Topology in Fundamental Physics

Abstract

In this thesis I present a mathematical tool for understanding the spin networks that arise from the study of the loop states of quantum gravity. The spin networks that arise in quantum gravity possess more information than the original spin networks of Penrose: they are embedded within a manifold and thus possess topological information. There are limited tools available for the study of this information. To remedy this I introduce a slightly modi ed mathematical object - Braided Ribbon Networks - and demonstrate that they can be related to spin networks in a consistent manner which preserves the di eomorphism invariant character of the loop states of quantum gravity.

Given a consistent de nition of Braided Ribbon Networks I then relate them back to previous trinion based versions of Braided Ribbon Networks. Next, I introduce a consistent evolution for these networks based upon the duality of these networks to simplicial complexes. From here I demonstrate that there exists an invariant of this evolution and smooth deformations of the networks, which captures some of the topological information of the networks.

The principle result of this program is presented next: that the invariants of the Braided Ribbon Networks can be transferred over to the original spin network states of loop quantum gravity.

From here we represent other advances in the study of braided ribbon networks, accompanied by comments of their context given the consistent framework developed earlier including: the meaning of isolatable substructures, the particular structure of the capped three braids in trivalent braided ribbon networks and their application towards emergent particle physics, and the implications of the existence of microlocal topological structures in spin networks.

Lastly we describe the current state of research in braided ribbon networks, the implications of this study on quantum gravity as a whole and future directions of research in the area.