Classical and Quantum Theories Share a Common Structure

Scientists have long recognised differing representations within classical and quantum theories, each linked to potential values of generalised charges. Benjamin H. Feintzeig from the University of Washington demonstrates a preservation of structure between these representation theories as one transitions from the classical to the quantum realm. Feintzeig achieves this by examining specific representation-theory preserving morphisms, utilising Lagrangian relations in the classical context and Hilbert bimodules within the quantum framework. This research is significant because it establishes a categorical equivalence between quantization and its classical limit, treating quantization as a strict deformation of a Poisson algebra and the classical limit as an extension of uniformly continuous C-algebras. By proving these functors are “almost-inverse” to each other, the work offers novel insights into the fundamental relationship between classical and quantum physics.

New work reveals a surprising structural harmony underlying these seemingly disparate realms, offering a fresh perspective on how the predictable world gives way to the probabilistic one. Researchers investigate whether representation theories are preserved when transitioning from classical to quantum physics.

To demonstrate this, they discuss representation-theory preserving morphisms in both classical and quantum contexts, specifically considering categories where morphisms are Lagrangian relations classically and Hilbert bimodules quantumly. These morphisms are significant as they induce representations of classical and quantum theories respectively. Quantization and the classical limit are treated as determining functors between these categories, with quantization utilising strict deformation quantization of a Poisson algebra and the classical limit approached via extension.

Lagrangian relations and symplectic dual pairs define classical models for quantum correspondence

Scientists investigate a categorical equivalence relating classical and quantum physics through the quantization and classical limit of symmetries and their associated charges. Classical theories, formulated on Poisson manifolds, utilise representations as symplectic manifolds, while quantum theories employ C-algebras of observables with representations on Hilbert spaces.

Even with a shared Lie group of symmetries, classical and quantum representations may not correspond one-to-one due to the continuous nature of classical charge versus the quantized nature of quantum charge. This prompts an exploration of a precise structural correspondence between these theories using category theory, considering a category of classical models with Poisson manifolds associated with Lie groupoids as objects and Lagrangian relations as arrows, preserving classical representation-theoretic structure.

These Lagrangian relations are restricted to those determined by symplectic dual pairs. Similarly, a category of quantum models is defined with C-algebras associated with the quantization of a Lie groupoid as objects and (equivalence classes of) Hilbert bimodules as arrows, preserving quantum representation-theoretic structure. Landsman previously defined a quantization functor Q from the classical category to the quantum category, mapping symplectic dual pairs to Hilbert bimodules.

Feintzeig and Steeger defined a classical limit functor L from the quantum category to the classical category, associating quantum morphisms with classical morphisms. The central claim is that the classical limit functor is almost inverse to the quantization functor, meaning L ◦ Q and Q ◦ L are naturally isomorphic to the identity functor, substantiating the shared representation-theoretic structures between classical and quantum theories.

The paper defines the objects of the classical and quantum categories for models constructed from Lie groupoids. A classical object is the dual of a Lie algebroid associated with a Lie groupoid, possessing a natural Poisson structure. A quantum object is a (reduced) Lie groupoid C-algebra associated with a Lie groupoid. Quantization is achieved via strict deformation quantization, while the classical limit is understood through the extension of a uniformly continuous bundle of C-algebras.

Next, the classical and quantum arrows are defined. Classical arrows are Lagrangian relations determined by symplectic dual pairs, while quantum arrows are (equivalence classes of) Hilbert bimodules. The transformations involving the quantization and classical limit of these arrows, as provided by Landsman and Feintzeig and Steeger, are reviewed. The core result, the categorical equivalence of the quantization and classical limit functors, is then proven.

A Lie groupoid, a small category where every arrow has an inverse, is central to these models. The objects in G are identified with identity arrows, denoted G0, with source and target projections sG and tG respectively. Groupoid multiplication is composition of arrows. For a Lie groupoid, both G and G0 are smooth manifolds, and sG and tG are surjective submersions.

Examples include the pair groupoid, the action groupoid, and the gauge groupoid. A Lie groupoid is associated with a Poisson manifold, specifically the dual of the Lie algebroid. The Lie algebroid G is a vector bundle over G0, with projection τG and anchor map τaG. The dual Lie algebroid G∗ carries a Poisson bracket on C∞(G∗), defined by relations among special cases of functions.

A Lie groupoid also defines a non-commutative C-algebra of observables, the reduced groupoid C-algebra C∗(G), defined as a completion of the algebra C∞c(G) with the convolution product. This involves right/left Haar systems and a quasi-invariant measure ν0 on G0. The left regular representation πG of C∞c(G) on L2(G, ν0 × νs) is defined for f ∈ C∞c(G) and ψ ∈ L2(G, ν0×νs).

The reduced groupoid C-algebra C∗(G) is the completion of C∞c(G) in the operator norm for the representation πG. The classical model (Poisson manifold G∗) is related to the quantum model (C-algebra C∗(G)) by strict deformation quantization, involving a family of C-algebras (Aħ)ħ∈ and quantization maps (Qħ: P → Aħ)ħ∈, with Q0: P → Cb(M) the inclusion map.

These maps must satisfy von Neumann’s, Dirac’s, and Rieffel’s conditions. If Qħ is one-to-one and Qħ(P) is dense in Aħ, the structure is a strict deformation quantization. A continuous bundle of C-algebras is crucial, particularly with properties of uniform continuity. A strict quantization of a Poisson algebra P ⊆ C0(P).

Category theory reveals consistent classical to quantum transitions

Scientists have long sought a consistent bridge between the smooth determinism of classical physics and the probabilistic nature of the quantum world. This recent work offers a subtle yet powerful contribution, not by proposing a new physical theory, but by clarifying the mathematical structures underpinning the transition between the two. For decades, a key challenge has been to find a way to relate classical and quantum descriptions without losing information or introducing inconsistencies during the process of ‘quantization’.

Previous attempts often involved approximations or ad-hoc procedures, leaving open the possibility of hidden discrepancies. This research demonstrates a preservation of mathematical relationships, specifically within representation theory, as one moves from classical to quantum descriptions. By employing advanced category theory, researchers have established an ‘almost-inverse’ relationship between the processes of quantization and taking the classical limit.

This means information isn’t simply lost in translation; the quantum and classical theories are more closely connected than previously understood. Such a connection is valuable because it provides a firmer foundation for exploring the boundaries between these two fundamental frameworks. The work remains largely abstract, dealing with mathematical categories rather than specific physical systems.

While the categorical equivalence established is elegant, applying it to complex physical models will require considerable effort. Moreover, the reliance on specific mathematical tools, like Hilbert bimodules and strict deformation quantization, may limit its direct applicability to all quantum theories. Despite these limitations, this development could inspire new approaches to problems in quantum gravity, where reconciling classical spacetime with quantum mechanics is a central goal. Future work might focus on extending these categorical relationships to more realistic physical scenarios, or exploring how they can be used to develop new computational methods for quantum systems.