In the summer semester 2025, we will discuss deformation quantization in the Poisson geometry reading seminar. Formal deformation quantization is a mathematical framework that describes the transition from classical to quantum mechanics by deforming the algebra of functions on phase space. It replaces the classical pointwise product with a noncommutative star product parameterized by Planck’s constant, capturing quantum effects within a formal expansion. This approach provides a systematic way to study the relationship between classical and quantum mechanics while connecting to areas like symplectic geometry and Poisson structures.
We meet on Tuesdays at 14:00h in Seminarraum 1. Here is a tentative schedule:
08.04. (Jorn) Introduction and star products [Esp14, §3.1–3.3], [Wal07, §6.1].
15.04 (Christoph) Quantizations of linear Poisson structures [Sch22, §3.3], [Gut83].
22.04. (Christoph) Deformations and differential graded Lie algebras [Esp14, §3.4], [Wal07, §6.2.3], [CKTB05, §2.1].
29.04 & 06.05. (Rodrigo) Deformation theory of associative algebras [Esp14, §3.5–3.6], [Wal07, §6.2], [CKTB05, §3.1–3.5].
13.05. (Christoph) Fedosov’s construction I [Sch22, §3.4], [Wal07, §6.4], [Fed95, Ch. 3–5].
20.05. (Ioan) Fedosov’s construction II [Sch22, §3.4], [Wal07, §6.4], [Fed95, Ch. 3–5].
27.05 & 03.06. (Jorn) Kontsevich’s formality theorem [Esp14, §3.7–4.2], [CKTB05, Ch. 2], [Kon03], [CFT02].
10.06. No seminar due to lecture-free week of Pentecost.
17.06 & 24.06. (TBD) Operads and Tamarkin’s approach [CKTB05, §4], [MSS02, Ch. 1].
01.07. No seminar due to the “Interactions of Poisson Geometry, Lie Theory and Symmetry” conference in Lisbon.
08.07. Buffer/Guest lecture
15.07. Buffer/Guest lecture
References
[CFT02] Alberto Cattaneo, Giovanni Felder and Lorenzo Tomassini. ‘From local to global deformation quantization of poisson manifolds’. Duke Mathematical Journal, 115(2), November 2002.
[CKTB05] Alberto Cattaneo, Bernhard Keller, Charles Torossian and Alain Bruguieres. ‘Déformation, Quantification, Théorie de Lie’. Panoramas et Synthese, 2005.
[Esp14] Chiara Esposito. ‘Formality Theory. From Poisson Structures to Deformation Quantization’, volume 2 of SpringerBriefs in Mathematical Physics. Springer Cham, 2014.
[Fed95] Boris Fedosov. ‘Deformation Quantization and Index Theory’. Wiley, 1995.
[Gut83] S. Gutt. ‘An explicit *-product on the cotangent bundle of a Lie group‘. Letters in Mathematical Physics, 7:249–258, 1983.
[Kon03] Maxim Kontsevich. ‘Deformation Quantization of Poisson Manifolds’. Letters in Mathematical Physics, 66:157–216, 2003.
[MSS02] Martin Markl, Steve Shnider and Jim Stasheff. ‘Operads in Algebra, Topology and Physics’, volume 96 of Mathematical Surveys and Monographs. American Mathematical Society, 2002.
[Sch22] Jonas Schnitzer. ‘Poisson Geometry and Deformation Quantization’. https://home.mathematik.uni-freiburg.de/geometrie/lehre/ws2022/ DQ/pdf/skript.pdf, 2022.
[Wal07] Stefan Waldmann. ‘Poisson-Geometrie und Deformationsquantisierung. Eine Einführung’. Springer Berlin, Heidelberg, 2007.