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Field Theory from a Bundle Point of View

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Field Theory from a Bundle Point of View Field theory from a bundle point of view Laurent Claessens December 8, 2008 2 Contents 1 Differential geometry 9 1.1 Differentiable manifolds ................................ 9 1.1.1 Definition and examples ............................ 9 Example: the sphere .............................. 10 Example: projective space ........................... 10 1.1.2 Topology on manifold and submanifold ................... 10 1.1.3 Tangent vector ................................. 12 1.1.4 Differential of a map .............................. 14 1.1.5 Tangent and cotangent bundle ........................ 15 Tangent bundle ................................. 15 Commutator of vector fields .......................... 16 Some Leibnitz formulas ............................ 16 Cotangent bundle ............................... 17 Exterior algebra ................................ 17 Pull-back and push-forward .......................... 18 Differential of k-forms ............................. 19 Hodge operator ................................. 19 Volume form and orientation ......................... 20 1.1.6 Musical isomorphism .............................. 20 1.1.7 Lie derivative .................................. 20 1.2 Example: Lie groups .................................. 22 1.2.1 Connected component of Lie groups ..................... 22 Õ 1.2.2 The Lie algebra of SUÔ2 ........................... 23 1.2.3 What is g ¡1dg ? ................................ 24 1.2.4 Exponential map ................................ 24 1.2.5 Invariant vector fields ............................. 24 1.2.6 Adjoint map .................................. 25 1.3 Fundamental vector field ................................ 26 1.4 Vector bundle ...................................... 27 1.4.1 Transition functions .............................. 28 1.4.2 Inverse construction .............................. 28 1.4.3 Equivalence of vector bundle ......................... 29 1.4.4 Sections of vector bundle ........................... 30 1.5 Vector valued differential forms ............................ 30 1.6 Lie algebra valued differential forms ......................... 31 1.7 Principal bundle .................................... 32 1.7.1 Transition functions .............................. 33 3 4 CONTENTS 1.7.2 Morphisms and such. ............................ 35 1.7.3 Frame bundle: first ............................... 36 1.7.4 Frame bundle: second ............................. 37 Basis ....................................... 37 Construction .................................. 37 1.7.5 Sections of principal bundle .......................... 38 1.7.6 Equivalence of principal bundle ........................ 40 Transition functions .............................. 40 1.7.7 Reduction of the structural group ...................... 41 1.7.8 Density ..................................... 42 1.8 Associated bundle ................................... 42 1.8.1 Transition functions .............................. 43 1.8.2 Sections on associated bundle ......................... 43 Equivariant functions ............................. 43 For the endomorphism of sections of E .................... 44 Local expressions ................................ 44 1.8.3 Associated and vector bundle ......................... 45 General construction .............................. 45 1.8.4 Equivariant functions for a vector field .................... 45 1.8.5 Gauge transformations ............................. 46 1.9 Adjoint bundle ..................................... 46 1.10 Connection on vector bundle: local description ................... 47 1.10.1 Connection and transition functions ..................... 48 1.10.2 Torsion and curvature ............................. 49 1.10.3 Divergence, gradient and Laplacian ...................... 50 1.11 Connexion on vector bundle: algebraic view ..................... 50 1.11.1 Exterior derivative ............................... 51 Covariant exterior derivative ......................... 52 Soldering form and torsion .......................... 52 Example : Levi-Civita ............................. 53 1.12 Connection on principal bundle ............................ 54 1.12.1 First definition: 1-form ............................ 54 1.12.2 Horizontal space ................................ 54 1.12.3 Curvature .................................... 56 1.13 Exterior covariant derivative and Bianchi identity .................. 57 1.14 Covariant derivative on associated bundle ...................... 57 1.14.1 Curvature on associated bundle ........................ 60 1.14.2 Connection on frame bundle .......................... 60 General framework ............................... 60 Levi-Civita connection ............................. 