Biquaternion Analysis

Introduction

This article introduces the analysis of biquaternion-valued functions. It follows Biquaternion Algebra, which defined the algebra $\mathbb{B}$, its conjugations and its six distinguished subspaces, and Biquaternion Norm and Invertibility, which defined the norm, the Hermitian form and the Euclidean norm. The aim is to define limits, continuity and differentiation for biquaternion-valued functions, and to build the differential operators the rest of the framework runs on.

The treatment is mathematical. The independent variables are four real parameters — the coordinates of a four-dimensional real subspace of $\mathbb{B}$ — and the complex structure of the coefficients and the noncommutativity of the quaternion units are the only algebraic ingredients. Every claim is either proved or stated as a definition, and long computations are shown in full.

Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with basis $e_0=1,e_1,e_2,e_3$, $e_1e_2=e_3$, and central scalar imaginary $i$; a general element is $\tilde{Q}=\sum_{\mu=0}^{3}Q_\mu e_\mu$ with $Q_\mu\in\mathbb{C}$, written $\tilde{Q}=Q_0e_0+\mathbf{Q}$. The six distinguished subspaces $\mathbb{H}_{\mathbb{B}}$, $i\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_-$ and $\mathbb{M}_+$ are those of Biquaternion Algebra. The integral theory of biquaternion-valued functions is Biquaternion Integration; the class of functions the operators below single out is Biquaternion Regular Functions and Fueter Theory for Biquaternions.

The Metric Structure of $\mathbb{B}$

The Euclidean Norm

The biquaternion algebra $\mathbb{B}$ is a real vector space of dimension $8$. A biquaternion is written

$$ \tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3, \qquad Q_\mu \in \mathbb{C}. $$

Writing each complex coefficient as $Q_\mu = q_\mu + i q'_\mu$ with $q_\mu, q'_\mu \in \mathbb{R}$, the Euclidean norm is

$$ \|\tilde{Q}\|_E = \sqrt{q_0^2 + (q'_0)^2 + q_1^2 + (q'_1)^2 + q_2^2 + (q'_2)^2 + q_3^2 + (q'_3)^2} = \sqrt{\sum_{\mu=0}^{3} |Q_\mu|^2}. $$

Equivalently, it is the square root of the scalar part of the Hermitian form, $\|\tilde{Q}\|_E = \sqrt{\mathrm{Sc}(\tilde{Q}\tilde{Q}^{*})}$, defined in Biquaternion Norm and Invertibility.

The Euclidean norm is a genuine norm on the real vector space $\mathbb{B}\cong\mathbb{R}^8$: positive-definite, subadditive and homogeneous of degree one,

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

The biquaternion norm is not multiplicative with respect to the biquaternion product: $\|\tilde{Q}\tilde{R}\|_E = \|\tilde{Q}\|_E\|\tilde{R}\|_E$ fails in general. This is in contrast to the multiplicative but complex-valued norm $N(\tilde{Q}) = \tilde{Q}\tilde{Q}^{\natural} = \sum_\mu Q_\mu^2$, which satisfies $N(\tilde{Q}\tilde{R}) = N(\tilde{Q})N(\tilde{R})$ but vanishes on the zero divisors, so it cannot serve as a norm.

As a concrete example take $\tilde{Q} = \tilde{R} = e_1 + ie_2$. Then $\|\tilde{Q}\|_E = \|\tilde{R}\|_E = \sqrt{2}$, but

$$ \tilde{Q}\tilde{R} = (e_1 + ie_2)^2 = e_1^2 + e_1(ie_2) + (ie_2)e_1 + (ie_2)^2 = -1 + ie_3 - ie_3 + 1 = 0, $$

so $\|\tilde{Q}\tilde{R}\|_E = 0 \neq \|\tilde{Q}\|_E \|\tilde{R}\|_E = 2$. The Euclidean norm is the ordinary norm on the underlying real vector space and does not respect the algebra structure.

The Distance

The distance between two biquaternions is

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

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

Balls and Neighbourhoods

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

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

A neighbourhood 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 $\tilde{F} : \mathbb{B} \to \mathbb{B}$ be a function, $\tilde{Q}_0 \in \mathbb{B}$ a point and $\tilde{L} \in \mathbb{B}$ a biquaternion. Then $\tilde{F}(\tilde{Q})$ tends to $\tilde{L}$ as $\tilde{Q}$ tends to $\tilde{Q}_0$, written

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

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 $\mathbb{B} \setminus \{\tilde{Q}_0\}$ with $\tilde{Q}_n \to \tilde{Q}_0$, we have $\tilde{F}(\tilde{Q}_n) \to \tilde{L}$.

