Gauge Fields, Knots And Gravity
This 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.
Autres éditions - Tout afficher
3-dimensional basis Bianchi identity called Chapter Chern Chern-Simons theory components compute coordinates cotangent covariant derivative curvature define diagram diffeomorphism differential forms dimensions Einstein Einstein's equation electric field electromagnetism End(E End(E)-valued 1-form example Exercise exterior derivative fact fiber formula frame field function f gauge theory gauge transformation geometry given Hamiltonian Hodge star operator holonomy integral isomorphism isotopy Kauffman bracket Lagrangian Lie algebra Lie group linear link invariant linking number loop Lorentz connection magnetic field manifold mathematics Maxwell's equations Minkowski Note open set oriented p-forms particle path physics polynomial quantum field theory quantum gravity reader Reidemeister moves relativity representation Riemann tensor rotation self-dual Show skein relations smooth spacetime Suppose symmetry tangent bundle tangent space tangent vector theorem topology trivial bundle vector bundle vector fields vector potential vector space volume form wedge product wormhole write Yang-Mills equations