[Paper Review] Convergence of Star Product: From Examples to a General Framework

Stefan Waldmann|arXiv (Cornell University)|Jan 31, 2019

Homotopy and Cohomology in Algebraic Topology116 references4 citations

TL;DR

This paper develops a general framework for convergence of star products in deformation quantization by analyzing three key examples: the Weyl, Gutt, and Wick-type star products. It establishes that convergence can be achieved through careful analytic control of formal power series, enabling unitary representations and GNS constructions in Hilbert spaces, thus bridging formal deformation quantization with physically realizable quantum theories.

ABSTRACT

We recall some of the fundamental achievements of formal deformation quantization to argue that one of the most important remaining problems is the question of convergence. Here we discuss different approaches found in the literature so far. The recent developments of finding convergence conditions are then outlined in three basic examples: the Weyl star product for constant Poisson structures, the Gutt star product for linear Poisson structures, and the Wick type star product on the Poincaré disc.

Motivation & Objective

• To address the long-standing problem of convergence in formal deformation quantization, where star products are defined as formal power series in ħ.
• To demonstrate that convergence is achievable in concrete, physically relevant examples, thereby validating the feasibility of a convergent framework.
• To establish a bridge between formal star products and physically realizable quantum theories through Hilbert space representations and GNS constructions.
• To explore the analytic structure of star products beyond formal power series, particularly in non-compact and symmetric spaces.
• To lay the groundwork for extending deformation quantization to infinite-dimensional systems and less regular Poisson structures.

Proposed method

• Analyzes the Weyl star product on constant Poisson structures, showing convergence via control of Taylor coefficients and growth conditions.
• Examines the Gutt star product on linear Poisson structures, proving that the product extends continuously to a Fréchet ∗-algebra, enabling self-adjointness of Lie algebra representations.
• Studies the Wick-type star product on the Poincaré disc, leveraging real-analytic structure and positivity to ensure convergence and unitary representation theory.
• Uses GNS representations of continuous positive linear functionals to construct Hilbert space completions and prove essential self-adjointness of momentum map components.
• Applies spectral theory and ∗-representation techniques to analyze the completed algebras, enabling applications in differential geometry and representation theory.
• Proposes a general convergence scheme based on growth conditions on Taylor coefficients and topological control via Fréchet structures, informed by concrete examples.

Experimental results

Research questions

• RQ1 Can formal star products in deformation quantization be promoted to convergent series in a way that preserves physical consistency and allows for unitary representations?
• RQ2 How do convergence properties of star products depend on the underlying Poisson structure, particularly in non-compact or symmetric spaces?
• RQ3 To what extent can GNS representations be used to establish self-adjointness of quantum generators in the completed algebra?
• RQ4 What are the analytic and topological conditions under which formal power series in ħ converge to bounded or continuous operators?
• RQ5 Can the convergence framework be generalized beyond locally multiplicatively convex algebras to include more general locally convex algebras with richer analytic structure?

Key findings

• The Weyl star product on constant Poisson structures converges when the Taylor coefficients of the functions involved satisfy appropriate decay conditions, enabling a continuous ∗-product on a Fréchet space.
• The Gutt star product on linear Poisson structures extends to a continuous ∗-product on a Fréchet algebra, allowing the momentum map components to be essentially self-adjoint and to integrate to a strongly continuous unitary representation of SU(1,n).
• The Wick-type star product on the Poincaré disc converges due to the real-analytic structure of the manifold and the positivity of the product, leading to a well-behaved ∗-algebra with GNS representations.
• In all three examples, the completed star product algebras support GNS representations that yield analytic vectors and essential self-adjointness, enabling a direct link to quantum mechanics.
• The convergence of the star product is not solely determined by the formal parameter ħ but depends on the interplay between function space growth and Poisson structure geometry.
• The examples suggest that non-compact Hermitian symmetric spaces and real-analytic manifolds are ideal candidates for further study due to favorable convergence and positivity properties.