mu \tilde{F}$ for all $\mu, \nu$.

The Differential Operators

The Split-Biquaternion Gradient

The split-biquaternion gradient is the operator

$$ \tilde{\nabla} = e_0 \partial_0 + e_1 \partial_1 + e_2 \partial_2 + e_3 \partial_3 = \sum_{\mu=0}^{3} e_\mu \partial_\mu. $$

It acts on a split-biquaternion-valued function $\tilde{F}$ on $V$ by the rule

$$ \tilde{\nabla} \tilde{F} = \left(\sum_{\mu=0}^{3} e_\mu \partial_\mu\right)\left(\sum_{\nu=0}^{3} F_\nu e_\nu\right) = \sum_{\mu=0}^{3} \sum_{\nu=0}^{3} (\partial_\mu F_\nu) \, e_\mu e_\nu. $$

The tilde on $\tilde{\nabla}$ signals that it is a split-biquaternion-valued operator: its coefficients are $e_\mu$, which are quaternion units, and its "scalar" parts are the partial derivatives $\partial_\mu$.

Writing $\tilde{F} = F_0 + \mathbf{F}$ with $\mathbf{F} = F_1 e_1 + F_2 e_2 + F_3 e_3$, we can compute $\tilde{\nabla}\tilde{F}$ in scalar-vector form. Separating the cases in the double sum, we obtain

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

where

$$ \mathrm{div}\,\mathbf{F} = \sum_{k=1}^{3} \partial_k F_k, \qquad \mathrm{grad}\,F_0 = \sum_{k=1}^{3} (\partial_k F_0) e_k, \qquad \mathrm{rot}\,\mathbf{F} = \sum_{j,k,l=1}^{3} \epsilon_{jkl} (\partial_j F_k) e_l. $$

The formula is identical to the biquaternion case. The reason is that the operator acts on functions of four real variables, and the multiplication of the coefficients is the same as in the biquaternion case up to the sign of the extra unit.

The Quaternion Conjugate of the Gradient

The quaternion conjugate of $\tilde{\nabla}$ is obtained by negating the vector part:

$$ \tilde{\nabla}^{\natural} = e_0 \partial_0 - e_1 \partial_1 - e_2 \partial_2 - e_3 \partial_3. $$

The product $\tilde{\nabla}^{\natural}\tilde{F}$ is computed in the same way:

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

The d'Alembertian

The d'Alembertian is the second-order operator obtained by composing the gradient with its quaternion conjugate:

$$ \Box = \tilde{\nabla} \tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural} \tilde{\nabla}. $$

We compute both products and show that they are equal.

Computation of $\tilde{\nabla}\tilde{\nabla}^{\natural}$. We compute $\tilde{\nabla}\tilde{\nabla}^{\natural}$ as a product of operators, using the fact that the $e_\mu$ are constants:

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = \left(\sum_{\mu=0}^{3} e_\mu \partial_\mu\right)\left(\sum_{\nu=0}^{3} e_\nu^{\natural} \partial_\nu\right), $$

where $e_0^{\natural} = e_0$ and $e_k^{\natural} = -e_k$ for $k = 1, 2, 3$. Expanding and separating the cases, the cross terms cancel, and we obtain

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = e_0 (\partial_0^2 + \Delta), $$

where $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ is the ordinary three-dimensional Laplacian.

Computation of $\tilde{\nabla}^{\natural}\tilde{\nabla}$. By the same computation,

$$ \tilde{\nabla}^{\natural}\tilde{\nabla} = e_0 (\partial_0^2 + \Delta). $$

Equality. We have shown

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \left(\partial_0^2 + \Delta\right) e_0. $$

The operator $\partial_0^2 + \Delta$ is the four-dimensional Laplacian, and we write

$$ \Box = \partial_0^2 + \Delta. $$

The operator $\Box$ is a scalar operator: it multiplies a split biquaternion by the identity $e_0$. It acts component-wise on the coefficients of $\tilde{F}$.

