Global Quaternionic Fock Structure

• Global quaternionic Fock structure is an extension of classical Fock spaces to noncommutative quaternionic function spaces defined via slice regularity.
• It establishes norm-equivalence between global and slice-defined spaces through Gaussian-weighted integrals and explicit reproducing kernels.
• The framework underpins advanced operator theory in quaternionic analysis, enabling spectral studies, integral transforms, and symmetry representations.

The global quaternionic Fock structure encompasses a comprehensive analytic and operator-theoretic framework for function spaces of slice regular and related classes of quaternionic-valued functions, equipped with Gaussian-type measures, explicit reproducing kernels, and norm-equivalent formulations across slices of the quaternionic algebra. The theory generalizes the classical complex Bargmann–Fock space to higher-dimensional, noncommutative settings and provides the foundational Hilbert and Banach module structures for analysis, operator theory, and invariant function spaces over the quaternions.

1. Global Fock Norms and Function Spaces

Let $\mathbb H$ denote the quaternions and $\mathbb S = \{I \in \mathbb H : I^2=-1\}$ the 2-sphere of imaginary units. Any nonreal $z \in \mathbb H$ can be written as $x + I y$, and the corresponding "slice" is $\mathbb C_I = \mathbb R + I\mathbb R$. For $\alpha > 0$ and $0 < p \leq \infty$, a global Gaussian $L^p$-norm for a measurable slice function $f:\mathbb H \to \mathbb H$ is defined by

$$\|f\|_{p,\alpha}^p = \int_\mathbb{H} |f(z) e^{-\frac{\alpha}{2}|z|^2}|^p\,dV(z) = \int_\mathbb{H} |f(z)|^p\, d\lambda_{\alpha p/2}(z),$$

with $$d\lambda_\alpha(z) = \frac{2\alpha}{\pi} e^{-\alpha|z|^2} dV(z)$$, where $dV(z)$ is the product of area on each slice $\mathbb C_I$ and the uniform measure $d\sigma(I)$ on $\mathbb S$. For $p=\infty$, the essential supremum is taken with the same Gaussian weight.

The associated global quaternionic Fock space is

$$F^p_\alpha = \left\{ f\in \mathrm{SR}(\mathbb H) : \|f\|_{p,\alpha} < \infty \right\},$$

where $\mathrm{SR}(\mathbb H)$ denotes the set of slice-regular functions. Parallel to this, a slice-defined norm on each $\mathbb C_I$ leads to the slice Fock space $\mathfrak F^p_\alpha$.

2. Equivalence of Global and Slice Fock Structures

A fundamental result establishes the equivalence (both as sets and up to norm equivalence) between $F^p_\alpha$ and the slice-constructed space

$$\mathfrak F^p_\alpha = \left\{ f \in \mathrm{SR}(\mathbb H) : \sup_{I\in\mathbb{S}} \|f\|_{p,\alpha,I} < \infty\right\}, \qquad \|f\|_{p,\alpha,I}^p = \frac{\alpha p}{2\pi} \int_{\mathbb{C}_I} |f(z) e^{-\frac{\alpha}{2}|z|^2}|^p \, dm_{2,I}(z).$$

Specifically, for $0 < p \leq \infty$ and all $\alpha > 0$,

$$F^p_\alpha = \mathfrak F^p_\alpha, \qquad \|f\|_{p,\alpha} \approx \sup_{I\in\mathbb S} \|f\|_{p,\alpha,I}.$$

If $p = 2$, this identification is isometric, and the inner products agree via a quaternionic polarization identity (Lin et al., 15 Jan 2026).

This equivalence relies fundamentally on the slice representation formula for slice-regular functions, which allows one to recover the global $L^p$ norm from the aggregate of slice-based norms and guarantees global independence from the choice of slice.

3. Reproducing Kernel Hilbert Space Structure

In the Hilbert space case $p=2$, $F^2_\alpha$ is a right quaternionic Hilbert space endowed with the inner product

$$\langle f, g \rangle_\alpha = \int_\mathbb{H} \overline{g(z)} f(z) d\lambda_\alpha(z).$$

This space possesses a unique reproducing kernel

$$K_\alpha(z, w) = \sum_{n=0}^\infty \frac{\alpha^n}{n!} z^n \overline{w}^n = e_\star^{\alpha z \overline{w}},$$

where $e_\star^{u} = \sum_{n=0}^\infty u^n / n!$ is the quaternionic star-exponential. Thus,

$$f(z) = \langle f(\cdot), K_\alpha(\cdot, w)\rangle_\alpha,$$

for all $f \in F^2_\alpha$.

The full projection $P_\alpha: L_{s,\alpha}^2 \to F^2_\alpha$ is given globally by

$$(P_\alpha h)(z) = \int_\mathbb H K_\alpha(z, w) h(w) d\lambda_\alpha(w),$$

yielding a slice-independent orthogonal projection.