Proof. The forward direction is immediate. 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 $\tilde{F} : \mathbb{B} \to \mathbb{B}$ is continuous at a point $\tilde{Q}_0 \in \mathbb{B}$ if the limit exists and equals the value 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 $\|\tilde{R} - \tilde{F}(\tilde{Q}_0)\|_E < \eta$, then $\delta > 0$ 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 complex-valued coefficient $F_\mu$ is continuous at $\tilde{Q}_0$.

Proof. The Euclidean norm is equivalent to the maximum of the moduli of the coefficients: $\max_\mu |F_\mu| \leq \|\tilde{F}\|_E \leq 2 \max_\mu |F_\mu|$. So convergence in $\mathbb{B}$ is equivalent to convergence of each coefficient.

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

Proof. 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 $\tilde{F} : \mathbb{B} \to \mathbb{B}$ 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 biquaternion and the quotient is a biquaternion quotient. This definition runs into two difficulties, both specific to the noncommutative and non-division structure of $\mathbb{B}$.

The First Obstruction: Left and Right Division

In a noncommutative 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.

The 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 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 $\mathbb{B}$ there are nonzero elements $\tilde{H}$ with vanishing norm, $\tilde{H} \tilde{H}^{\natural} = 0$, for which no inverse exists. These are the zero divisors of the algebra, treated in Biquaternion Zero Divisors. For such $\tilde{H}$ the quotient $\tilde{A} / \tilde{H}$ is undefined, and the limit defining the derivative cannot be evaluated along directions in which $\tilde{H}$ is a zero divisor.

The Standard Approach

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

We adopt this approach. The function is defined on a four-dimensional real subspace of $\mathbb{B}$, the independent variables are four real parameters, and the biquaternion structure is used to define the differential operators.

Physical reading: why the framework differentiates on a slice. The two obstructions are not technicalities to be wished away; they are the reason the framework's calculus is built on a four-dimensional real subspace and not on the full eight-dimensional algebra. On a slice with four real coordinates the division ambiguity disappears, because the derivatives are taken with respect to real parameters, and the null directions are a measure-zero set on which the field equations are simply singular — the light cone. That a biquaternion-valued field theory is a field on $\mathbb{R}^{1,3}$ and not a holomorphic function of a single biquaternion variable is therefore a theorem about the algebra, not a modelling choice.

Functions on a Four-Dimensional Subspace

The Four Named Subspaces

In each of the four subspaces treated in this section the complex coefficients $Q_0, Q_1, Q_2, Q_3$ take specific forms.

The quaternion subspace $\mathbb{H}_{\mathbb{B}}$. An element is written

$$ \tilde{Q} = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3, \qquad q_0, q_1, q_2, q_3 \in \mathbb{R}, $$

so $Q_0 = q_0$, $Q_1 = q_1$, $Q_2 = q_2$, $Q_3 = q_3$. All four coefficients are real.

The anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$. An element is written

$$ \tilde{Q} = i q'_0\, e_0 + i q'_1\, e_1 + i q'_2\, e_2 + i q'_3\, e_3, \qquad q'_0, q'_1, q'_2, q'_3 \in \mathbb{R}, $$

so all four coefficients are purely imaginary. Every partial derivative on this subspace carries a factor of $-i$, and the d'Alembertian is the negative of the Euclidean Laplacian — the mirror of the situation on $\mathbb{H}_{\mathbb{B}}$.

The anti-Hermitian subspace $\mathbb{M}_-$. An element is written

$$ \tilde{Q} = i q'_0\, e_0 + q_1\, e_1 + q_2\, e_2 + q_3\, e_3, \qquad q'_0, q_1, q_2, q_3 \in \mathbb{R}, $$

so the first coefficient is purely imaginary and the three spatial coefficients are real.

The Hermitian subspace $\mathbb{M}_+$. An element is written

$$ \tilde{Q} = q_0\, e_0 + i q'_1\, e_1 + i q'_2\, e_2 + i q'_3\, e_3, \qquad q_0, q'_1, q'_2, q'_3 \in \mathbb{R}, $$

so the first coefficient is real and the three spatial coefficients are purely imaginary.

