Strict Convergent Quantization of Poisson Structures

Definitions and mathematical statement

A Poisson algebra over a field $\mathbf{k}$ is a commutative associative $\mathbf{k}$-algebra $A$ with a bilinear bracket $\{-,-\}:A\times A\to A$ that is antisymmetric, satisfies the Jacobi identity, and obeys the Leibniz rule \[ \{a,bc\}=\{a,b\}c+b\{a,c\}\qquad(a,b,c\in A). \] This supplies algebraic terminology; the source does not specify the underlying field, algebra, geometric space, or regularity class of its Poisson structures.

A formal star product is conventionally written \[ a\star b=ab+\sum_{r\geq 1}\hbar^r C_r(a,b), \] with bilinear maps $C_r:A\times A\to A$, and is associative on $A[[\hbar]]$. In one standard normalization its first-order commutator satisfies \[ C_1(a,b)-C_1(b,a)=\{a,b\}. \] This normalization is explanatory only: the excerpt does not specify its parameter or commutator normalization. Formal existence alone does not specify convergence after assigning numerical values to $\hbar$.

Source question. Determine which Poisson structures, or classes of Poisson structures, admit "strict" convergent quantizations.

Unresolved scope. The supplied record does not define "strict," specify a topology or norm for convergence, identify the algebra of observables, give an allowed parameter set, or state continuity, involution, or representation requirements. Consequently, it does not determine a single precise existence or classification problem. These choices must be recovered from the source context before a definitive formulation can be given; the formal-star-product description above does not supply the missing definition of strict quantization.

Supplied background qualifications. The excerpt reports that [BS23] combines results of [BWa, Chapter 10] with functional analysis to extend convergent star products obtained from algebraizations of an unspecified Theorem 3, from polynomial functions to analytic functions with infinite radius of convergence. It further reports that, in examples, the resulting algebras of quantum observables admit representations by adjointable operators on pre-Hilbert spaces [BS23, Section 3.4], relating this to the standard formulation of quantum mechanics. These are contextual claims from the supplied excerpt, not requirements asserted for every quantization sought. The excerpt also contains a disconnected clause describing $C_{\Lambda}$ as a formal power series of differential operators associated with a graph $\Lambda$; neither its formula nor the graph construction is supplied.

Short English statement

Determine which Poisson structures or classes admit "strict" convergent quantizations. The source leaves the analytic setting and meaning of "strict" unspecified, so a fully precise existence criterion cannot yet be stated. Its partially solved label and claims about polynomial examples are supplied dated curation, requiring independent review.

Sources

• Local UnsolvedMath registry, permanent ID $30005903$, title: Strict Convergent Quantization of Poisson Structures; exact source problem code: catalogue entry. The supplied record attributes the question to the Oberwolfach Reports Open Problems collection, with proposed year 2024; no individual proposer is identified.
• Hochschild (Co)Homology and Applications (2024), Oberwolfach Reports, report pages 1097--1162, DOI 10.4171/owr/2024/20. Supplied source report and literature reference: https://doi.org/10.4171/owr/2024/20.
• Barmeier and Schmitt, "strict quantization" (supplied reference label): https://doi.org/10.1007/s00220-022-04541-4. The excerpt cites [BS23], including Section 3.4, but supplies no full bibliographic expansion of that citation key.
• The excerpt also cites [BWa, Chapter 10] and an internal Theorem 3. Their bibliographic identification and mathematical contents are not supplied.
• Literature assessment checked on 2026-08-22: the supplied curation labels the problem partially solved and reports strict convergent quantizations for broad polynomial Poisson classes, while reporting no known general construction for arbitrary Poisson structures or classes. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The source's description of the problem as widely open is likewise a supplied characterization.