Gauge Fields and Cartan-Ehresmann Connections: Part A
By addebook • Oct 3rd, 2008 • Category: Mathematics •
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Gauge Fields and Cartan-Ehresmann Connections: Part A (Interdisciplinary Mathematics Series, No 10)
Gauge Fields and Cartan-Ehresmann Connections: Part A (Interdisciplinary Mathematics Series, No 10)
By Robert Hermann
Publisher: Math Science Pr
Number Of Pages:
Publication Date: 1975-06
ISBN-10 / ASIN: 0915692090
ISBN-13 / EAN: 9780915692095
Binding: Paperback
Contents REVIEW OF THE THEORY OF FIELD LAGRANGIANS 1 JET SPACES, coordinate system, equivalence relation Lagrangians Cartan Forms and Symmetries 11 vector field, diffeomorphism, Lagrangian Lagrangians Invariant Under a Given 18 vector fields, LAGRANGIANS, infinitesimal Affine Gauge Vector Field System 25 affine map, linear map, gauge transformations Operators Which Intertwine Affine Groups 33 linear map, gauge transformation, affine maps THE GENERAL FORM OF THE MAXWELLYANG 45 Lie algebra, magnetic potential, vector space GaugeInvariance of the MaxwellYangMills 52 Jacobi identity, algebra structure, Yang-Mil EHRESMANN CONNECTIONS FOR FIBF R SPACES 59 EHRESMANN CONNECTIONS, complete vector field, integral curve INTEGRABILITY AND CURVATURE OF EHRESMANN 85 EHRESMANN CONNECTIONS, complete vector field, covering map Curvature in Terms of Forms on the Base 93 curvature forms, moving frame, differential forms AFFINE CONNECTIONS 101 affine connection, vector bundle, linear connection THE AFFINE CONNECTION G STRUCTURE 129 G-structure, frame bundle, Riemannian metric Affine Connections Associated with Cartan 139 torsion tensor, structure group, skew-symmetric Groups of Matrices with the Weyl 145 orthogonal groups, bilinear forms, skew-symmetric Real Matrix Groups Which Have the Weyl 156 linear subspace, Lie groups, non-singular Groups with the WeylCartan Property 163 simple Lie algebra, tensor product, fundamental representations GEODESICS OF HYPERBOLIC RIEMANNIAN 173 covariant derivative, tangent vector, RIEMANNIAN METRICS The First and Second Variation 184 space-like, submanifolds, homotopy The Exponential Map and Local Convexity 193 EXPONENTIAL MAP, open subset, geodesic Convex Functions and Convex Regions 200 convex functions, Whitehead’s theorem, open subsets Conjugate Points and Local Maximization 208 Riemannian manifolds, conjugate point, DGCV The HamiltonJacobi Equation 217 HAMILTON-JACOBI EQUATION, Jacobi equation, grad f PROJECTIVE CONNECTIONS 231 Ehresmann connection, linear connection, projective connection PRCJECTIVE HOMOMORPHISMS OF AFFINE 275 linear subspace, connection homomorphism, equivalence class Projective Homomorphisms of Affine 289 homomorphism, morphism, Ybca The Relation Between Projective 296 connection on T(N, homomorphism, T(N x Equivalence of the Cartan and Weyl 303 conformal connection, Ricci tensor, scalar curvature CONFORMAL GEOMETRY THE HYDROGEN 323 Poisson bracket, Lie derivative, symmetry group The Resolvent of Greens Function of 330 GREEN’S FUNCTION, Hilbert space, conformal map Conformal Transformations of the Planar 342 conformal geometry, harmonic oscillator, Hamiltonian POISSON BRACKET AND QUANTIZATION 349 linearly independent, POISSON BRACKET, Euler-Lagrange equations The Quantization Problem for Degenerate 368 Poisson brackets, classical mechanics, quantization Hamiltonian Formalism for a Spinning 374 symplectic structure, Hamilton-Jacobi theory, Hamilton equations Classical and Quantum Mechanical Non 382 Hilbert space, linear representation, Lie subalgebra The Poisson Bracket Lie Algebra Associated 391 product bundle, cotangent bundle, symplectic structure The Poisson Bracket Lie Algebra and 399 Casimir operator, UNIVERSAL ENVELOPING ALGEBRA, dual space Cosymplectic Manifolds 406 dual space, Jacobi identity, contravariant Homomorphisms of Cosymplectic Manifolds 412 cosymplectic structure, algebra homomorphism, tensor DIRAC EQUATIONS AND SPINORIAL CONNECTIONS 419 Dirac equation, flat connection, non-singular Definition of a Dirac Equation on 429 Dirac equation, Dirac operator, MINKOWSKI SPACE Differential Operators with Solutions 438 existence theorem, linear differential operator, solution of D The Dirac Equation and the KleinGordon 447 Dirac operator, Laplace-Beltrami operator, KLEIN-GORDON EQUATION Holonomy Group and Curvature 457 holonomy group, parallel transport, Riemannian manifold Remarks on Quantum Gravitation Theory 502 quantum gravitation, Bianchi identities, Einstein equations Bibliography 511 Quantum field theories, Current algebra, Fourier Analysis
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