Theories des champs topologiques et mecanique quantique en espace non-commutatif

Abstract : In particle physics, the Standard Model describes the interactions between fundamental particles. However, it was not able until now to unify quantum field theory and general relativity. This PhD thesis focuses on two different unification approaches, though they might show some compatibility: topological field theories and quantum mechanics on noncommutative space.
Topological field theories have been introduced by Witten some twenty years ago and have a very strong link to mathematics: their observables are topological invariants of the manifold they are defined on. In this thesis, we firs give interest to topological Yang-Mills. We develop a superspace formalism and give a systematic method of determination of the observables. This approach allows, once projected on a particular supergauge (of Wess-Zumino type), to recover the existing results but it also gives a generalisation to the case of an unspecified supergauge. We have then be able to show that the up-to-now known observables correspond to the most general form of the solutions. This superspace formalism can be applied to more complex models; the case of topological gravity is given here in example.
Quantum mechanics on noncommutative space provides an extension of the Heisenberg algebra of ordinary quantum mechanics. What differs here is that the components of the position or momentum operators do not commute with each other anymore. This implies to introduce a fundamental length. The second part of this thesis focuses on the description of the commutation algebra. Applications are made to low-dimensional quantum systems (Landau system, harmonic oscillator...) and to supersymmetric systems.
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M. Lefrançois. Theories des champs topologiques et mecanique quantique en espace non-commutatif. Physique Nucléaire Théorique [nucl-th]. Université Claude Bernard - Lyon I, 2005. Français. ⟨tel-00012196⟩

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