Symplectic Geometry and Quantizaton

Annotation

The course gives a comprehensive introduction into the basics of symplectic geometry and quantization. These deeply interrelated subjects provide a geometrical insight into the classical and quantum description of fundamental physical systems. We begin with basic structures of symplectic geometry such as Poisson brackets, hamiltonian fields, momentum maps, Atiyah-Bott theory etc. and their properties. After this we give an introduction to the concept of quantization in general and illustrate it by standard simple examples. A particular attention is paid to the concept of symbols and deformations. We then consider Fedosov’s construction of deformation quantization of general symplectic manifolds. Finally, a more general approach of Kontsevich is presented together with the elements of general deformation theory such as Hochschild cohomology, homotopy algebras etc.

Course plan

1. Symplectic structures on manifolds, Poisson brackets, Hamiltonian vector fields.
2. Darboux theorem; Liouville’s integrability theory.
3. Schouten-Nijenhuis brackets and Poisson structures. Weinstein theorem (analog of Darboux’s theorem). Symplectic leaves and foliations.
4. Complex analytic symplectic structures, elements of Kaehler geometry.
5. Symplectic group actions and momentum maps; Atiyah-Bott theorem. Examples.
6. General concepts of quantization. Operator algebras and representation spaces. Canonical and Weyl quantization. Symbols of operators and star-products.
7. Deformation quantization. Weyl-Moyal star-product, quantization of cotangent bundles and algebras of differential operators. 
8. Fedosov’s quantization and algebraic index formula.
9. General deformation quantization problem for Poisson manifolds. Universal enveloping algebra as the quantization of the dual Lie algebra. Deformations of associative algebras, Hochschild complex.
10. Kontsevich’s universal quantization formula, graph complex and higher homotopy structures.

Literature

1. V. Arnold, Mathematical methods in classical mechanics. Springer-Verlag New York, 1989
2. V. Guillemin, Sh. Sternberg, Symplectic techniques in Physics.Cambridge University Press, 1984
3. L. A. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008 
4. A. Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2008
5. F. A. Berezin, General concept of quantization. Comm. Math. Phys. 40(2): 153-174 (1975).
6. B. Fedosov, Deformation Quantization and Index Theory. Akademie-Verl., 1996 
7. M. Kontsevich (2003), Deformation Quantization of Poisson Manifolds, Letters of Mathematical Physics 66, pp. 157–216. arXiv:q-alg/9709040