Sur quelques problèmes de quantification : en espace-temps de de Sitter et par états cohérents

Abstract : This manuscript is devoted to quantization problems in mathematical and theoretical physics. It is divided into two subjects: conformally invariant fields on de Sitter spacetime and quantization through coherent states. • The first subject follows a comprehensive program of covariant quantization of fields in de Sitter spacetime, whose spirit is close to the Wightman axiomatic. We paid a particular attention to the quantization of fields that could be (naturally) extended to the conformal group. We developed a geometric point of view enabling us to link fields on (anti-)de Sitter and Minkowski spacetimes while keeping transparent the action of the conformal group. This geometric viewpoint led us to the expression of a new propagator for the vector potential. We thus reached the simplest and the most compact form ever seen for this object. Finally, this approach allowed us to set up a general framework to compute the propagators of higher rank conformally invariant tensors in de Sitter spacetime. • The second subject deals with coherent states (CS) and their use in quantization problems. It is well known that many classical observables cannot be quantized by following canonical quantization rules, depending on the geometry or topology of the phase space. Quantization through coherent states and their generalizations avoid to a certain extent those drawbacks, or at least might give us hints on how to circumvent them. The particle in an infinite well is a toy model for this type of CS quantization since the momentum operator, in spite of being symmetric, is not self-adjoint and there is impossibility of canonical commutation rule (Pauli Theorem). Thanks to a new family of 2-component vector-valued coherent states, we were able, to quantize, in a consistent way, the particle in the infinite well. Finally, we began the fuzzyfication of the hyperboloid, that is the quantization of the de Sitter spacetime itself, through generalized coherent states.
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Julien Queva. Sur quelques problèmes de quantification : en espace-temps de de Sitter et par états cohérents. Physique mathématique [math-ph]. Université Paris-Diderot - Paris VII, 2009. Français. ⟨tel-00503186⟩

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