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Aspects semi-classiques de la quantification géométrique

Abstract : In this thesis we study the Berezin-Toeplitz operators on Kähler manifolds and their generalisation to compact symplectic manifolds. The first chapter is devoted to the Feynman integral: we compute the quantum propagator of a Toeplitz operator as a limit of path integrals in terms of the classical action. In the second chapter, we give an ansatz for the Schwartz kernel of the Berezin-Toeplitz operators. From this, we recover the known results on these operators and we describe the calculus of the covariant and contravariant symbols in terms of the Kähler metric. This leads to the definition of some star-products on the Kähler manifolds by universal formulas. In the third chapter, we generalize the previous ansatz to quantize the Lagrangian submanifolds of Kähler manifolds. We apply this to construct quasi-modes, state the Bohr-Sommerfeld conditions and quantize the symplectomorphims. To compare with the theory of pseudo-differential operators, Kähler invariants replace the invariants of the cotangent spaces. In the last chapter, we generalize the previous results to compact symplectic manifolds.
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Contributor : Laurent Charles <>
Submitted on : Thursday, March 28, 2002 - 4:27:46 PM
Last modification on : Wednesday, September 23, 2020 - 4:30:53 AM
Long-term archiving on: : Tuesday, September 11, 2012 - 5:10:48 PM


  • HAL Id : tel-00001289, version 1



Laurent Charles. Aspects semi-classiques de la quantification géométrique. Mathématiques [math]. Université Paris Dauphine - Paris IX, 2000. Français. ⟨tel-00001289⟩



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