University of Zagreb. Faculty of Science. Department of Physics.
Abstract
Otvoreno pitanje u modernoj teorijskoj fizici predstavlja ujedinjenje kvantne mehanike i gravitacije unutar istog matematičkog formalizma. Mogući pristup problemu je da pretpostavimo da je na malim skalama prostorvrijeme kvantizirano te pokušamo konstruirati teoriju polja na takvoj strukturi. U ovom radu konstruiramo kvantizirane, odnosno nekomutativne varijante homogenih prostora, uzimajući u obzir zahtjev, po uzoru na princip korespondancije u kvantnoj mehanici, da postoji odgovarajući limes u kojem se nekomutativni prostor reducira na glatku mnogostrukost. Konkretna konstrukcija nekomutativnih analogona homogenih prostora ilustrirana je na primjerima kvantne mehanike i kvantizirane sfere. U kontekstu kvantizirane varijante klasične sfere uvodimo dinamičke nekomutativne prostore, baždarnu teoriju polja te osnovne koncepte kvantne teorije polja na nekomutativnim prostorima.An open question in modern theoretical physics is the union of quantum mechanics and gravity within the same mathematical framework. One approach to this problem is that we assume that spacetime is quantized on small scales and attempt to construct a field theory on such a structure. In this dissertation we construct quantized, that is, noncommutative analogues of homogeneous spaces, taking into consideration the request that, similarly to the correspondence principle in quantum mechanics, there is a corresponding limit in which the noncommutative space reduces to a smooth manifold. An explicit construction of noncommutative analogues of homogeneous spaces is illustrated on the examples of quantum mechanics and the quantized sphere. In the context of a quantized version of the classical sphere, we introduce dynamical noncommutative spaces, gauge field theory and basic concepts of quantum field theory on noncommutative spaces