62 1.14.3 Holonomy .................................... 63 1.14.4 Connection and gauge transformation .................... 63 Local description ................................ 64 Covariant derivative .............................. 65 1.15 Product of principal bundle .............................. 65 1.15.1 Putting together principal bundle ....................... 65 1.15.2 Connections ................................... 66 1.15.3 Representations ................................. 67 CONTENTS 5 2 Decompositions of Lie algebras 69 2.1 Root spaces ....................................... 69 2.1.1 Cartan subalgebra ............................... 69 2.1.2 Cartan-Weyl basis ............................... 70 2.1.3 Cartan matrix ................................. 71 2.1.4 Dynkin diagram ................................ 73 2.1.5 Chevalley basis ................................. 73 2.2 Representations ..................................... 74 2.2.1 About group representations ......................... 75 2.2.2 Weyl group ................................... 76 2.2.3 List of the weights of a representation .................... 76 Finding all the weights of a representation .................. 77 2.2.4 Tensor product of representations ....................... 79 Tensor and weight ............................... 79 Decomposition of tensor products of representations ............ 80 Symmetrization and anti symmetrization .................. 81 2.3 Verma module ..................................... 82 Õ 2.4 The group SOÔ3 and its Lie algebra ......................... 83 2.4.1 Rotations of functions ............................. 84 Õ 2.4.2 Representations of SOÔ3 ........................... 84 Õ Ô Õ 2.4.3 Representations of the algebra suÔ2 so 3 ................. 85 Determination of the representations ..................... 85 Ladder operators ................................ 85 Weight vectors ................................. 87 Õ 2.5 Thinks about suÔ3 ................................... 88 3 From Clifford algebra to spin manifold 89 3.1 Invitation : Clifford algebra in quantum field theory ................ 89 3.1.1 Schrödinger, Klein-Gordon and Dirac .................... 89 3.1.2 Lorentz algebra ................................. 90 3.2 Clifford algebra ..................................... 91 3.2.1 Definition and universal problem ....................... 91 3.2.2 First representation .............................. 93 3.2.3 Some consequences of the universal property ................ 93 3.2.4 Trace ...................................... 94 3.3 Spinor representation .................................. 96 3.3.1 Explicit representation ............................. 99 3.3.2 A remark .................................... 100 3.3.3 General two dimensional Clifford algebra .................. 101 3.4 Spin group ....................................... 101 3.4.1 Studying the group structure ......................... 102 Õ 3.4.2 Redefinition of Spin ÔV ............................ 108 3.4.3 A few about Lie algebra ............................ 109 3.4.4 Grading ΛW .................................. 110 2 Ê 3.4.5 Clifford algebra for V .......................... 110 General definitions ............................... 110 The maps α and τ ............................... 112 The spin group ................................. 112 3.5 Clifford modules .................................... 113 6 CONTENTS 3.6 Spin structure ...................................... 115 3.6.1 Example: spin structure on the sphere S2 .................. 116 3.6.2 Spinor bundle .................................. 117 3.7 Dirac operator ..................................... 119 3.7.1 Preliminary definition ............................. 119 3.7.2 Definition of Dirac ............................... 121 2 3.8 Dirac operator on Ê .................................. 123 2 Ê Õ 3.8.1 Connection on SOÔ ............................. 124 3.8.2 Construction of γ ................................ 126 Õ 3.8.3 Covariant derivative on ΓÔS ......................... 126 2 3.8.4 Dirac operator on the euclidian Ê ...................... 127 4 Relativistic field theory 129 4.1 Mathematical framework of field theory ....................... 129 4.1.1 Axioms of the (quantum) relativistic field theory mechanics ........ 129 4.1.2 Symmetries and Wigner’s theorem ...................... 130 4.1.3 Projective representations ........................... 134 4.1.4 Representations and power expansions .................... 135 4.2 The symmetry group of nature ............................ 137 4.2.1 Spin and double covering ........................... 137 4.2.2 How to implement the Poincaré group .................... 139 4.2.3 Momentum operator .............................. 140 4.2.4 Pure Lorentz transformation ......................... 141 4.2.5 Rebuilding of a basis for H .........................
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