| keyword(s) : Equation de Klein-Gordon – géométrie multisymplectique – EDP variationnelles – théorie des champs quantiques – quantification géométrique – quantification par déformation – arbres plans – théorie des perturbations – séries de Butcher – contrôle non--linéaire. |
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| Field theory : multisymplectic approach of quantization procedure, perturbative theory and application |
The main subject of this thesis is the study of the Klein--Gordon equation together with an interaction term and the quantization of this theory from the multisymplectic point of view. Multisymplectic geometry provides a general framework for a covariant finite dimensional Hamiltonian formulation of variational problems with several variables.
In the first part we study the linear Klein--Gordon equation (free fields). We propose a description of the canonical quantization of free field from the multisymplectic point of view. We investigate three approachs : the algebraic approach by giving a representation of the Lie algebra of the symetries, the deformation point of view and finally we introduce a notion of multisymplectic geometric quantization.
In the second part we study the classical phi^p-theory. First we define explicitely a conserved quantity using a perturbative expansion based on planar trees and a kind of Feynman rule. Then we link this expansion with Butcher series which describe the perturbative expansion of the solutions of some PDE and we show how Butcher series can be related to perturbative quantum theory. Finally we see how we can apply our result in order to solve problems from control theory. |
| english keyword(s) : Klein-Gordon equation – variational PDE's – quantum field theory – multisymplectic geometry – geometric quantization – deformation quantization – planar trees – perturbatives theory – Butcher series – nonlinear control. |