The Product $\tilde{\nabla}^2$

We now compute the product $\tilde{\nabla}\tilde{\nabla}$, which is different from the d'Alembertian because it uses the gradient twice, without the quaternion conjugate.

$$ \tilde{\nabla}\tilde{\nabla} = \sum_{\mu=0}^{3}\sum_{\nu=0}^{3} e_\mu e_\nu \partial_\mu \partial_\nu. $$

Separating the cases as before, we obtain

$$ \tilde{\nabla}\tilde{\nabla} = e_0 \left(\partial_0^2 - \Delta\right) + 2\sum_{k=1}^{3} e_k \partial_0 \partial_k. $$

In scalar-vector form,

$$ \tilde{\nabla}\tilde{\nabla} = \left(\partial_0^2 - \Delta\right) + 2\sum_{k=1}^{3} e_k \partial_0 \partial_k. $$

The scalar part is $\partial_0^2 - \Delta$, which is the wave operator, and the vector part is $2\sum_k e_k \partial_0 \partial_k$.

The two products differ:

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \left(\partial_0^2 + \Delta\right) e_0, $$

$$ \tilde{\nabla}\tilde{\nabla} = \left(\partial_0^2 - \Delta\right) + 2\sum_{k=1}^{3} e_k \partial_0 \partial_k. $$

The d'Alembertian is a scalar operator with $\partial_0^2 + \Delta$. The product $\tilde{\nabla}\tilde{\nabla}$ is a split-biquaternion-valued operator with scalar part $\partial_0^2 - \Delta$ and vector part $2\sum_k e_k \partial_0 \partial_k$.

This is a key structural fact: the gradient has two natural second-order products, the d'Alembertian (via the conjugate) and the square (via the product with itself). The formulas are identical to the biquaternion case.

The Convective Derivative

Let

$$ \tilde{U} = \sum_{\mu=0}^{3} u_\mu e_\mu, \qquad u_\mu \in \mathbb{R}, $$

be a split biquaternion with real coefficients. The split-biquaternion convective derivative is the operator

$$ \tilde{D} = \tilde{U}^{\natural} \tilde{\nabla}, $$

where $\tilde{U}^{\natural} = u_0 e_0 - u_1 e_1 - u_2 e_2 - u_3 e_3$ is the quaternion conjugate.

We compute it explicitly:

$$ \tilde{D} = \left(\sum_{\mu=0}^{3} \bar{u}_\mu e_\mu\right)\left(\sum_{\nu=0}^{3} e_\nu \partial_\nu\right) = \sum_{\mu=0}^{3}\sum_{\nu=0}^{3} \bar{u}_\mu e_\mu e_\nu \partial_\nu, $$

where $\bar{u}_0 = u_0$ and $\bar{u}_k = -u_k$.

Separating the cases and collecting terms, we obtain

$$ \tilde{D} = \left(u_0 \partial_0 + \mathbf{u}\cdot\mathrm{grad}\right) + \sum_{k=1}^{3} e_k (u_0 \partial_k - u_k \partial_0) - \mathrm{rot}(\mathbf{u}), $$

where $\mathbf{u}\cdot\mathrm{grad} = \sum_k u_k \partial_k$ is a scalar and $\mathrm{rot}(\mathbf{u}) = \sum_{j,k,l} \epsilon_{jkl} u_j \partial_k e_l$ is a split biquaternion.

The scalar part of $\tilde{D}$ is

$$ \mathrm{Sc}(\tilde{D}) = u_0 \partial_0 + \mathbf{u}\cdot\mathrm{grad}. $$

This is the convective derivative in four dimensions. When $u_0 = 1$, it reduces to the ordinary convective derivative $\partial_0 + \mathbf{u}\cdot\mathrm{grad}$ for a field advected by a flow with velocity $\mathbf{u}$.

