Complex Weyl Correspondence in Fock Space

• Complex Weyl correspondence is the holomorphic quantization mapping that represents operators on Bargmann–Fock spaces using complex symbols with parity-based quantizers.
• It utilizes the Bargmann transform to relate real Weyl quantization with Berezin calculus, enabling Gaussian symbol formulas and Moyal-type star products.
• Its covariance under group actions and connections to the orbit method provide explicit formulas for representation operators and bridge diverse quantization frameworks.

Complex Weyl correspondence is the holomorphic or Fock-space form of Weyl quantization in which operators on a Bargmann–Fock space are represented by symbols on a complex phase space $\mathbb C^n$. In the works of Cahen and collaborators, it is defined through a parity-based Stratonovich–Weyl quantizer and is closely related to Berezin calculus, the Bargmann transform, and Moyal-type star products (Cahen, 9 Sep 2025). The term is not uniform across the literature: in some contexts “Weyl” refers instead to Weyl semimetals, Weyl groups, or Weyl-covariant geometry, and in one nearby but distinct case the relevant notion is actually the algebro-geometric Weil correspondence rather than any Weyl correspondence (Cecotti, 2024).

1. Terminology, scope, and historical placement

Within quantization theory, the complex Weyl correspondence is the correspondence between operators on a holomorphic Fock space and functions on $\mathbb C^n$, obtained by transporting the ordinary Weyl correspondence on $L^2(\mathbb R^n)$ through the Bargmann transform. In this setting the symbol lives on complex coordinates $z\in\mathbb C^n$, coherent states provide reproducing kernels, and the relevant quantizer is built from parity conjugated by the Heisenberg representation (Cahen, 2023).

This usage should be separated from several unrelated meanings of “Weyl.” In Seiberg–Witten geometry, the relevant phrase is the Weil correspondence, not Weyl correspondence; it concerns extensions of Abelian varieties and Seiberg–Witten differentials, and the cited paper explicitly warns that many distinct mathematical notions circulate under the name “Weil correspondence” (Cecotti, 2024). In condensed-matter physics, “complex Weyl” can instead refer to interaction-induced Weyl points with complex frequencies, where the correspondence is between a Green’s-function tetrad and an emergent effective metric, not a quantization map on Fock space (Nissinen et al., 2017). Other papers use “Weyl” for homological bulk–edge correspondence in Weyl semimetals (Gomi, 2020), Weyl-covariant holography (Ciambelli et al., 2019), spin representations of Weyl groups (Ciubotaru, 2010), or Weyl chamber actions in higher-rank dynamics (Hilgert et al., 2021).

A precise encyclopedic use of the term therefore refers most naturally to the holomorphic-symbol calculus developed on Bargmann–Fock space and its representation-theoretic extensions. This is the sense in which explicit “complex Weyl symbols” are computed for metaplectic, Jacobi-type, generalized diamond, and harmonic representations (Cahen, 14 May 2026).

2. Fock-space formulation and defining quantizer

The basic setting is the Bargmann–Fock space of holomorphic functions on $\mathbb C^n$. For the Heisenberg-group constructions used in the generalized diamond-group paper, one works with

$$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$

with

$d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$

and coherent states

$$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$

which satisfy the reproducing property $f(z)=\langle f,e_z\rangle_{\mathcal F_\lambda}$ (Cahen, 9 Sep 2025).

The complex Weyl correspondence is defined through a parity operator and its Heisenberg translates. In the generalized diamond-group formulation,

$$(R_0 f)(z)=2^n f(-z), \qquad \Omega_0(z)=\rho_\lambda(z,0)\,R_0\,\rho_\lambda(z,0)^{-1},$$

and explicitly

$\mathbb C^n$0

For a trace-class operator $\mathbb C^n$1 on $\mathbb C^n$2, the complex Weyl symbol is then

$\mathbb C^n$3

This is the defining formula in the Fock-space approach (Cahen, 9 Sep 2025).

An equivalent formulation appears in the metaplectic paper, where the same construction is presented as a Stratonovich–Weyl quantizer on Fock space, with $\mathbb C^n$4 identified as the holomorphic counterpart of the classical Weyl symbol. The kernel formula

$\mathbb C^n$5

gives a direct midpoint/difference-coordinate analogue of the real Weyl transform (Cahen, 2023).