In each case the four real parameters ($q_0, q_1, q_2, q_3$ for $\mathbb{H}_{\mathbb{B}}$; $q'_0, q'_1, q'_2, q'_3$ for $i\mathbb{H}_{\mathbb{B}}$; $q'_0, q_1, q_2, q_3$ for $\mathbb{M}_-$; $q_0, q'_1, q'_2, q'_3$ for $\mathbb{M}_+$) are the independent variables.

Physical reading: the four slices of the dictionary. The four subspaces are the four slices on which the framework's fields live, and the choice between them is the choice of signature. On $\mathbb{M}_-$ the four real parameters are a time and a three-space, and this is the material four-position: $\tilde{Q} = ict\,e_0 + \mathbf{x}$ with the imaginary time coordinate in the scalar slot and the three real spatial coordinates in the vector slots. On $\mathbb{M}_+$ the roles are exchanged: the real scalar is the informational time and the imaginary spatial slots are the informational coordinates. The $\mathbb{H}_{\mathbb{B}}$ slice, all coefficients real, is the Euclidean four-space in which no causal structure has been put in by hand; the $i\mathbb{H}_{\mathbb{B}}$ slice is its mirror. The six subspaces of Biquaternion Algebra are what a reader must keep in view here, because a statement about "the" gradient is a statement about one slice, and the sign of the resulting operator depends on which one.

Functions on a Subspace

A biquaternion-valued function on a four-dimensional subspace is a map

$$ \tilde{F} : V \to \mathbb{B}, \qquad \tilde{Q} \mapsto \tilde{F}(\tilde{Q}), $$

where $V$ is one of the four subspaces above. Because $V$ is four-dimensional, $\tilde{F}$ is determined by four complex-valued functions of the four real parameters that parametrise $V$. For example, on $\mathbb{M}_-$,

$$ \tilde{F}(\tilde{Q}) = F_0(q'_0, q_1, q_2, q_3) e_0 + F_1(q'_0, q_1, q_2, q_3) e_1 + F_2(q'_0, q_1, q_2, q_3) e_2 + F_3(q'_0, q_1, q_2, q_3) e_3. $$

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

Partial Derivatives

The partial derivatives are taken with respect to the complex coefficients $Q_\mu$ of the standard basis. When $Q_\mu$ is real, the partial derivative is the ordinary real partial derivative with respect to $Q_\mu$. When $Q_\mu$ is purely imaginary, $Q_\mu = i q'_\mu$ with $q'_\mu \in \mathbb{R}$, and the derivative with respect to $Q_\mu$ is defined by the chain rule:

$$ \frac{\partial}{\partial Q_\mu} = \frac{\partial}{\partial (i q'_\mu)} = \frac{1}{i} \frac{\partial}{\partial q'_\mu} = -i \frac{\partial}{\partial q'_\mu}. $$

So, on each subspace, the partial derivatives take the following forms.

Subspace	$\partial/\partial Q_0$	$\partial/\partial Q_k$
$\mathbb{H}_{\mathbb{B}}$	$\partial/\partial q_0$	$\partial/\partial q_k$
$i\mathbb{H}_{\mathbb{B}}$	$-i\,\partial/\partial q'_0$	$-i\,\partial/\partial q'_k$
$\mathbb{M}_-$	$-i\,\partial/\partial q'_0$	$\partial/\partial q_k$
$\mathbb{M}_+$	$\partial/\partial q_0$	$-i\,\partial/\partial q'_k$

The partial derivatives act component-wise on the coefficients:

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

Since the quaternion units $e_\nu$ are constants, the rules of ordinary differential calculus apply to each coefficient separately, and the partial derivatives commute: $\partial_\mu \partial_\nu \tilde{F} = \partial_\nu \partial_\mu \tilde{F}$ for all $\mu, \nu$.

The Differential Operators

The Biquaternionic Gradient

The biquaternionic gradient is the operator

$$ \tilde{\nabla} = e_0 \frac{\partial}{\partial Q_0} + e_1 \frac{\partial}{\partial Q_1} + e_2 \frac{\partial}{\partial Q_2} + e_3 \frac{\partial}{\partial Q_3} = \sum_{\mu=0}^{3} e_\mu \frac{\partial}{\partial Q_\mu}. $$

This is the Cauchy–Riemann operator of the biquaternion theory, classically the Dirac operator. The tilde signals that it is biquaternion-valued: its coefficients are the quaternion units and its scalar parts are the partial derivatives.