The vector part of $\tilde{D}$ is

$$ \mathrm{Vec}(\tilde{D}) = \sum_{k=1}^{3} e_k (u_0 \partial_k - u_k \partial_0) - \mathrm{rot}(\mathbf{u}). $$

This contains terms that couple the four components of $\tilde{U}$ to the partial derivatives.

The Operators in the Idempotent Basis

Because the split biquaternion algebra is the direct sum of two copies of the quaternion algebra, the differential operators can be expressed in the idempotent basis. For a function $\tilde{F} = \tilde{F}_+ \tilde\Pi_+ + \tilde{F}_- \tilde\Pi_-$ with $\tilde{F}_\pm \in \mathbb{H}$, the gradient acts componentwise:

$$ \tilde{\nabla} \tilde{F} = (\tilde{\nabla} \tilde{F}_+) \tilde\Pi_+ + (\tilde{\nabla} \tilde{F}_-) \tilde\Pi_-, $$

where $\tilde{\nabla}$ on the right is the quaternion gradient acting on each component. The d'Alembertian and the convective derivative act in the same way.

So the split biquaternion analysis is the quaternion analysis applied to each of the two idempotent components separately. This is the cleanest way to understand the structure of the theory, and it is a consequence of the semisimple structure of the algebra.

The Relation to Fueter and Clifford Analysis

The approach in this article is the split biquaternion analogue of the quaternionic analysis of Fueter. The key features are:

The function is defined on a four-dimensional real subspace $V \subset \mathbb{H}_{\mathbb{D}}$, with four real coordinates.
The differential operators are split-biquaternion-valued: the gradient $\tilde{\nabla}$ and its quaternion conjugate $\tilde{\nabla}^{\natural}$.
The "regular" functions are those satisfying $\tilde{\nabla}\tilde{F} = 0$, the split biquaternion analogue of the Cauchy–Riemann equations.

This approach is closely related to Clifford analysis, which generalizes the theory to $\mathbb{R}^n$ with Clifford algebra coefficients. The split biquaternion algebra is isomorphic to the even subalgebra of a definite (Euclidean) Clifford algebra, $\mathbb{H}_{\mathbb{D}} \cong \mathrm{Cl}_{0,4}^+ \cong \mathrm{Cl}_{4,0}^+ \cong \mathrm{Cl}_{0,3}$, and the split biquaternion analysis developed here is the four-dimensional case of the general Clifford analysis.

The main difference from the quaternion Fueter theory is that the functions are split-biquaternion-valued rather than quaternion-valued, and the coefficients are split complex. This allows for the idempotent decomposition, which reduces the analysis to two copies of the quaternion analysis.

The Role of the Zero Divisors

The zero divisor set of $\mathbb{H}_{\mathbb{D}}$ is the union of two four-dimensional linear subspaces $Z_+$ and $Z_-$. In the analysis, the zero divisors play a role in the following ways.

In the definition of differentiability. The naive derivative is not defined along directions in $Z_+$ or $Z_-$, because the inverse of the increment does not exist.

In the restriction to four-dimensional subspaces. On a four-dimensional subspace $V$, the intersection with the zero divisor set is: - $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}} \cap (Z_+ \cup Z_-) = \{0\}$ (the quaternion subspace contains no zero divisors). - $\mathbb{M}_+ \cap (Z_+ \cup Z_-) = \{0\}$ (a nonzero element $q_0 e_0 + \sum_k q'_k (j e_k)$ has idempotent components $q_0 \pm \sum_k q'_k e_k$, both nonzero). - $\mathbb{M}_- \cap (Z_+ \cup Z_-) = \{0\}$ (a nonzero element $j q'_0 e_0 + \sum_k q_k e_k$ has idempotent components $\pm q'_0 + \sum_k q_k e_k$, both nonzero). - $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cap (Z_+ \cup Z_-)$ is a one-dimensional subset (the zero divisors of the split complex algebra).

