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
2 - Loop coordinates and the extended group of loops
- 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
Continuing with the idea of describing gauge theories in terms of loops, we will now introduce a set of techniques that will aid us in the description of loops themselves. The idea is to represent loops with a set of objects that are more amenable to the development of analytical techniques. The advantages of this are many: whereas there is limited experience in dealing with functions of loops, there is a significant machinery to deal with analytic functions. They even present advantages for treatment with computer algebra.
Surprisingly, we will see that the end result goes quite beyond our expectations. The quantities we originally introduced to describe loops immediately reveal themselves as having great potential to replace loops altogether from the formulation and go beyond, allowing the development of a reformulation of gauge theories that is entirely new. This formulation introduces new perspectives with respect to the loop formulation that have not been fully developed yet, though we will see in later chapters some applications to gauge theories and gravitation.
The plan for the chapter is as follows: in section 2.2 we will start by introducing a set of tensorial objects that embody all the information that is needed from a loop to construct the holonomy and therefore to reconstruct any quantity of physical relevance for a gauge theory. In section 2.3 we will show how the group of loops is a subgroup of a Lie group with an associated Lie algebra, the extended loop group.
- Type
- Chapter
- Information
- Loops, Knots, Gauge Theories and Quantum Gravity , pp. 29 - 51Publisher: Cambridge University PressPrint publication year: 1996