Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics

Mark J. D. Hamilton - Collection Yellow Sale 2023

658 pages, parution le 09/01/2018

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Résumé

This book explains the mathematical background behind the Standard Model, translating ideas from physics into a mathematical language and vice versa.The Standard Model is the foundation of modern particle and high energy physics. This book explains the mathematical background behind the Standard Model, translating ideas from physics into a mathematical language and vice versa.Part I Mathematical foundations1 Lie groups and Lie algebras: Basic concepts1.1 Topological groups and Lie groups1.2 Linear groups and symmetry groups of vector spaces1.3 Homomorphisms of Lie groups1.4 Lie algebras1.5 From Lie groups to Lie algebras1.6 From Lie subalgebras to Lie subgroups1.7 The exponential map1.8 Cartan's Theorem on closed subgroups1.9 Exercises for Chapter 12 Lie groups and Lie algebras: Representations and structure theory2.1 Representations2.2 Invariant metrics on Lie groups2.3 The Killing form2.4 Semisimple and compact Lie algebras2.5 Ad-invariant scalar products on compact Lie groups2.6 Homotopy groups of Lie groups2.7 Exercises for Chapter 23 Group actions3.1 Transformation groups3.2 Definition and first properties of group actions3.3 Examples of group actions3.4 Fundamental vector fields3.5 The Maurer-Cartan form and the differential of a smooth group action3.6 Left or right actions?3.7 Quotient spaces3.8 Homogeneous spaces3.9 Stiefel and Grassmann manifolds3.10 The exceptional Lie group G23.11 Godement's Theorem on the manifold structure of quotient spaces3.12 Exercises for Chapter 34 Fibre bundles4.1 General fibre bundles4.2 Principal fibre bundles4.3 Formal bundle atlases4.4 Frame bundles4.5 Vector bundles4.6 The clutching construction4.7 Associated vector bundles4.8 Exercises for Chapter 45 Connections and curvature5.1 Distributions and connections5.2 Connection 1-forms5.3 Gauge transformations5.4 Local connection 1-forms and gauge transformations5.5 Curvature5.6 Local curvature 2-forms5.7 Generalized electric and magnetic fields on Minkowski spacetime of dimension 45.8 Parallel transport5.9 The covariant derivative on associated vector bundles5.10 Parallel transport and path-ordered exponentials5.11 Holonomy and Wilson loops5.12 The exterior covariant derivative5.13 Forms with values in Ad(P)5.14 A second and third version of the Bianchi identity5.15 Exercises for Chapter 56 Spinors6.1 The pseudo-orthogonal group O(s