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Meyer, Frank (2006): Gauge-Field Theories and Gravity on Noncommutative Spaces. Dissertation, LMU München: Faculty of Physics
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Abstract

In this thesis gauge-field theories and gravity on noncommutative spaces are studied. We start with an introduction to the concepts underlying the construction of field theories on noncommutative spaces. By a noncommutative space we mean a noncommutative algebra, which replaces the algebra of functions on ordinary space. We construct derivatives and deformed symmetries ("Quantum Group" symmetries) acting on noncommutative spaces. Consistency requires us to change the action on a product of representations ("deformed coproducts"); this gives rise in particular to deformed Leibniz rules. We also show how a noncommutative space and the generators of deformed symmetries acting on it can be represented on the ordinary algebra of functions; the commutative, point-wise product is substituted by a noncommutative one ("star-product"). One possible way to define gauge-field theories on noncommutative spaces is to construct "Seiberg--Witten maps". In this approach it is possible to express all noncommutative quantities in terms of their commutative counterparts. We illustrate this by two examples, the two-dimensional q -deformed Euclidean plane and the \kappa -deformed Minkowski space-time. In addition gauge-field theory on "fuzzy" S^{2}\times S^{2} is discussed as a multi-matrix model. We show that this model reduces in an appropriate limit to gauge-field theory on noncommutative \mathbb{R}^{4} . We also present a new approach to deformed gauge theories, which is based on "twisted" gauge transformations. In this setting new fields occur in addition to the usual gauge fields. Consistent equations of motion and conserved currents are obtained. This is the first time that conservation laws have been derived from a generalized, Quantum Group symmetry. We discuss in detail how to construct deformed infinitesimal diffeomorphisms by deformations via generic "twists". Then we construct gravity as a theory, which is covariant with respect to these diffeomorphisms. This leads to a deformation of Einstein's equations. For canonically deformed spaces, a deformed Einstein--Hilbert action can be even defined. It reduces to the usual Einstein--Hilbert action in the commutative limit. All relevant quantities are expanded in terms of the usual, commutative fields up to second order in the deformation parameter.