[Paper Review] On the classical and quantum momentum map

Chiara Esposito|arXiv (Cornell University)|Mar 19, 2012

Algebraic structures and combinatorial models8 references4 citations

TL;DR

This thesis develops a generalized momentum map theory in Poisson geometry, introducing an infinitesimal momentum map and proving conditions for its reconstruction into a full momentum map. It extends classical Poisson reduction to quantum settings via deformation quantization, defining a quantum momentum map that factorizes quantum actions and enabling quantum reduction—demonstrated through examples like SU(2) actions on deformation quantized manifolds.

ABSTRACT

In this thesis we study the classical and quantum momentum maps and the theory of reduction. We focus on the notion of momentum map in Poisson geometry and we discuss the classification of the momentum map in this framework. Furthermore, we describe the so-called Poisson Reduction, a technique that allows us to reduce the dimension of a manifold in presence of symmetries implemented by Poisson actions.

Motivation & Objective

• To generalize the classical momentum map to Poisson geometry, particularly within Lu’s framework of Poisson-Lie group actions.
• To clarify the conditions under which an infinitesimal momentum map uniquely determines a full momentum map.
• To develop a Poisson reduction theory based on Lu’s momentum map and Weinstein’s local normal form for Poisson manifolds.
• To define a quantum momentum map as a deformation of the classical one, ensuring compatibility with quantum group actions.
• To establish a quantum reduction procedure via deformation quantization, illustrated on examples such as Uℏ(su(2)) actions.

Proposed method

• Introduces the infinitesimal momentum map as a weaker structure derived from the Lie algebra action on a Poisson manifold.
• Uses Weinstein’s local normal form for Poisson manifolds to describe the infinitesimal generators of Poisson actions.
• Applies deformation quantization techniques to lift classical momentum maps to quantum momentum maps that factorize quantum actions.
• Constructs quantum reduction by considering invariant ideals in deformation-quantized algebras, such as the ideal generated by H in C∞ℏ(M).
• Utilizes symplectic groupoids and the symplectization functor to lift Poisson actions to Hamiltonian actions with well-defined momentum maps.
• Compares deformation quantization with geometric quantization, particularly in the context of integrable systems like the Gelfand-Cetlin system.

Experimental results

Research questions

• RQ1 Under what conditions does an infinitesimal momentum map uniquely determine a classical momentum map in Poisson geometry?
• RQ2 How can Poisson reduction be systematically constructed using Lu’s momentum map and the local structure of Poisson manifolds?
• RQ3 What is the correct definition of quantum reduction, and how does it relate to classical reduction via deformation quantization?
• RQ4 Can the quantum momentum map be consistently defined as a deformation of the classical momentum map that preserves factorization of the quantum action?
• RQ5 How do different quantization schemes—deformation quantization versus geometric quantization—compare in the context of Poisson-Lie group actions and integrable systems?

Key findings

• The infinitesimal momentum map determines a full momentum map if and only if the cohomological obstruction vanishes, which is shown explicitly in two cases.
• A Poisson reduction space μ−1(x)/Gx is isomorphic to μ−1(Ox)/G as a Poisson manifold, where Ox is the dressing orbit, establishing a canonical Poisson structure on the quotient.
• The quantum momentum map is constructed such that it factorizes the quantum action, ensuring consistency between classical and quantum symmetries.
• For the Uℏ(su(2)) action on C∞ℏ(M), the ideal I generated by H is Uℏ(su(2))-invariant, and the quotient (C∞ℏ(M)/I)^Uℏ(su(2)) is a deformation quantization of the Poisson reduction M//SU(2).
• The symplectization functor lifts Poisson actions to Hamiltonian actions on symplectic groupoids, allowing the construction of momentum maps in cases where they do not exist classically.
• Geometric quantization of the Gelfand-Cetlin system yields representation-theoretic results, suggesting a path to compare geometric and deformation quantization in Poisson-Lie group settings.