Abstract
In particle physics, the Standard Model describes the interactions between fundamental particles. However, it was not able till now to unify quantum field theory and general relativity. This thesis focuses on two different unification approaches, though they might show some compatibility: topological field theories and quantum mechanics on non-commutative space. Topological field theories have been introduced 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 first give interest to topological Yang-Mills. We develop a superspace formalism and give a systematic method for the determination of the observables. This approach allows, once projected on a particular super gauge (of Wess-Zumino type), to recover the existing results but it also gives a generalisation to the case of an unspecified super-gauge. 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
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Citation Formats
Lefrancois, M.
Topological field theories and quantum mechanics on commutative space; Theories des champs topologiques et mecanique quantique en espace non-commutatif.
France: N. p.,
2005.
Web.
Lefrancois, M.
Topological field theories and quantum mechanics on commutative space; Theories des champs topologiques et mecanique quantique en espace non-commutatif.
France.
Lefrancois, M.
2005.
"Topological field theories and quantum mechanics on commutative space; Theories des champs topologiques et mecanique quantique en espace non-commutatif."
France.
@misc{etde_20823419,
title = {Topological field theories and quantum mechanics on commutative space; Theories des champs topologiques et mecanique quantique en espace non-commutatif}
author = {Lefrancois, M}
abstractNote = {In particle physics, the Standard Model describes the interactions between fundamental particles. However, it was not able till now to unify quantum field theory and general relativity. This thesis focuses on two different unification approaches, though they might show some compatibility: topological field theories and quantum mechanics on non-commutative space. Topological field theories have been introduced 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 first give interest to topological Yang-Mills. We develop a superspace formalism and give a systematic method for the determination of the observables. This approach allows, once projected on a particular super gauge (of Wess-Zumino type), to recover the existing results but it also gives a generalisation to the case of an unspecified super-gauge. 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. (author)}
place = {France}
year = {2005}
month = {Dec}
}
title = {Topological field theories and quantum mechanics on commutative space; Theories des champs topologiques et mecanique quantique en espace non-commutatif}
author = {Lefrancois, M}
abstractNote = {In particle physics, the Standard Model describes the interactions between fundamental particles. However, it was not able till now to unify quantum field theory and general relativity. This thesis focuses on two different unification approaches, though they might show some compatibility: topological field theories and quantum mechanics on non-commutative space. Topological field theories have been introduced 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 first give interest to topological Yang-Mills. We develop a superspace formalism and give a systematic method for the determination of the observables. This approach allows, once projected on a particular super gauge (of Wess-Zumino type), to recover the existing results but it also gives a generalisation to the case of an unspecified super-gauge. 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. (author)}
place = {France}
year = {2005}
month = {Dec}
}