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

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

Writing $\tilde{F} = F_0 + \mathbf{F}$ with $\mathbf{F} = F_1 e_1 + F_2 e_2 + F_3 e_3$, and separating the cases in the double sum,

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

where

$$ \mathrm{div}\,\mathbf{F} = \sum_{k=1}^{3} \frac{\partial F_k}{\partial Q_k}, \qquad \mathrm{grad}\,F_0 = \sum_{k=1}^{3} \left(\frac{\partial F_0}{\partial Q_k}\right) e_k, \qquad \mathrm{rot}\,\mathbf{F} = \sum_{j,k,l=1}^{3} \epsilon_{jkl} \left(\frac{\partial F_k}{\partial Q_j}\right) e_l. $$

The Quaternion Conjugate of the Gradient

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

$$ \tilde{\nabla}^{\natural} = e_0 \frac{\partial}{\partial Q_0} - e_1 \frac{\partial}{\partial Q_1} - e_2 \frac{\partial}{\partial Q_2} - e_3 \frac{\partial}{\partial Q_3}. $$

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

$$ \tilde{\nabla}^{\natural}\tilde{F} = \left(\frac{\partial F_0}{\partial Q_0} + \mathrm{div}\,\mathbf{F}\right) + \left(\frac{\partial \mathbf{F}}{\partial Q_0} - \mathrm{grad}\,F_0 - \mathrm{rot}\,\mathbf{F}\right). $$

Physical reading: the four-gradient and the Dirac operator. On the material slice $\mathbb{M}_-$, with $\tilde{Q} = ict\,e_0 + \mathbf{x}$, the gradient is the four-gradient of the physics convention: the scalar slot carries $\partial/\partial(ict) = -ic^{-1}\partial_t$ and the three vector slots carry the spatial derivatives. The scalar-and-vector split of $\tilde{\nabla}\tilde{F}$ above is then the split of the Dirac operator into its time and space parts, and the pairing of $\tilde{\nabla}$ with $\tilde{\nabla}^{\natural}$ is the pairing of the massless Dirac operator with its conjugate. The fact that $\tilde{\nabla}$ is biquaternion-valued — one operator, four components, and the same operator acting on every component of the field — is the algebraic content of the statement that one field equation carries the four components of a four-spinor. The massless Dirac equation is $\tilde{\nabla}\tilde{F}=0$; the functions satisfying it are Biquaternion Regular Functions, and the passage from a one-component to a four-component object is Fueter Theory for Biquaternions.

The Bigradients

The gradient has a complexified companion, and it is the operator the parallel literature uses. Multiplying $\tilde{\nabla}$ by the central imaginary $i$ on the material slice $\mathbb{M}_-$, where the scalar derivative is $\partial/\partial Q_0 = -i\,\partial_\tau$ with $\tau = ct$ and the three vector slots carry the ordinary spatial gradient $\nabla = \sum_{k=1}^{3} e_k \partial_k$, exchanges the time derivative and the space gradient:

$$ \nabla^{+} := i\,\tilde{\nabla} = \partial_\tau + i\nabla , \qquad \nabla^{-} := i\,\tilde{\nabla}^{\natural} = \partial_\tau - i\nabla . $$

These are the mutual complex gradients — the bigradients, in the short name used by L. A. Alexeyeva's differential algebra of biquaternions, where they are the standing operator of the biwave equation $\nabla^{\pm}\tilde{K} = \tilde{G}$. They are conjugates of one another, $(\nabla^{+})^{*} = \nabla^{-}$, and they compose to the wave operator,

$$ \nabla^{+}\nabla^{-} = \nabla^{-}\nabla^{+} = -\Box = \partial_\tau^{2} - \Delta , $$

the sign following from the $ict$ convention of the section above, in which $\Box = \Delta - c^{-2}\partial_t^{2}$ and the wave operator is $-\Box$. On a field $\tilde{F} = f + \mathbf{F}$ the action separates into scalar and vector parts exactly as the gradient's does,

$$ \nabla^{\pm}\tilde{F} = \bigl(\partial_\tau f \mp i\,\mathrm{div}\,\mathbf{F}\bigr) + \bigl(\partial_\tau \mathbf{F} \pm i\,\mathrm{grad}\,f \pm i\,\mathrm{rot}\,\mathbf{F}\bigr), $$

with the upper signs for $\nabla^{+}$: the complexified counterpart of the split of $\tilde{\nabla}\tilde{F}$ above. The pair is the operator of the electro-gravimagnetic programme's biwave equation, and the corpus develops it — with its non-central shifted forms $\nabla^{\pm} + f \pm F$ and the light-cone solution of the biwave equation — in The Biquaternion D'Alembertian and Its Green's Functions, and its shock fronts in Shock Electromagnetic Waves.