In the integration theory. The fundamental solution of the gradient has singularities on the zero divisor set, and the integral formulas require careful treatment on or near the zero divisors.

The Root Set as a Manifold

The root set of $-1$ in $\mathbb{H}_{\mathbb{D}}$ is the product of two copies of the unit sphere $\mathbb{S}^2$:

$$ \{\xi \in \mathbb{H}_{\mathbb{D}} : \xi^2 = -1\} \cong \mathbb{S}^2 \times \mathbb{S}^2. $$

The isomorphism is given by $\xi \mapsto (\mu_+, \mu_-)$, where $\xi = \mu_+ \tilde\Pi_+ + \mu_- \tilde\Pi_-$ and $\mu_\pm$ are unit pure real quaternions.

The parametrisation $\xi \mapsto (\mu_+, \mu_-)$ is a regular parametrisation: its differential has rank $4$ everywhere, because the two-sphere is a manifold and the parametrisation is a product of the two identity maps. So the root set of $-1$ in $\mathbb{H}_{\mathbb{D}}$ is a compact four-dimensional embedded submanifold, isomorphic to $\mathbb{S}^2 \times \mathbb{S}^2$. The proof of regularity uses the derivative, and for that reason the manifold statement is placed here in the analysis rather than in the algebraic article Split-Biquaternion Roots of Minus One, where the root set is determined only as a set.

Comparison with the Biquaternion Root Set

The biquaternion root set has dimension 4 (the non-trivial family is four-dimensional, and the degenerate families are lower-dimensional). So both root sets have dimension 4, but their structures are different: the biquaternion root set is a four-dimensional stratified space with a more complicated topology, while the split biquaternion root set is the product of two two-spheres.

Open Questions

The following questions are not answered in this article and are left for later work:

Invariance. Under what transformations of the four-dimensional subspace $V$ do the operators $\tilde{\nabla}$, $\Box$, and $\tilde{\nabla}^2$ transform in a simple way?
Solutions of $\tilde{\nabla}\tilde{F} = 0$. What is the structure of split-biquaternion-valued functions on $V$ that satisfy $\tilde{\nabla}\tilde{F} = 0$? This is the split biquaternion analogue of the Cauchy–Riemann equations.
Solutions of $\Box\tilde{F} = 0$. What is the structure of solutions of the four-dimensional Laplace equation?
The convective operator. What is the complete structure of $\tilde{D}\tilde{\nabla}$, and what does it represent mathematically?
The general case. Can the analysis be extended from a four-dimensional subspace $V$ to the full split biquaternion algebra $\mathbb{H}_{\mathbb{D}}$?
Differentiability on the split complex subspace. How does the analysis extend to the two-dimensional split complex subspace $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$, and what replaces the quaternion gradient?
The relation to the polar decomposition. How does the polar decomposition of the split biquaternion algebra interact with the analysis on the four-dimensional subspaces?

Summary

The split biquaternion algebra $\mathbb{H}_{\mathbb{D}}$ is a real vector space of dimension 8, with a Euclidean norm and the associated metric topology. Limits and continuity are defined in the usual way, and the basic properties hold.

A direct definition of differentiability with respect to the split biquaternion variable is problematic, because of the ambiguity of left and right division and the presence of zero divisors. The standard approach is to restrict to a four-dimensional real subspace $V \subset \mathbb{H}_{\mathbb{D}}$, with four real coordinates, and to define the differential operators on this subspace.

The split-biquaternion gradient $\tilde{\nabla} = \sum_\mu e_\mu \partial_\mu$ is a split-biquaternion-valued first-order operator. Its quaternion conjugate $\tilde{\nabla}^{\natural}$ is obtained by negating the vector part. The d'Alembertian $\Box = \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = (\partial_0^2 + \Delta)e_0$ is a scalar second-order operator. The square $\tilde{\nabla}^2 = (\partial_0^2 - \Delta) + 2\sum_k e_k \partial_0 \partial_k$ is a split-biquaternion-valued second-order operator. The convective derivative $\tilde{D} = \tilde{U}^{\natural}\tilde{\nabla}$ is a split-biquaternion-valued first-order operator whose scalar part is the four-dimensional convective derivative.

