The Weyl Quantization and the Quantum Group Quantization of the Moduli Space of Flat SU(2)-Connections on the Torus are the Same

2002/01/28 by Razvan Gelca, Răzvan Gelca, Alejandro Uribe · 11 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algorithm #Canonical quantization #Geometric quantization #Geometry #Mathematical physics #Mathematics #Moduli space #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Physics #Pure mathematics #Quantization (signal processing) #Quantum #Quantum gravity #Quantum mechanics #Torus #math-ph #math.MP #math.QA #msc:57M27 #msc:81S10 #msc:81T45

paper · pdf · doi:10.1007/s00220-002-0759-3

published in Communications in Mathematical Physics 233(3), 493-512 (Springer Science+Business Media) · 18 pages, latex format, 2 figures

arxiv created 2002/01/28 · openalex publication_date 2003/02/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We prove that, for the moduli space of flat SU(2)-connections on the torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) are unitarily equivalent. This is done by comparing the matrices of the operators associated by the two quantization to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus.

Citations

• The Geometry and Physics of Knots
• A product-to-sum formula for the quantum group of SL(2,C)
• Quantum field theory and the Jones polynomial
• Invariants of 3-manifolds via link polynomials and quantum groups