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 $\tilde{\nabla}\tilde{\nabla}^{\natural}$. Using that the $e_\mu$ are constants,

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

where $\bar{e}_0 = e_0$ and $\bar{e}_k = -e_k$ for $k = 1, 2, 3$. Expanding and separating the cases, the cross terms cancel, and

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = e_0 \left(\frac{\partial^2}{\partial Q_0^2} + \Delta_Q\right), $$

where $\Delta_Q = \partial^2/\partial Q_1^2 + \partial^2/\partial Q_2^2 + \partial^2/\partial Q_3^2$ is the three-dimensional Laplacian in the remaining coordinates.

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

$$ \tilde{\nabla}^{\natural}\tilde{\nabla} = e_0 \left(\frac{\partial^2}{\partial Q_0^2} + \Delta_Q\right). $$

Equality. We have shown

$$ \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \left(\frac{\partial^2}{\partial Q_0^2} + \Delta_Q\right) e_0. $$

The operator $\partial^2/\partial Q_0^2 + \Delta_Q$ is the four-dimensional Laplacian in the formal coordinates $Q_0, Q_1, Q_2, Q_3$, and we write

$$ \Box = \frac{\partial^2}{\partial Q_0^2} + \Delta_Q. $$

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

Remark on the signature. The formal expression $\partial^2/\partial Q_0^2 + \Delta_Q$ has the interpretation of a Laplacian in the coordinates $Q_\mu$, but when some of the $Q_\mu$ are purely imaginary the operator takes a Lorentzian form in the underlying real coordinates. On $\mathbb{M}_-$, where $Q_0 = iq'_0$ and $Q_k = q_k$, the derivative $\partial^2/\partial Q_0^2 = -\partial^2/\partial (q'_0)^2$, so

$$ \Box\big|_{\mathbb{M}_-} = -\frac{\partial^2}{\partial (q'_0)^2} + \frac{\partial^2}{\partial q_1^2} + \frac{\partial^2}{\partial q_2^2} + \frac{\partial^2}{\partial q_3^2}, $$

the Lorentzian wave operator on $\mathbb{R}^{1,3}$ up to an overall sign convention. Similarly, on $\mathbb{M}_+$, where $Q_0 = q_0$ and $Q_k = iq'_k$,

$$ \Box\big|_{\mathbb{M}_+} = \frac{\partial^2}{\partial q_0^2} - \frac{\partial^2}{\partial (q'_1)^2} - \frac{\partial^2}{\partial (q'_2)^2} - \frac{\partial^2}{\partial (q'_3)^2}, $$

also Lorentzian, with the opposite sign convention. On $\mathbb{H}_{\mathbb{B}}$, where all $Q_\mu$ are real, $\Box$ is the ordinary Euclidean four-dimensional Laplacian.

The d'Alembertian is the natural second-order operator in the biquaternion framework: it is symmetric under the exchange of $\tilde{\nabla}$ and $\tilde{\nabla}^{\natural}$, it is scalar-valued, and it is the operator that appears in the second-order equations, in the Cauchy integral formula and in the factorisation of the Cauchy–Riemann operator.

Physical reading: the wave operator. On the material slice the identification of the time parameter is $q'_0 = ct$, so $\partial^2/\partial(q'_0)^2 = c^{-2}\partial_t^2$ and

$$ \Box = \Delta - c^{-2}\,\partial_t^2, $$

the wave operator in the series' mostly-minus sign convention. The factorisation $\Box = \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla}$ is therefore the algebraic statement that the wave operator is the square of a first-order operator, and that first-order operator is the Dirac operator: this is the same factorisation of the Klein–Gordon operator by which a second-order field equation for a scalar amplitude is rewritten as a first-order equation for a spinor. The sign of $\Box$ on $\mathbb{M}_-$ — a plus on the spatial part and a minus on the time part — is the mostly-minus metric, and the fact that the same formal expression $\partial^2/\partial Q_0^2 + \Delta_Q$ becomes Lorentzian or Euclidean according to which slice one uses is the algebraic reason the framework can present one operator in two signatures.

