Sykora, Andreas (2004): The application of starproducts to noncommutative geometry and gauge theory. Dissertation, LMU München: Faculty of Physics 

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Abstract
Due to the singularities arising in quantum field theory and the difficulties in quantizing gravity it is often believed that the description of spacetime by a smooth manifold should be given up at small length scales or high energies. In this work we will replace spacetime by noncommutative structures arising within the framework of deformation quantization. The ordinary product between functions will be replaced by a *product, an associative product for the space of functions on a manifold. We develop a formalism to realize algebras defined by relations on function spaces. For this porpose we construct the Weylordered *product and present a method how to calculate *products with the help of commuting vector fields. Concepts developed in noncommutative differential geometry will be applied to this type of algebras and we construct actions for noncommutative field theories. In the classical limit these noncommutative theories become field theories on manifolds with nonvanishing curvature. It becomes clear that the application of *products is very fruitful to the solution of noncommutative problems. In the semiclassical limit every *product is related to a Poisson structure, every derivation of the algebra to a vector field on the manifold. Since in this limit many problems are reduced to a couple of differential equations the *product representation makes it possible to construct noncommutative spaces corresponding to interesting Riemannian manifolds. Derivations of *products makes it further possible to extend noncommutative gauge theory in the SeibergWitten formalism with covariant derivatives. The resulting noncommutative gauge fields may be interpreted as one forms of a generalization of the exterior algebra of a manifold. For the Formality *product we prove the existence of the abelian SeibergWitten map for derivations of these *products. We calculate the enveloping algebra valued non abelian SeibergWitten map pertubatively up to second order for the Weylordered *product. A general method to construct actions invariant under noncommutative gauge transformations is developed. In the commutative limit these theories are becoming gauge theories on curved backgrounds. We study observables of noncommutative gauge theories and extend the concept of so called open Wilson lines to general noncommutative gauge theories. With help of this construction we give a formula for the inverse abelian SeibergWitten map on noncommutative spaces with nondegenerate *products.
Item Type:  Thesis (Dissertation, LMU Munich) 

Subjects:  600 Natural sciences and mathematics 600 Natural sciences and mathematics > 530 Physics 
Faculties:  Faculty of Physics 
Language:  English 
Date Accepted:  2. December 2004 
1. Referee:  Wess, Julius 
Persistent Identifier (URN):  urn:nbn:de:bvb:1928937 
MD5 Checksum of the PDFfile:  acebd6daa8dbf05a8235378d423694fe 
Signature of the printed copy:  0001/UMC 14257 
ID Code:  2893 
Deposited On:  16. Feb 2005 
Last Modified:  16. Oct 2012 07:44 