4. Generalizations: Polyanalytic, Cholewinski, and Fueter–Regular Structures

(a) Polyanalytic and Poly-Fueter Extensions

The theory extends to slice-polyanalytic Fock spaces of order $N$: $$F_N^{\mathrm{slice}}(\mathbb H) = \left\{ f \text{ slice-polyanalytic of order } N \mid \int_{\mathbb C_I} |f(p)|^2 d\mu_I(p) < \infty \right\},$$ where, for each $f$, the reproducing kernel is

$$K_N(q, r) = e^{\overline{r} q} \sum_{k=0}^{N-1} (-1)^k \binom{N}{k+1} \frac{(\overline{q} r)^k}{k!} = e^{\overline{r} q}\, L_{N-1}^{(1)}(|q - r|^2),$$

with $L_{N-1}^{(1)}$ the generalized Laguerre polynomial (Alpay et al., 2018).

Poly-Fueter Fock spaces are constructed as images under higher-order analogues of the Fueter mapping theorem, producing a global ladder of Fock-theoretic structures relating slice-analytic, polyanalytic, and monogenic function classes (Diki, 12 Oct 2025).

(b) Cholewinski-Fock Spaces

Cholewinski–Fock spaces introduce further generality by weighting norms with Macdonald (modified Bessel) functions,

$$\|f\|^2_\alpha = \int_{\mathbb C_I} |f_e(q)|^2 d\lambda_{\alpha,I}(q) + \int_{\mathbb C_I} |f_o(q)|^2 |q|^{-2} d\lambda_{\alpha+1,I}^*(q),$$

where $f_e$ and $f_o$ denote even and odd components (Diki, 2019). The associated monotonic inclusion chain in $\alpha$ provides a continuous family of RKHSs interpolating between the classical slice Fock space ($\alpha = -1$) and more singularly weighted spaces.

5. Orthonormal Bases and Power Series Expansions

In all global quaternionic Fock spaces $F^2_\alpha$ (and their generalizations), the space admits an explicit orthonormal basis formed by the monomials

$e_n(q) = \frac{q^n}{\sqrt{n!}} , \qquad n \geq 0,$

so that any $f \in F^2_\alpha$ can be expanded globally as

$$f(q) = \sum_{n=0}^\infty q^n a_n, \quad \sum_{n=0}^\infty |a_n|^2 n! < \infty,$$

where the coefficients $a_n$ are quaternionic. Similar bases exist for polyanalytic and Cholewinski spaces, often involving composite monomials or Hermite polynomials (Kumar et al., 2016, Hamyani et al., 2017, Diki, 2019).

6. Operators and Symmetry Properties

The global quaternionic Fock structure naturally accommodates creation and annihilation operators: $$(a^* f)(q) = q f(q), \qquad (a f)(q) = \partial_S f(q),$$ satisfying $[a, a^*] = I$ and their Cholewinski generalizations $[D_\alpha, M] = I + (2\alpha+1)A$ with $A$ the parity operator (Diki, 2019, Alpay et al., 2014). Invariant structures under Heisenberg–Weyl and rotation groups are manifest via the kernel forms and power series.

Symmetries include $SU(2)$- and $SU(2,2)$-actions, slice independence of the inner product and kernel, and explicit intertwining with quaternionic Segal–Bargmann and Bargmann–Fock–Fueter transforms (Diki et al., 2019, Frenkel et al., 2024). In the quasi-regular function formalism, the global Fock structure is realized as a fermionic Fock space over Clifford algebra generators, carrying representations of the quaternionic conformal group GL$(2, \mathbb H^{\mathbb C})$ with pseudo-unitary inner products and explicit Green's function–type reproducing kernels $(Z - W)^{-1}$ (Frenkel et al., 2024).

7. Applications and Operator Theory

The global quaternionic Fock framework enables the rigorous development of operator theory in quaternionic analysis, including:

• Kernel-induced integral operators and their boundedness/compactness via Fock–Carleson measure characterizations.
• Berezin transforms and Toeplitz operators with slice-function or measure symbols, with boundedness and compactness characterized in terms of Berezin-type averages and slice/symmetrized BMO$^1$ conditions (Lin et al., 15 Jan 2026).
• Embedding theorems, duality relations, and norm isomorphisms across global and slice-Fock spaces, informing spectral theory, representation decompositions, and explicit integral transforms.

The global structure thus knits together slice-regularity, complex-analytic techniques, and noncommutative harmonic analysis, extending the Bargmann–Fock theory into the quaternionic and hypercomplex domains with complete analytic, algebraic, and operator-theoretic infrastructure (Lin et al., 15 Jan 2026, Kumar et al., 2016, Diki, 12 Oct 2025, Alpay et al., 2014).