This framework makes the adjective “complex” substantive: the symbols live on $\mathbb C^n$6, the operators act on a holomorphic Hilbert space, and the quantizer is formulated intrinsically in complex coordinates rather than by merely complexifying a real phase-space formula.

3. Relation to Berezin calculus, Bargmann transport, and star products

The complex Weyl correspondence is tightly linked to Berezin calculus. For an operator $\mathbb C^n$7 on $\mathbb C^n$8, the Berezin symbol is

$\mathbb C^n$9

and the double Berezin symbol is

$L^2(\mathbb R^n)$0

The papers stress that $L^2(\mathbb R^n)$1 is the unitary part in the polar decomposition of the Berezin correspondence, while the Berezin transform smooths the Weyl symbol. In the metaplectic treatment this relation is written as

$L^2(\mathbb R^n)$2

so the Berezin symbol is the heat-regularized version of the complex Weyl symbol (Cahen, 2023).

The correspondence with ordinary real Weyl calculus is mediated by the Bargmann transform. In the generalized diamond-group paper, if $L^2(\mathbb R^n)$3 is the Bargmann transform, then

$L^2(\mathbb R^n)$4

and for the classical Weyl correspondence $L^2(\mathbb R^n)$5,

$L^2(\mathbb R^n)$6

This identifies the complex Weyl symbol as the holomorphic avatar of the ordinary Weyl symbol (Cahen, 9 Sep 2025).

The same structure transports operator composition into a star product. Defining $L^2(\mathbb R^n)$7 by

$L^2(\mathbb R^n)$8

the generalized diamond-group paper shows that $L^2(\mathbb R^n)$9 is related to the real Moyal product through the map $z\in\mathbb C^n$0, and consequently admits the expansion

$z\in\mathbb C^n$1

The paper further derives explicit Gaussian star-product identities, including

$z\in\mathbb C^n$2

which it also relates to the real Moyal formula on $z\in\mathbb C^n$3 (Cahen, 9 Sep 2025).

A torus analogue is established in the paper on $z\in\mathbb C^n$4. There the Berezin–Toeplitz operator $z\in\mathbb C^n$5 on the torus is represented, up to $z\in\mathbb C^n$6, by a complex Weyl operator $z\in\mathbb C^n$7 with symbol

$z\in\mathbb C^n$8

This shows that the Toeplitz-to-Weyl heat transform familiar on the plane survives in the periodic setting after imposing Floquet conditions and adapting the Bargmann model to the torus (Rouby, 2017).

4. Covariance, orbit method, and Stratonovich–Weyl structure

A central structural property is covariance under the relevant representation. For the generalized diamond group

$z\in\mathbb C^n$9

with generic Fock representation $\mathbb C^n$0, the paper proves

$\mathbb C^n$1

where the induced action on phase space is

$\mathbb C^n$2

The parallel Berezin covariance identity also holds, and these formulas show that conjugation on operators becomes pullback by the geometric group action on $\mathbb C^n$3 (Cahen, 9 Sep 2025).

This covariance is one ingredient in identifying the complex Weyl correspondence as a genuine Stratonovich–Weyl correspondence. The same paper constructs an equivariant map

$\mathbb C^n$4

satisfying

$\mathbb C^n$5

with explicit formula

$\mathbb C^n$6

The map $\mathbb C^n$7 identifies $\mathbb C^n$8 with the coadjoint orbit $\mathbb C^n$9, and $$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$0 thereby becomes a Stratonovich–Weyl correspondence both on $$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$1 and on the orbit model (Cahen, 9 Sep 2025).

The metaplectic-symbol paper presents the same idea in a related guise. There the complex Weyl symbol is defined through the quantizer $$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$2, and the correspondence is described as the Bargmann/Fock realization of classical Weyl calculus. This work emphasizes that the complex-symbol calculus, the Berezin calculus, and the ordinary real Weyl calculus are three tightly connected symbol theories rather than separate constructions (Cahen, 2023).

A plausible implication is that the complex Weyl correspondence is best viewed not merely as a computational device for kernels on Fock space, but as a covariant quantization scheme tied to coherent states, group actions, and coadjoint-orbit geometry.

5. Explicit symbol formulas for representation operators

One of the main technical virtues of the complex Weyl correspondence is that it yields explicit closed formulas for representation operators.