The Square of the Gradient

The square of the gradient is the second-order operator

$$ \tilde{\nabla}^2 = \tilde{\nabla}\tilde{\nabla} = \sum_{\mu=0}^{3}\sum_{\nu=0}^{3} e_\mu e_\nu \frac{\partial^2}{\partial Q_\mu \partial Q_\nu}, $$

which uses the gradient twice, without the quaternion conjugate. Separating the cases,

$$ \tilde{\nabla}^2 = \left(\frac{\partial^2}{\partial Q_0^2} - \Delta_Q\right) + 2\sum_{k=1}^{3} e_k \frac{\partial^2}{\partial Q_0 \partial Q_k}. $$

The scalar part is $\partial^2/\partial Q_0^2 - \Delta_Q$ and the vector part is $2\sum_k e_k \partial^2/(\partial Q_0 \partial Q_k)$.

The square is related to the d'Alembertian by the identity

$$ \boxed{\;\tilde{\nabla}^2 = 2\,\frac{\partial}{\partial Q_0}\,\tilde{\nabla} - \Box.\;} $$

The conjugate square satisfies the analogous identity

$$ (\tilde{\nabla}^{\natural})^2 = 2\,\frac{\partial}{\partial Q_0}\,\tilde{\nabla}^{\natural} - \Box. $$

The two squares are related to each other and to $\Box$ by

$$ \tilde{\nabla}^2 + (\tilde{\nabla}^{\natural})^2 = 2\left(\frac{\partial^2}{\partial Q_0^2} - \Delta_Q\right) e_0, \qquad \tilde{\nabla}^2 - (\tilde{\nabla}^{\natural})^2 = 4\sum_{k=1}^{3} e_k \frac{\partial^2}{\partial Q_0 \partial Q_k}. $$

Verification. We compute directly:

$$ 2\,\frac{\partial}{\partial Q_0}\,\tilde{\nabla} - \Box = 2\,\frac{\partial}{\partial Q_0}\left(e_0 \frac{\partial}{\partial Q_0} + \sum_k e_k \frac{\partial}{\partial Q_k}\right) - \left(\frac{\partial^2}{\partial Q_0^2} + \Delta_Q\right) e_0 $$

$$ = 2\,\frac{\partial^2}{\partial Q_0^2}\, e_0 + 2\sum_k e_k \frac{\partial^2}{\partial Q_0 \partial Q_k} - \frac{\partial^2}{\partial Q_0^2}\, e_0 - \Delta_Q\, e_0 $$

$$ = \left(\frac{\partial^2}{\partial Q_0^2} - \Delta_Q\right) e_0 + 2\sum_k e_k \frac{\partial^2}{\partial Q_0 \partial Q_k} = \tilde{\nabla}^2. $$

So the identity holds, and the other identities follow by conjugation and by combining.

Status of the two operators. The d'Alembertian $\Box$ is the natural second-order operator: scalar-valued, symmetric under conjugation, and the operator of the standard second-order equations. The square $\tilde{\nabla}^2$ arises when the gradient is applied twice without conjugation. It is a valid second-order operator, and the identity above expresses it in terms of $\Box$ and the first derivative in the first coordinate; the same expression rewrites any equation involving $\tilde{\nabla}^2$ as one involving $\Box$ and $\tilde{\nabla}$.

The Convective Derivative

Let

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

be a biquaternion. The biquaternionic 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:

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

where $\bar{u}_0 = u_0$ and $\bar{u}_k = -u_k$. Separating the cases and collecting terms,

$$ \tilde{D} = \left(u_0 \frac{\partial}{\partial Q_0} + \mathbf{u}\cdot\mathrm{grad}\right) + \sum_{k=1}^{3} e_k \left(u_0 \frac{\partial}{\partial Q_k} - u_k \frac{\partial}{\partial Q_0}\right) - \mathrm{Curl}(\mathbf{u}), $$

where the scalar operator $\mathbf{u}\cdot\mathrm{grad} = \sum_k u_k \partial/\partial Q_k$ acts by multiplication by the components of $\mathbf{u}$ followed by the partial derivatives, and

$$ \mathrm{Curl}(\mathbf{u}) = \sum_{j,k,l=1}^{3} \epsilon_{jkl} \, u_j \, \frac{\partial}{\partial Q_k} \, e_l. $$

The operator $\mathrm{Curl}(\mathbf{u})$ is not the same as the $\mathrm{rot}$ of a biquaternion-valued function defined earlier: $\mathrm{rot}\,\mathbf{F} = \sum_{j,k,l}\epsilon_{jkl}(\partial F_k/\partial Q_j) e_l$ acts on the coefficient functions of $\mathbf{F}$, whereas $\mathrm{Curl}(\mathbf{u})$ acts on a fixed real quaternion $\mathbf{u}$ and the derivative operators fall on the function to its right. In particular $\mathrm{Curl}(\mathbf{u})$ does not vanish for constant $\mathbf{u}$, unlike the classical curl of a constant vector field, because the partial derivatives are still there to act on the function.