The formulas for the operators are identical to the biquaternion case, because the operators act on functions of four real variables and the multiplication of the coefficients is the same up to the sign of the extra unit. What is different is the algebraic structure of the domains on which the operators are defined:

The zero divisor set is the union of two four-dimensional linear subspaces $Z_+$ and $Z_-$, in contrast to the complex cone of complex dimension 3 (real dimension 6) of the biquaternion case.
The Hermitian form is indefinite of signature $(4, 4)$, not positive-definite.
The invertibility criterion is linear in the idempotent basis, not quadratic.
The idempotent decomposition reduces the analysis to two copies of the quaternion analysis.

The idempotent decomposition is the most important structural tool in the split biquaternion analysis. In the idempotent basis, the differential operators act componentwise, and the split biquaternion analysis is the quaternion analysis applied to each of the two components separately. This is a consequence of the semisimple structure of the algebra.

The specialization to specific four-dimensional subspaces, including the quaternion subspace $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, the Hermitian subspace $\mathbb{M}_+$, and the anti-Hermitian subspace $\mathbb{M}_-$, is not covered here. The integral theory, including the Cauchy integral formula, is the subject of the companion article on split biquaternion integration.

Summary of Notation

Symbol	Meaning
$\mathbb{D}$	Split complex algebra, unit $j$, $j^2 = +1$
$\mathbb{H}$	Quaternion algebra
$\mathbb{H}_{\mathbb{D}}$	Split biquaternion algebra, $\mathbb{D} \otimes_{\mathbb{R}} \mathbb{H}$
$e_0 = 1, e_1, e_2, e_3$	Quaternion basis
$j$	Split complex unit, central, $j^2 = +1$
$\tilde\Pi_+ = \tfrac{1}{2}(1 + j)$, $\tilde\Pi_- = \tfrac{1}{2}(1 - j)$	Idempotents of $\mathbb{D}$
$\tilde{Q} = \sum_{\mu=0}^{3} Q_\mu e_\mu$	General split biquaternion
$Q_\mu = q_\mu + j q'_\mu$	Split complex coefficient
$\mathbf{Q} = Q_1 e_1 + Q_2 e_2 + Q_3 e_3$	Split vector part
$\tilde{Q}^{\natural}, \bar{\tilde{Q}}, \tilde{Q}^{*}, \tilde{Q}^\flat$	The four conjugations
$N(\tilde{Q}) = \tilde{Q} \tilde{Q}^{\natural}$	Split-Biquaternion norm
$\tilde{Q}_\pm = \tilde{Q} \tilde\Pi_\pm$	Idempotent components
$\\|\tilde{Q}\\|_E$	Euclidean norm on $\mathbb{H}_{\mathbb{D}} \cong \mathbb{R}^8$
$d(\tilde P, \tilde{Q}) = \\|\tilde P - \tilde{Q}\\|_E$	Distance
$V$	A four-dimensional real subspace, coordinates $Q_0, Q_1, Q_2, Q_3$
$\tilde{F}$	Split-biquaternion-valued function on $V$
$\tilde{\nabla}$	Split-biquaternion gradient
$\tilde{\nabla}^{\natural}$	Quaternion conjugate of the gradient
$\Box$	d'Alembertian (wave operator)
$\tilde{\nabla}^2$	Square of the split-biquaternion gradient
$\tilde{D}$	Split-biquaternion convective derivative
$\{\xi \in \mathbb{H}_{\mathbb{D}} : \xi^2 = -1\} \cong \mathbb{S}^2\times\mathbb{S}^2$	Root set of $-1$, a compact four-dimensional manifold