Quaternionic Clifford Analysis

Updated 20 February 2026

Quaternionic Clifford analysis is a hypercomplex function theory defined on Euclidean spaces with a quaternionic structure, extending classical monogenic concepts.
It introduces novel Dirac-type operators and refined Fischer decompositions under Sp(p) symmetry and Howe dual pairs like osp(4|2).
The framework integrates advanced spectral theory, S-functional calculus, and integral formulas, facilitating applications in boundary value problems and quantum mechanics.

Quaternionic Clifford analysis is the function theory and operator calculus on Euclidean spaces equipped with a hypercomplex (quaternionic) structure, formulated within the framework of Clifford algebras. It extends classical Euclidean (orthogonal) and Hermitian Clifford analysis by incorporating the maximal quaternionic symmetry group $\mathrm{Sp}(p)$ , leading to new systems of Dirac-type operators, refined notions of monogenicity, and multiplicity-free polynomial decompositions governed by dual pairs such as $\mathfrak{so}(4,\mathbb{C})\times\mathrm{Sp}(p)$ or superalgebras like $\mathfrak{osp}(4|2)$ . Quaternionic Clifford analysis now encompasses spectral theory on the $S$ -spectrum, block operator calculi relevant in quaternionic quantum mechanics, and connections to slice-regular and polyharmonic function theories (Colombo et al., 24 Jan 2025, Brackx et al., 2014, Brackx et al., 2014, Brackx et al., 2015, Colombo et al., 2021).

1. Algebraic and Geometric Foundations

A quaternionic structure on $\mathbb{R}^{4p}$ consists of three commuting, anti-involutive orthogonal transformations $I, J, K = IJ$ with $I^2 = J^2 = K^2 = -\mathrm{Id}$ and $IJ=-JI=K$ . The subgroup of $\mathrm{SO}(4p)$ commuting with the full quaternionic structure is isomorphic to $\mathrm{Sp}(p)$ ; thus, $\mathbb{R}^{4p}$ admits a realisation as $\mathbb{H}^p$ where quaternionic multiplication corresponds to explicit $4p\times 4p$ real matrices. The quaternionic Clifford algebra $\mathbb{R}_{0,4p}$ is generated by an orthonormal basis $\{e_1,\ldots,e_{4p}\}$ with $e_a e_b + e_b e_a = -2\delta_{ab}$ , and functions are most naturally considered with values in minimal left ideals (spinor spaces) of its complexification (Brackx et al., 2014, Brackx et al., 2014).

Spinor space $S$ admits a refined decomposition into symplectic cells (irreducible $\mathrm{Sp}(p)$ -modules), governed by operators $P=\sum_{k=1}^p f_{2k}f_{2k-1}$ and $Q=P^\dagger$ , acting on the Witt basis $f_k$ . These intertwine with a natural $\mathfrak{sl}_2$ -action $(P, Q, \beta)$ , leading to the full symmetry $\mathrm{Sp}(p)\times\mathfrak{sl}_2$ at the heart of the harmonic and monogenic decompositions (Brackx et al., 2014).

2. Quaternionic Dirac Systems and Monogenic Functions

Quaternionic Clifford analysis refines classical notions by introducing a tetrad of mutually commuting Dirac-type operators. On $\mathbb{R}^{4p}$ :

The standard Dirac operator: $\partial = \sum_{a=1}^{4p} e_a\,\partial_{x_a}$ ,
Its quaternionic rotations: $\partial_I = I[\partial], \ \partial_J = J[\partial], \ \partial_K = K[\partial]$ .

A function $F: U \subset \mathbb{R}^{4p} \to S$ is quaternionic monogenic if it is annihilated by all four: $\partial F = \partial_I F = \partial_J F = \partial_K F = 0$ . In complex notation, introducing Hermitian Dirac operators $\partial_{\underline{z}},\partial_{\underline{z}}^\dagger$ (and their $J$ -twists), this system is equivalent to a four-operator null system:

$\partial_{\underline{z}} F = 0, \quad \partial_{\underline{z}}^\dagger F = 0, \quad \partial_{\underline{z}}^J F = 0, \quad (\partial_{\underline{z}}^\dagger)^J F = 0.$

Quaternionic monogenic systems are $\mathrm{Sp}(p)$ -invariant and can be characterized as the kernel of certain generalized gradients in the sense of Stein--Weiss (Brackx et al., 2015, Brackx et al., 2019). Inclusion relations hold: every quaternionic monogenic function is Hermitian monogenic ( $U(2p)$ -invariant), and thus also Euclidean monogenic ( $\mathrm{Spin}(4p)$ -invariant).

3. Fischer Decompositions, Symmetries, and Howe Duality

Quaternionic Clifford analysis admits a refined Fischer decomposition of the space of spinor-valued homogeneous polynomials. The full decomposition, multiplicity-free under $\mathrm{Sp}(p)$ , is governed by Howe dual pairs:

For scalar polynomials: $(\mathfrak{sl}_2\oplus\mathfrak{sl}_2)\times \mathrm{Sp}(p)$ ,
For spinor-valued polynomials: $\mathfrak{osp}(4|2)\times\mathrm{Sp}(p)$ .

Let $\mathcal{P}_{a,b}(\mathbb{C}^{2p})$ denote bihomogeneous polynomials of bidegree $(a,b)$ . The $\mathrm{Sp}(p)$ -harmonic decomposition reads

$\mathcal{H}_{a,b} = \bigoplus_{j=0}^{\min(a,b)} \mathcal{E}^{\dagger\,j}\mathcal{H}_{a+j, b-j}^0,$

where $\mathcal{H}_{a+j, b-j}^0$ are kernels of the "spin-Euler" operators $\mathcal{E}$ or $\mathcal{E}^\dagger$ . The Fischer decomposition aligns with the joint representations of $\mathrm{Sp}(p)$ and the (super)algebra, the former acting on symplectic cells in spinor space, the latter shifting polynomial degrees and building explicit projections (Brackx et al., 2014, Brackx et al., 2015).