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

$$ \mathrm{Sc}(\tilde{D}) = u_0 \frac{\partial}{\partial Q_0} + \mathbf{u}\cdot\mathrm{grad}, $$

the convective derivative in four variables, which for $u_0 = 1$ reduces to the ordinary convective derivative $\partial/\partial Q_0 + \mathbf{u}\cdot\mathrm{grad}$ for a function advected by a flow of velocity $\mathbf{u}$.

The vector part is

$$ \mathrm{Vec}(\tilde{D}) = \sum_{k=1}^{3} e_k \left(u_0 \frac{\partial}{\partial Q_k} - u_k \frac{\partial}{\partial Q_0}\right) - \mathrm{Curl}(\mathbf{u}), $$

which couples the four components of $\tilde{U}$ to the partial derivatives.

Physical reading: advection in the four-position. The convective derivative is the material derivative of continuum mechanics written in the algebra: its scalar part is $\partial_t + \mathbf{u}\cdot\nabla$ once the time slot is taken to be $ict$, that is, the rate of change following a flow of four-velocity built from $\tilde{U}$. The extra vector part has no counterpart in the scalar theory: it is the piece that mixes the components of the field with the direction of motion, and it is the algebraic origin of the coupling between a flow and the spin of what it carries. In the framework's language an element $\tilde{U}$ with $u_0$ in the scalar slot and $\mathbf{u}$ in the vector slots is a four-velocity in $\mathbb{M}_-$ when the spatial part is real, and $\tilde{D}$ is the derivative along it.

The Relation to Fueter and Clifford Analysis

The approach in this article is the quaternionic analysis of Fueter, generalised to the biquaternion algebra. The key features are:

the function is defined on a four-dimensional real subspace of $\mathbb{B}$, with four real parameters;
the differential operators are 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 biquaternion analogue of the Cauchy–Riemann equations.

This is closely related to Clifford analysis, which generalises the theory to Clifford algebras over arbitrary base rings. The biquaternion algebra is isomorphic to the even subalgebra of the Clifford algebra $\mathrm{Cl}_{1,3}$, as The Clifford Structure of the Biquaternion Algebra records, and the analysis developed here is the four-dimensional case of the general Clifford analysis.

The main difference from classical Fueter theory is that the functions are biquaternion-valued rather than quaternion-valued, and the coefficients are complex. This allows a richer structure, including the two polar forms and the four conjugations, which do not appear in the quaternion case.

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 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 biquaternion-valued functions on the subspace that satisfy $\tilde{\nabla}\tilde{F} = 0$? This is the biquaternion analogue of the Cauchy–Riemann equations, begun in Biquaternion Regular Functions.
Solutions of $\Box\tilde{F} = 0$. What is the structure of solutions of the four-dimensional wave 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 to the full algebra $\mathbb{B}$, with a precise definition of the partial derivatives with respect to genuinely complex coefficients?
Differentiability on other subspaces. How does the analysis extend to the other fixed-point subspaces, and in particular to the relation between $\mathbb{M}_+$ and $\mathbb{M}_-$?
The square of the gradient. The identity $\tilde{\nabla}^2 = 2\partial_{Q_0}\tilde{\nabla} - \Box$ expresses the square in terms of $\Box$ and the first derivative. What is the natural setting in which the square, as opposed to $\Box$, is the operator that appears?

Summary

The algebra $\mathbb{B}$ 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: uniqueness of limits, the sequential criterion, the algebra of limits, continuity of sums, products and compositions, component-wise continuity, and uniform continuity on compact sets.

A direct definition of differentiability with respect to the biquaternion variable is problematic, because of the ambiguity of left and right division and the presence of zero divisors. The standard approach restricts to a four-dimensional real subspace of $\mathbb{B}$, with four real parameters as independent variables, and defines the differential operators on that subspace.

The four subspaces used here are the quaternion subspace $\mathbb{H}_{\mathbb{B}}$, with all complex coefficients real; the anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$, with all coefficients purely imaginary; the Hermitian subspace $\mathbb{M}_+$, with the first coefficient real and the three spatial coefficients purely imaginary; and the anti-Hermitian subspace $\mathbb{M}_-$, with the first coefficient purely imaginary and the three spatial coefficients real. In each case the partial derivatives are taken with respect to the complex coefficients, with a factor of $-i$ when the coefficient is purely imaginary.

