Gauge Fields, Knots And GravityWorld Scientific Publishing Company, 24 oct. 1994 - 480 pages 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. |
Expressions et termes fréquents
3-dimensional basis Bianchi identity called Chapter Chern Chern-Simons theory components compute coordinates cotangent covariant derivative curvature define diffeomorphism differential forms dimensions Einstein Einstein's equation 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 IR³ isomorphism isotopy Kauffman bracket Lagrangian Lie algebra Lie group linear link invariant linking number loop Lorentz connection magnetic field manifold mathematics Maxwell's equations metric Minkowski Note open set oriented p-forms particle path physics polynomial quantum field theory quantum gravity reader Reidemeister moves representation Riemann tensor rotation self-dual Show simply skein relations smooth spacetime Suppose symmetry tangent vector theorem topology trivial bundle vector bundle vector fields vector potential vector space wedge product wormhole write Yang-Mills equation αβ