Loops, Knots, Gauge Theories and Quantum Gravity

Abstract

Now in paperback, this text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text begins with a detailed review of loop representation theory. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed in detail. Following chapters move on to consider knot theories, braid theories and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research.

A partial elucidation of the gauge principle

“…D) The resultant quantum theory could be very difficult to renormalize. For details on these problems and others see (Gambini & Pullin, 1996).…”

Geometric foundations of classical Yang–Mills theory

“…Hence, inequivalent flat connections will define different holonomy groups. Since the so-called Wilson loops are gauge invariant observables obtained from the holonomy defined by a connection, inequivalent flat connections will define different physical observables (Gambini & Pullin, 1996). Hence, different elections of a background flat connection might have different physical consequences.…”

“…In this case, the minimal ontology of matter fields is extended by introducing additional degrees of freedom that describe the dynamics of the physical connection. 15 Since in Yang-Mills theory the degrees of freedom that describe the geometry of the fiber bundle are coupled to matter fields (instead of being completely determined by them), Yang-Mills theory can be understood to extend the Einstenian programme to physical interactions other than gravity. 16 It is worth noting that the background independence at stake in Yang-Mills theory exclusively concerns the geometry of the fiber bundle.…”

“…This problem can be solved by using framed connections (Donaldson & Kronheimer, 1990). It can be shown 15 In order to consider the spatiotemporal connection A as a dynamic entity, it it necessary to foliate space-time M by spacelike hypersurfaces. In other words, it is necessary to choose a diffeomorphism ι : M → M × R, where M is a smooth 3-dimensional riemannian manifold.…”

Loop and Spin Foam Quantum Gravity: A Brief Guide for Beginners

“…The situation becomes considerably more complicated for wave functions which contain a spin network vertex which lies in the surface S; in this case the area operator does not necessarily act diagonally anymore (see figure 4). Expression (9) lies at the core of the statement that areas are quantised in LQG. The construction of the volume operator follows similar logic, although it is substantially more involved.…”