[Paper Review] A Counter Example of Invariant Deformation Quantization

Xiang Tang|ArXiv.org|Nov 29, 2004

Homotopy and Cohomology in Algebraic Topology15 references3 citations

TL;DR

This paper presents a counterexample to the conjecture that every Hamiltonian Lie algebra action on a symplectic manifold admits an invariant star product. Using explicit computation of bidifferential operators in a deformation quantization framework, the author demonstrates a contradiction in the structure constants of a putative invariant star product on ℝ² with the standard symplectic form, proving that no such geometrically invariant star product can exist for this action.

ABSTRACT

In this note, we will show one example of hamiltonian Lie algebra action which has no invariant star product.

Motivation & Objective

• To challenge the long-standing conjecture that every Hamiltonian Lie group or algebra action on a symplectic manifold admits an invariant star product.
• To investigate whether the existence of an invariant symplectic connection—previously thought sufficient—guarantees an invariant star product.
• To construct a concrete example where the necessary conditions for invariance lead to a contradiction in the star product's structure constants.
• To provide a negative result analogous to Van Hove's no-go theorem in the context of invariant deformation quantization.

Proposed method

• Constructs a specific Hamiltonian action of a Lie algebra on ℝ² equipped with the standard symplectic form dx∧dy.
• Assumes the existence of a geometrically G-invariant star product and expresses it as a formal power series ∑ħʳCʳ(f,g), where Cʳ are local bidifferential operators.
• Analyzes the coefficient C¹(f,g) in the star product expansion using multi-index notation for derivatives, focusing on terms involving fyygxhx and fxxgyhy.
• Applies the associativity condition (f⋆g)⋆h = f⋆(g⋆h) to derive a system of equations constraining the structure constants Cᵢⱼ;ₖₗ of the bidifferential operators.
• Uses Proposition 2.1 to eliminate certain terms based on index ordering (e.g., i > l implies Cᵢⱼ;ₖₗ = 0), reducing the number of possible non-zero coefficients.
• Derives a contradiction by showing that the required antisymmetry condition [u,v] = −iħ{u,v} + o(ħ) forces C¹₁₀;₀₁ − C¹₀₁;₁₀ = −i, but both terms are forced to zero, violating the Poisson bracket realization.

Experimental results

Research questions

• RQ1 Does every Hamiltonian Lie algebra action on a symplectic manifold admit an invariant star product?
• RQ2 Is the existence of an invariant symplectic connection sufficient for the existence of an invariant star product?
• RQ3 Can a geometrically invariant star product be consistently defined on ℝ² with the standard symplectic structure under a specific Lie algebra action?
• RQ4 What are the structural constraints on bidifferential operators in an invariant star product that lead to contradictions?
• RQ5 Does the absence of a non-degenerate invariant connection imply the non-existence of an invariant star product in general?

Key findings

• The paper constructs a specific Hamiltonian Lie algebra action on ℝ² with the standard symplectic form dx∧dy.
• It proves that no geometrically invariant star product can exist for this action by deriving a contradiction in the structure constants of the bidifferential operators.
• The contradiction arises from the requirement that C¹₁₀;₀₁ − C¹₀₁;₁₀ = −i to realize the Poisson bracket, while both coefficients are forced to zero by index constraints.
• The analysis shows that C¹₀₁;₁₀ = 0 and C¹₁₀;₀₁ = 0 due to the condition i > l implying vanishing coefficients, which violates the fundamental commutator relation.
• This result serves as a counterexample to the conjecture that invariant star products always exist for Hamiltonian actions, analogous to Van Hove’s no-go theorem.
• The work demonstrates that the existence of an invariant symplectic connection is not sufficient to guarantee an invariant star product, challenging a widely assumed principle in deformation quantization.