The biquaternionic gradient $\tilde{\nabla} = \sum_\mu e_\mu \partial/\partial Q_\mu$ is a biquaternion-valued first-order operator, and its quaternion conjugate $\tilde{\nabla}^{\natural}$ is obtained by negating the vector part. The d'Alembertian $\Box = \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \partial^2/\partial Q_0^2 + \Delta_Q$ is the natural second-order operator: scalar-valued, symmetric under conjugation, and the operator of the standard second-order equations. On $\mathbb{H}_{\mathbb{B}}$ it is the ordinary four-dimensional Euclidean Laplacian, on $i\mathbb{H}_{\mathbb{B}}$ its negative, and on $\mathbb{M}_+$ and $\mathbb{M}_-$ it takes a Lorentzian form in the underlying real coordinates, $\Box = \Delta - c^{-2}\partial_t^2$ on the material slice. The square of the gradient $\tilde{\nabla}^2 = (\partial^2/\partial Q_0^2 - \Delta_Q) + 2\sum_k e_k \partial^2/(\partial Q_0 \partial Q_k)$ is a related second-order operator, expressed through the identity $\tilde{\nabla}^2 = 2\partial_{Q_0}\tilde{\nabla} - \Box$. The convective derivative $\tilde{D} = \tilde{U}^{\natural}\tilde{\nabla}$ is a biquaternion-valued first-order operator whose scalar part is the four-dimensional convective derivative. The bigradients $\nabla^\pm = \partial_\tau \pm i\nabla$ are the complexified pair $i\tilde{\nabla}$, $i\tilde{\nabla}^{\natural}$ on the material slice: conjugate to one another, composing to the wave operator, $\nabla^+\nabla^- = -\Box = \partial_\tau^2 - \Delta$, and standing as the operator of the biwave equation used by the electro-gravimagnetic programme.

The approach is closely related to Fueter's quaternionic analysis and to Clifford analysis; the generalisation to the biquaternion algebra includes the complex coefficients and the four conjugations. The class of functions singled out by $\tilde{\nabla}\tilde{F} = 0$, the integral theory and the Cauchy formula are the subjects of Biquaternion Regular Functions, Fueter Theory for Biquaternions and Biquaternion Integration.

Summary of Notation

Symbol	Meaning
$\mathbb{B}$	Biquaternion algebra
$\mathbb{H}_{\mathbb{B}}$	Quaternion subspace: all coefficients real; the Euclidean slice
$i\mathbb{H}_{\mathbb{B}}$	Anti-quaternion subspace: all coefficients purely imaginary
$\mathbb{M}_-$	Anti-Hermitian subspace: $Q_0$ imaginary, $Q_k$ real; the material four-position $ict\,e_0+\mathbf{x}$
$\mathbb{M}_+$	Hermitian subspace: $Q_0$ real, $Q_k$ imaginary; the informational slice
$\tilde{Q} = \sum_\mu Q_\mu e_\mu$	Point in a subspace
$Q_\mu = q_\mu + i q'_\mu$	Complex coefficient
$\\|\tilde{Q}\\|_E = \sqrt{\sum_\mu \|Q_\mu\|^2}$	Euclidean norm; the metric of $\mathbb{R}^8$; not multiplicative
$\partial/\partial Q_\mu$	Partial derivative with respect to $Q_\mu$; equals $-i\partial/\partial q'_\mu$ on an imaginary coefficient
$\tilde{\nabla} = \sum_\mu e_\mu \partial/\partial Q_\mu$	Biquaternionic gradient; the massless Dirac operator
$\tilde{\nabla}^{\natural}$	Quaternion conjugate of the gradient
$\nabla^{\pm} = \partial_\tau \pm i\nabla$	Bigradients (mutual complex gradients); $\nabla^{+} = i\tilde{\nabla}$, $\nabla^{-} = i\tilde{\nabla}^{\natural}$, $\nabla^{+}\nabla^{-} = -\Box$
$\Box = \tilde{\nabla}\tilde{\nabla}^{\natural} = \tilde{\nabla}^{\natural}\tilde{\nabla} = \Delta - c^{-2}\partial_t^2$	d'Alembertian; the wave operator on the material slice
$\tilde{\nabla}^2 = 2\partial_{Q_0}\tilde{\nabla} - \Box$	Square of the gradient
$\tilde{D} = \tilde{U}^{\natural}\tilde{\nabla}$	Convective derivative