Quantization as a Categorical Equivalence for Hilbert Bimodules and Lagrangian Relations

Abstract: It is well known that classical and quantum theories carry distinct types of representations, each type of representation corresponding to possible values of generalized charges in the classical or quantum context. This paper demonstrates a sense in the structure of these representation theories is preserved from classical to quantum physics. To show this, I discuss distinct representation-theory preserving morphisms in the classical and quantum contexts. Specifically, I consider categories whose morphisms are Lagrangian relations in the classical context and Hilbert bimodules in the quantum context. These morphisms are significant because they give rise to induced representations of classical and quantum theories, respectively. I consider quantization and the classical limit as determining functors between these categories. I treat quantization via the strict deformation quantization of a Poisson algebra and the classical limit via the extension of a uniformly continuous bundle of C*-algebras. With these tools, I prove that the quantization and classical limit functors are "almost-inverse" to each other, thus establishing a categorical equivalence.

• The paper proves that quantization and the classical limit are quasi-inverse functors between Lie-groupoid-based Poisson systems with Lagrangian relations and C*-algebras with Hilbert bimodules.
• It constructs functorial continuous-bundle lifts of Hilbert bimodules and shows their classical-limit spaces coincide with the corresponding Lagrangian relations, without requiring those limits to be symplectic.
• The result identifies the precise scope and remaining gap of classical–quantum correspondence: integrable, source-simply connected groupoids are covered, while tensor-product kernels may not fully determine quantum induction functors.

Overview

The paper establishes a categorical equivalence between a category of classical Hamiltonian systems and a category of quantum systems, both built from Lie groupoids. The central claim is that quantization and the classical limit are not merely mutually inverse on objects but also on morphisms that carry representation-theoretic content: symplectic dual pairs (equivalently, Lagrangian relations) on the classical side, and Hilbert bimodules on the quantum side. The author proves that the quantization functor $Q$ and the classical limit functor $L$ are "almost inverse" — that $L \circ Q$ and $Q \circ L$ are naturally isomorphic to the respective identity functors — thereby giving precise mathematical content to the idea that classical and quantum theories share the same structure for representing symmetries and charges (2602.15188).

This result generalizes earlier work in which Landsman constructed the quantization functor from dual pairs to Hilbert bimodules, and in which Feintzeig and Storme constructed the classical limit of Hilbert bimodules under conditions guaranteeing a symplectic limit space.

Classical and quantum objects from Lie groupoids

Both categories are anchored in Lie groupoids $G$. A classical object is the dual $\mathfrak{G}^*$ of the associated Lie algebroid, equipped with its canonical Poisson structure; a quantum object is the reduced groupoid C*-algebra $C^*(G)$, defined as the completion of $C_c^\infty(G)$ under the left regular representation. The two are related by Landsman's strict deformation quantization: with $\mathcal{P}_G = C^\infty_{PW}(\mathfrak{G}^*)$ the Paley–Wiener functions, quantization maps built from the Weyl exponential map and fiberwise Fourier transform satisfy von Neumann's, Dirac's, and Rieffel's conditions, yielding a strict deformation quantization with fibers $\mathfrak{A}_0 = C_0(\mathfrak{G}^*)$ and $L$0 for $L$1 (2602.15188).

The classical limit is formulated through uniformly continuous bundles of C*-algebras over $L$2, whose definition strengthens Kirchberg–Wassermann's by requiring uniform rather than vanishing continuity of sections. Such a bundle extends uniquely to $L$3, with limiting fiber $L$4, where $L$5 is the ideal of sections vanishing at $L$6. For bundles generated by strict deformation quantization, this recovers $L$7 up to isomorphism — so the classical geometry is uniquely determined by the quantum data at $L$8. Notably, equivalent strict quantizations determine the same bundle of continuous sections, making the framework insensitive to the choice among equivalent quantization schemes such as Weyl versus Berezin quantization.

Morphisms: Lagrangian relations and Hilbert bimodules

The morphisms are chosen to reflect representation-theoretic structure. On the classical side, a regular bibundle $L$9 between Lie groupoids induces a symplectic dual pair $L \circ Q$0 via explicit momentum maps. Since induction of symplectic realizations depends only on the image $L \circ Q$1, which is a Lagrangian submanifold of $L \circ Q$2, arrows in the category LPoisson are taken as equivalence classes of dual pairs determining the same Lagrangian relation. This is a deliberate departure from Landsman's category, which used isomorphism classes of dual pairs themselves.

On the quantum side, the same bibundle yields a Hilbert bimodule $L \circ Q$3, with middle space the completion of $L \circ Q$4 under a $L \circ Q$5-valued inner product induced by the Haar system. Induction of Hilbert space representations via the interior tensor product does not depend on the full bimodule structure; the invariant retained here is the tensor product kernel $L \circ Q$6. Arrows in LC* are accordingly equivalence classes of bimodules sharing the same tensor product kernel — again a coarser equivalence than unitary equivalence classes.

