Weyl Pseudo-Differential Operators

• Weyl pseudo-differential operators are generalized operators defined via an abstract representation-theoretic framework that extends classical Weyl quantization to infinite-dimensional settings.
• They employ Fourier-type transforms and twisted convolution to map symbols to operators, providing rigorous structures for operator composition.
• The framework unifies finite and infinite-dimensional analysis, recovering classical calculi while enabling applications in quantum mechanics and harmonic analysis.

Weyl pseudo-differential operators generalize the concept of Weyl quantization to infinite-dimensional settings by employing an abstract, representation-theoretic framework. The traditional calculus in finite dimensions—central to the analysis of quantum systems and Lie group representations—is thus extended to encompass operators acting on function spaces with infinitely many variables, including those arising from infinite-dimensional coadjoint orbits and infinite-dimensional Heisenberg groups (Beltita et al., 2010).

1. Abstract Formulation: Quasi-Localized Weyl Calculus

The general framework consists of a locally convex Lie group $M$ with Lie algebra $\mathfrak{m}$, a smooth exponential mapping $\exp_M: \mathfrak{m} \to M$, and a continuous unitary representation $T: M \to \mathcal{B}(\mathcal{Y})$ acting on a complex Hilbert space $\mathcal{Y}$. The role of phase space is played by the dual $\mathfrak{m}^*$, equipped with its weak-$*$ topology, and symbols are modeled as elements of a commutative unital C*-algebra $\operatorname{UC}_b(\mathfrak{m}^*)$.

Central to the construction is a triple:

• A locally convex space $E$ and measurable mapping $\theta: E \to \mathfrak{m}$,
• An auxiliary space $\mathcal{I}$ (continuously embedded in the dual of the algebra of bounded uniformly continuous functions on $\mathfrak{m}^*$),
• A test function space $\mathcal{Y}_{\Xi,00}$.

A Fourier-type transform $$\mathcal{F}_{\Xi}: \mathcal{I} \to \operatorname{UC}_b(E)$$ allows for the identification of symbols (measures on $E$) with bounded, uniformly continuous functions.

The Weyl calculus is defined via an operator-valued map

$$\operatorname{Op} : \mathcal{F}_{\Xi}^* \to \mathcal{B}(\mathcal{Y}, \mathcal{Y}_{\Xi, co}),$$

acting on test vectors $\phi, \psi$ by

$$\bigl(\operatorname{Op}(a)\,\phi \mid \psi\bigr) = \Bigl(\mathcal{F}_{\Xi}^{-1}(a),\, (T(\exp_M(\theta(\cdot)))\phi \mid \psi)\Bigr)_{\operatorname{UC}_b(E)}.$$

If $\mathcal{F}_{\Xi}^{-1}(a)$ is a complex Borel measure, this operator is given concretely by

$$\operatorname{Op}(a)\phi = \int_E T(\exp_M(\theta(x)))\phi \, d(\mathcal{F}_{\Xi}^{-1}(a))(x).$$

This structure generalizes the classical Weyl–Pedersen calculus for finite-dimensional nilpotent Lie groups to a broad functional analytic context.

2. Extension to Infinite-Dimensional Coadjoint Orbits

To accommodate representations tied to infinite-dimensional coadjoint orbits, the notion of a "predual" is introduced. For a nilpotent locally convex Lie algebra $\mathfrak{g}$ and coadjoint orbit $\mathcal{O} \subset \mathfrak{g}^*$, a predual consists of a pair $(E,\theta)$ with $E$ a locally convex real vector space and $\theta: E \to \mathfrak{g}$ a continuous linear map such that the induced $\theta^*$ is injective on $\mathcal{O}$. If $E$ is a closed subspace of $\mathfrak{g}$ with $\theta$ the inclusion, it plays the role of the predual.

For "flat" coadjoint orbits (i.e., those where the isotropy algebra is the center $\mathfrak{z}$), $E$ can be realized as $\ker(\xi_0)$, and the map $X \mapsto \operatorname{Ad}^*(X)\xi_0$ yields a diffeomorphism $E \simeq \mathcal{O}$.

Composition of Weyl-type operators is governed by twisted convolution products of measures:

$$\operatorname{Op}(a_1)\operatorname{Op}(a_2) = T(\mathcal{F}_{\Xi}^{-1}(a_1) *_\xi \mathcal{F}_{\Xi}^{-1}(a_2)).$$

This algebraic structure is crucial in extending the calculus to noncommutative, possibly infinite-dimensional, scenarios.

3. Example: Infinite-Dimensional Heisenberg Groups

Let $V$ be a real Hilbert space and $A \in \mathcal{B}(V)$ symmetric and injective. The infinite-dimensional Heisenberg algebra is defined as $b(V,A) = V \oplus V \oplus \mathbb{R}$ with bracket

$$[(x_1, y_1, t_1), (x_2, y_2, t_2)] = (0, 0, (Ax_1 \mid y_2)-(Ax_2 \mid y_1)).$$

The Heisenberg group $H(V,A)$ is formed via the BCH formula; $E = V \times V \times \{0\}$ acts as a predual space.

For the Schrödinger representation, Gaussian measure $y$ on a suitable space is constructed using covariance associated with $A$, and the representation is realized as

$$(T(x,y,t)\phi)(\cdot) = p(\cdot)^{1/2}\, \exp\left(i \left[ t + (\cdot \mid y)_0 + \frac{1}{2}(x \mid y)_0\right] \right) \phi(-x+\cdot),$$

where $p(\cdot)$ and the inner product $(\cdot \mid \cdot)_0$ are defined in terms of the Gaussian structure and $A$. This extends the finite-dimensional model to the infinite-dimensional setting and enables the pseudodifferential calculus therein.

4. Recovery of the Classical Weyl–Hörmander Calculus

Restricting to finite-dimensional Heisenberg groups, the framework reduces to traditional Weyl–Hörmander calculus. The Schrödinger representation in this regime yields the standard pseudo-differential operator theory, including Weyl quantization:

• Weyl–type quantization of magnetic operators is seen to coincide with the usual “magnetic” Weyl calculus, in agreement with the Hörmander theory for appropriate symbol classes.
• The approach thus subsumes the classical results as a special case and forms a bridge between finite and infinite-dimensional operator theory.

5. Summary of Structural and Analytical Implications

The abstract approach developed in this setting:

• Offers a unified framework for pseudo-differential Weyl calculi in both finite and infinite dimensions,
• Allows symbol/operator correspondences by means of Fourier–Laplace transforms, predual parametrizations, and integration against group representations,
• Handles operator composition via twisted convolution, providing a rigorous convolution algebra for symbols,
• Is robust to generalizations required by infinite-dimensional harmonic analysis, quantum mechanics with infinitely many degrees of freedom, and analysis on infinite-dimensional Lie groups,
• Recovers the classical Weyl calculus in finite-dimensional sub-cases without additional hypothesis,
• Enables construction and spectral analysis of operators in contexts previously inaccessible to classical pseudo-differential calculi.

This framework significantly expands the reach and flexibility of Weyl pseudo-differential operator theory, in particular by systematically incorporating the analytic, algebraic, and topological complexities inherent to infinite-dimensional scenarios (Beltita et al., 2010).