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Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theory

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 Added by Nima Moshayedi
 Publication date 2020
  fields Physics
and research's language is English




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These are lecture notes for the course Poisson geometry and deformation quantization given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes theorem, de Rhams theorem and Frobenius theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinsteins tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introduction to Poisson geometry where we also cover Courant structures, Dirac structures, the local splitting theorem, symplectic foliations and Poisson maps. The fourth chapter is about deformation quantization where we cover the Moyal product, $L_infty$-algebras, Kontsevichs formality theorem, Kontsevichs star product construction through graphs, the globalization approach to Kontsevichs star product and the operadic approach to formality. The fifth chapter is about the quantum field theoretic approach to Kontsevichs deformation quantization where we cover functional integral methods, the Moyal product as a path integral quantization, the Faddeev-Popov and BRST method for gauge theories, infinite-dimensional extensions, the Poisson sigma model, the construction of Kontsevichs star product through a perturbative expansion of the functional integral quantization for the Poisson sigma model for affine Poisson structures and the general construction.



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We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra, respectively. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with some conditions are categorical equivalence. On the other hand, the categories of Poisson enveloping algebra is not equivalent to the other categories.
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