Description
Book SynopsisThis is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.
Table of ContentsElectromagnetism: Maxwell's Equations; Manifolds; Vector Fields and Tangent Vectors; Differential Forms; Rewriting Maxwell's Equations; De Rham Theory in Electromagnetism; Braids, Knots and Electromagnetism; Quantizing the Free Electromagnetic Field; Gauge Theory.- Gauge Groups; Connections and Parallel Transport; Curvature and the Yang-Mills Equations; Chern-Simons Theory; Link Invariants from Gauge Theory. Gravity: Riemannian and Lorentzian Geometry; Einstein's Equations; The Lagrangian Approach to General Relativity; The ADM Formalism; The New Variables; Knots and Quantum Gravity.
