Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Holonomies and the group of loops
- 2 Loop coordinates and the extended group of loops
- 3 The loop representation
- 4 Maxwell theory
- 5 Yang–Mills theories
- 6 Lattice techniques
- 7 Quantum gravity
- 8 The loop representation of quantum gravity
- 9 Loop representation: further developments
- 10 Knot theory and physical states of quantum gravity
- 11 The extended loop representation of quantum gravity
- 12 Conclusions, present status and outlook
- References
- Index
10 - Knot theory and physical states of quantum gravity
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Holonomies and the group of loops
- 2 Loop coordinates and the extended group of loops
- 3 The loop representation
- 4 Maxwell theory
- 5 Yang–Mills theories
- 6 Lattice techniques
- 7 Quantum gravity
- 8 The loop representation of quantum gravity
- 9 Loop representation: further developments
- 10 Knot theory and physical states of quantum gravity
- 11 The extended loop representation of quantum gravity
- 12 Conclusions, present status and outlook
- References
- Index
Summary
Introduction
In the previous two chapters we developed several aspects of the loop representation of quantum gravity. One of the main consequences of these developments is a radically new description of one of the symmetries of the theory: because of diffeomorphism invariance wavefunctions in the loop representation must be invariant under deformations of the loops, they have to be knot invariants. This statement is much more than a semantical note. Knot invariants have been studied by mathematicians for a considerable time and recently there has been a surge in interest in knot theory. Behind this surge of interest is the discovery of connections between knot theory and various areas of physics, among them topological field theories. We will see in this chapter that such connections seem to play a crucial role in the structure of the space of states of quantum gravity in the loop representation. As a consequence we will discover a link between quantum gravity and particle physics that was completely unexpected and that involves in an explicit way the non-trivial dynamics of the Einstein equation. Such a link could be an accident or could be the first hint of a complete new sets of relationships between quantum gravity, topological field theories and knot theory.
We will start this chapter with a general introduction to the ideas of knot theory. We will then develop the notions of knot polynomials and the braid group.
- Type
- Chapter
- Information
- Loops, Knots, Gauge Theories and Quantum Gravity , pp. 238 - 274Publisher: Cambridge University PressPrint publication year: 1996