For the generalized diamond group, if $$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$3 and $$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$4, then under the assumption

$$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$5

the complex Weyl symbol of the representation operator is

$$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$6

The paper also provides the infinitesimal symbol

$$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$7

showing that the Fock-space symbol calculus remains explicit even for the semidirect-product deformation of the Heisenberg case (Cahen, 9 Sep 2025).

For metaplectic operators, the explicit formula has the standard Gaussian/Cayley-transform form. If

$$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$8

then

$$\mathcal F_\lambda = \left\{f\in \mathcal O(\mathbb C^n): \|f\|_\lambda^2 = \int_{\mathbb C^n}|f(z)|^2 e^{-\lambda |z|^2/2}\,d\mu_\lambda(z) <\infty \right\},$$9

with a scalar prefactor $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$0 determined by $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$1 and the relevant branch convention. The infinitesimal symbol for

$d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$2

is

$d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$3

These formulas recover known real-side Weyl-symbol identities after conjugation back through the Bargmann transform (Cahen, 2023).

The 2026 paper on $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$4 extends this pattern to harmonic representations. For the harmonic representation $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$5 of $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$6, when $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$7,

$d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$8

For the extended harmonic representation $d\mu_\lambda(z)=(2\pi)^{-n}\lambda^n\,dm(z),$9 of $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$0, the symbol acquires additional linear and scalar Gaussian terms in the Heisenberg variable $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$1 (Cahen, 14 May 2026).

Across these examples, a common pattern emerges: representation operators quantized by the complex Weyl correspondence produce explicit Gaussian-type symbols governed by determinant factors, Cayley transforms, and coherent-state shifts.

6. Extensions, analogues, and common misconceptions

The complex Weyl correspondence has been extended beyond the flat Fock-space setting in several directions. On the torus $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$2, the correspondence links Berezin–Toeplitz operators on a finite-dimensional holomorphic space satisfying Floquet periodicity to complex Weyl operators on the corresponding periodic IR-manifold $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$3, with the same heat-kernel symbol conversion law as in the complex plane (Rouby, 2017). On solvable groups such as the generalized diamond group, it remains covariant and compatible with the orbit method (Cahen, 9 Sep 2025). On reductive groups such as $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$4, it can be combined with oscillator-type harmonic representations to yield explicit symbols (Cahen, 14 May 2026).

Several misconceptions recur in adjacent literature. The first is terminological: the phrase “complex Weyl correspondence” should not be used for Cecotti’s 2024 paper on Seiberg–Witten geometry, whose central notion is the Weil correspondence in the sense of André Weil and Mazur–Messing, relating differentials of the second and third kind to extensions of Abelian varieties (Cecotti, 2024). A second misconception is to conflate complex Weyl correspondence with “complex Weyl” phenomena in Weyl semimetals. In the interacting-Weyl-point paper, “complex” refers to complex quasiparticle frequencies arising when $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$5, and the relevant correspondence is between the inverse Green’s function, an emergent tetrad $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$6, and an effective metric $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$7, not between operators and holomorphic symbols (Nissinen et al., 2017).

Other “Weyl correspondences” are likewise distinct in substance. The homological bulk–edge correspondence for Weyl semimetals is an equality of relative homology classes involving $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$8 and Fermi-arc cycles (Gomi, 2020). The holographic paper on Weyl connections concerns a Weyl-covariant reorganization of the bulk/boundary dictionary in asymptotically AdS geometry (Ciambelli et al., 2019). The higher-rank quantum–classical correspondence paper uses “Weyl” in the sense of Weyl chambers and Weyl-group spectral data on locally symmetric spaces (Hilgert et al., 2021). The spin-Springer paper studies projective spin representations of Weyl groups, not Weyl quantization (Ciubotaru, 2010).

Taken together, these distinctions suggest a useful editorial rule: unless the context is explicitly Bargmann–Fock quantization, operator symbols on $$e_z(w)=\exp\!\left(\frac{\lambda}{2}\bar z\, w\right),$$9, or a parity-based Stratonovich–Weyl quantizer, “complex Weyl correspondence” is likely to be a misleading label. In its established technical sense, it denotes a holomorphic Weyl calculus whose main features are the symbol map

$f(z)=\langle f,e_z\rangle_{\mathcal F_\lambda}$0

its covariance under representation-theoretic symmetries, its relation

$f(z)=\langle f,e_z\rangle_{\mathcal F_\lambda}$1

and its exact transport of operator composition into a Moyal-type star product (Cahen, 9 Sep 2025).