Integrability of the Poisson manifolds and source-simple-connectedness of the groupoids are assumed precisely so that identity arrows exist in both categories; these are substantive restrictions on the scope of the theorem.

Lifting bimodules to continuous bundles

The first technical contribution is a functorial lift: every arrow $L \circ Q$7 of LC* extends to a Hilbert bimodule $L \circ Q$8 between the algebras of uniformly continuous sections $L \circ Q$9 and $Q \circ L$0 (the normal groupoid C*-algebras). The construction takes sections smooth in $Q \circ L$1 that are uniformly continuous in $Q \circ L$2 relative to the bimodule norm. The key verification — that the inner product lands in $Q \circ L$3 — rests on two lemmas establishing uniform norm continuity of $Q \circ L$4 and norm continuity of the generating sections $Q \circ L$5, proved via dominated convergence in the Hilbert–Schmidt norm. The lift respects composition: $Q \circ L$6 is unitarily equivalent to the lift of $Q \circ L$7, via the fiberwise application of Landsman's intertwiner.

Strong nondegeneracy at $Q \circ L$8, the condition required by Feintzeig–Storme for the Hilbert classical limit to exist, is verified generally using a sequential approximate identity for the right action. Consequently each lifted bimodule has a well-defined classical limit, and the quotient algebra $Q \circ L$9 is abelian, hence of the form $G$0 by Gelfand duality. An important point of divergence from prior work: $G$1 need not be symplectic in this setting. Instead of imposing extra hypotheses forcing symplecticity, the paper shows directly that $G$2 is isomorphic to the Lagrangian relation determined by $G$3.

The categorical equivalence

The main theorem identifies the classical limit space with the original Lagrangian relation: $G$4. The proof proceeds through an asymptotic analysis of the tensor product kernels. At fixed $G$5, membership of $G$6 in $G$7 is characterized by an oscillatory integral operator $G$8 vanishing almost everywhere on $G$9. Taking $\mathfrak{G}^*$0, a stationary-phase-type computation shows the kernel converges to the pointwise product $\mathfrak{G}^*$1, so that $\mathfrak{G}^*$2 corresponds exactly to functions vanishing after composition with the momentum maps. Gelfand duality then forces $\mathfrak{G}^*$3 to be the image of the pair map — precisely the Lagrangian relation.

Two consequences follow immediately. First, $\mathfrak{G}^*$4 is well-defined on the tensor-product-kernel equivalence classes, since equal kernels imply equal Lagrangian relations. Second, $\mathfrak{G}^*$5 canonically; and because every arrow of LC* was defined as the quantization of some arrow of LPoisson, $\mathfrak{G}^*$6 as well. Functoriality of both maps follows from Landsman's and Feintzeig–Storme's composition-compatibility results, respectively. Hence:

Theorem: the functors $\mathfrak{G}^*$7 form a categorical equivalence.

The bold interpretive claim this substantiates is that the mismatch often observed between classical and quantum charge spectra — continuous versus discrete families of representations — does not indicate a failure of structural correspondence: the representation-theoretic morphisms stand in exact one-to-one correspondence once the appropriate equivalence relations (same Lagrangian relation; same tensor product kernel) are imposed.

Limitations and open questions

The equivalence holds only within the stated framework, and several assumptions constrain its scope. The objects are restricted to integrable Lie algebroids and source-simply connected Lie groupoids; integrability guarantees identity arrows, and source-simple-connectedness ensures the identity bibundle matches the identity dual pair. Systems outside this class — including many singular foliations and non-integrable Poisson structures — are not covered. The quantization relies on a Weyl exponential map requiring a connection and a cut-off supported where the exponential maps are diffeomorphisms, and the reduced groupoid C*-algebra is used throughout without addressing full-versus-reduced distinctions beyond amenability.

The paper itself flags a conceptual asymmetry it leaves open. In LPoisson, Lagrangian relations fully determine the induction functor on symplectic realizations; in LC*, the tensor product kernel is only an invariant of the induction functor, not obviously a complete determinant of it. Whether there exist conditions under which a tensor product kernel uniquely determines the induction functor — or alternative morphisms for categories of C*-algebras that do so — remains unresolved. A further question inherited from Landsman concerns generalizing the functors to broader classes of morphisms motivated by K-theory for C*-algebras.

Conclusion

The paper proves that quantization and the classical limit are quasi-inverse functors between a category of Poisson manifolds with Lagrangian-relation morphisms and a category of groupoid C*-algebras with Hilbert-bimodule morphisms, thereby exhibiting classical and quantum representation theory as categorically equivalent structures for the class of systems modeled by integrable, source-simply connected Lie groupoids. Technically, the work extends the Feintzeig–Storme classical-limit construction to bimodules whose limit spaces are merely Lagrangian relations, removing the requirement that the limit be symplectic. The residual gap between Lagrangian relations as complete determinants of classical induction and tensor product kernels as mere invariants of quantum induction is the principal open problem the paper leaves for subsequent investigation.