The introduction of the $\mathfrak{osp}(4|2)$ -monogenicity notion further refines the module structure and ensures irreducibility by imposing additional annihilation by lowering operators $\mathbb{E}_+,\,P$ (symplectic-harmonicity and spin-lowest weight) (Brackx et al., 2015).

4. Integral Formulas and Fueter Mapping Theorems

Quaternionic monogenic and related function classes possess integral representation formulas, notably generalizations of the classical Cauchy and Cauchy–Fueter formulas. In $\mathbb{R}^{4p}$ , the reproducing formula for quaternionic monogenic $G$ is given, for $X$ in the domain $D$ , by

$G(X) = (-4)^p \int_{\partial D} \{ E(Y-X)\,d\sigma(Y) + E_I(Y-X)\,d\sigma_I(Y) \} G(Y),$

with $E$ the Dirac fundamental solution and $d\sigma, d\sigma_I$ boundary forms constructed via the quaternionic structures $I, J, K$ . The integral identity is Sp $(p)$ -equivariant and, in cases reduced further by osp(4 $|$ 2) conditions, yields additional kernel vanishing properties (Brackx et al., 2019).

The polyharmonic, holomorphic Cliffordian, and polyanalytic function classes are constructed via explicit application of Laplacian and Dirac operators to slice-hyperholomorphic functions. The Fueter–Sce mapping theorem, in its operator and integral forms, provides surjective maps from slice-hyperholomorphic to axially monogenic functions:

$\Delta^{h+1} f(x) = \frac{1}{2\pi} \int_{\partial(U\cap\mathbb{C}_I)} F_L(s,x)\,ds_I\,f(s),$

with $F_L(s,x)$ an explicit integral kernel, and $h = (n-1)/2$ for $n$ odd. The resulting operator functional calculi (monogenic, F-calculus) are provably equivalent for intrinsic functions (Colombo et al., 24 Jan 2025).

5. Operator Calculi and $S$ -Spectrum Spectral Theory

Quaternionic Clifford analysis is the native setting for spectral theory over the $S$ -spectrum, generalizing the classical spectral calculus to slice-hyperholomorphic functions and non-commuting tuples of operators. For a paravector operator $T$ on a right Clifford module $V_n = V\otimes\mathbb{R}_n$ , the $S$ -spectrum is defined by

$\sigma_S(T) = \mathbb{R}^{n+1} \setminus \rho_S(T), \quad \rho_S(T) = \{ s : Q_{c,s}(T) \text{ invertible}\}$

with

$Q_{c,s}(T) = s^2 I - s\,(T + T^*) + T\,T^*.$

Functional calculus is provided via left- and right $S$ -resolvent operators and contour integrals over slice Cauchy domains, paralleling the quaternionic and complex cases. The existence of universal $S$ -functional calculus applies across all LSCS (locally slice complex symmetric) algebras, with quaternionic and Clifford cases as concrete instances (Colombo et al., 2021). Concrete applications include block-matrix operator spectra, spectral projections in quaternionic quantum mechanics, and spectral decompositions for systems with non-commuting symmetries (Colombo et al., 24 Jan 2025).

6. Function Spaces, RKHS, and Applications

The function-theoretic aspects of quaternionic Clifford analysis include the construction of reproducing kernel Hilbert spaces of monogenic (Fueter-regular) functions, built from Appell systems adapted to the quaternionic setting. For instance, Fueter–Appell polynomials $Q_k(q)$ on $\mathbb{H}$ , with $q = x_0 + x_1 i + x_2 j + x_3 k$ , yield orthogonal bases for Hardy and Fock spaces—complete with creation/annihilation operators, explicit reproducing kernels, and generalized Segal–Bargmann transforms (Alpay et al., 2020). The Fueter mapping theorem provides a direct link, as a differential or integral transform, between slice-regular/hyperholomorphic functions and their Fueter-regular (monogenic) images, with isometric correspondences at the RKHS level.

These analytic tools are foundational for boundary value problems, spectral analysis, and explicit kernel constructions (e.g., Green's functions, Szegő/Bergman kernels) within the $\mathrm{Sp}(p)$ -invariant framework (Brackx et al., 2014, Alpay et al., 2020).

7. Extensions: Dunkl Transforms, Fourier Analysis, and Computational Aspects

Recent work extends quaternionic Clifford analysis into Dunkl operator theory and harmonic analysis associated to reflection groups. The two-sided quaternionic Clifford–Dunkl transform is defined via two square roots of $-1$ in $Cl_{p,q}$ (e.g., $Cl_{0,2}\simeq\mathbb{H}$ ), simultaneously generalizing classical quaternionic Fourier transforms and Dunkl transforms. The associated inversion, Plancherel, and translation operators, as well as uncertainty principles (Miyachi-type), are established, embedding quaternionic analysis into a broad hypercomplex–reflection-group harmonic framework, with potential for new special-function theories and analytical techniques (Essenhajy et al., 9 Oct 2025).

Further, the structure of Clifford algebras of signature $(3,1)$ enables the definition of quadratic phase quaternionic Fourier transforms, their covariance properties, and applications to image recognition and fixed-point lattice actions, demonstrating computational utility for volumetric signal processing in $3+1$ dimensions (Furui, 2023).