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

فهرست مطالب :

Preface
Acknowledgements
Conventions
General
Linear Algebra
Groups
Manifolds
Diagrams
Contents
Part I Mathematical Foundations
1 Lie Groups and Lie Algebras: Basic Concepts
1.1 Topological Groups and Lie Groups
1.1.1 Normed Division Algebras and the Quaternions
1.1.2 Quaternionic Matrices
1.1.3 Products and Lie Subgroups
1.2 Linear Groups and Symmetry Groups of Vector Spaces
1.2.1 Isomorphism Groups of Vector Spaces
1.2.2 Automorphism Groups of Structures on Vector Spaces
1.2.3 Connectivity Properties of Linear Groups
1.3 Homomorphisms of Lie Groups
1.4 Lie Algebras
1.5 From Lie Groups to Lie Algebras
1.5.1 Vector Fields Invariant Under Diffeomorphisms
1.5.2 Left-Invariant Vector Fields
1.5.3 Induced Homomorphisms
1.5.4 The Lie Algebra of the General Linear Groups
1.5.5 The Lie Algebra of the Linear Groups
1.6 From Lie Subalgebras to Lie Subgroups
1.7 The Exponential Map
1.7.1 The Exponential Map for General Lie Groups
1.7.2 The Exponential Map of Tori
1.7.3 The Matrix Exponential
1.8 Cartan\'s Theorem on Closed Subgroups
1.9 Exercises for Chap. 1
2 Lie Groups and Lie Algebras: Representations and Structure Theory
2.1 Representations
2.1.1 Basic Definitions
2.1.2 Linear Algebra Constructions of Representations
2.1.3 The Weyl Spinor Representations of SL(2,C)
2.1.4 Orthogonal and Unitary Representations
Existence of Invariant Scalar Products
Decomposition of Representations
Unitary Representations of Non-Compact Lie Groups
2.1.5 The Adjoint Representation
2.2 Invariant Metrics on Lie Groups
2.3 The Killing Form
2.4 Semisimple and Compact Lie Algebras
2.4.1 Simple and Semisimple Lie Algebras in General
2.4.2 Compact Lie Algebras
2.4.3 Compact Lie Groups
2.5 Ad-Invariant Scalar Products on Compact Lie Groups
2.6 Homotopy Groups of Lie Groups
2.7 Exercises for Chap. 2
3 Group Actions
3.1 Transformation Groups
3.2 Definition and First Properties of Group Actions
3.3 Examples of Group Actions
3.4 Fundamental Vector Fields
3.5 The Maurer–Cartan Form and the Differential of a Smooth Group Action
3.5.1 Vector Space-Valued Forms
3.5.2 The Maurer–Cartan Form
3.5.3 The Differential of a Smooth Group Action
3.6 Left or Right Actions?
3.7 Quotient Spaces
3.7.1 Quotient Spaces Under Equivalence Relations on Topological Spaces
3.7.2 Quotient Spaces Under Equivalence Relations on Manifolds
3.7.3 Quotient Spaces Under Continuous Group Actions
3.7.4 Proper Group Actions
3.7.5 Quotient Spaces Under Smooth Group Actions
3.8 Homogeneous Spaces
3.8.1 Groups and Homogeneous Spaces
3.8.2 Topological Groups and Homogeneous Spaces
3.8.3 Lie Groups and Homogeneous Spaces
3.9 Stiefel and Grassmann Manifolds
3.10 The Exceptional Lie Group G2
3.10.1 Definition of the 3-Form ϕ and the Lie Group G2
3.10.2 G2 as a Compact Subgroup of SO(7)
3.10.3 An SU(2)-Subgroup of G2
3.10.4 The Dimension of G2
3.11 Godement\'s Theorem on the Manifold Structure of Quotient Spaces
3.11.1 Preliminary Facts
3.11.2 The Slice Theorem
3.11.3 Slices for Saturated Neighbourhoods and Proof of Godement\'s Theorem
3.12 Exercises for Chap. 3
4 Fibre Bundles
4.1 General Fibre Bundles
4.1.1 Definition of Fibre Bundles
4.1.2 Bundle Maps
4.1.3 Bundle Atlases
4.1.4 Pullback Bundle
4.1.5 Sections of Bundles
4.2 Principal Fibre Bundles
4.2.1 Definition of Principal Bundles
4.2.2 Principal Bundles Defined by Principal Group Actions
4.2.3 Bundle Morphisms, Reductions of the Structure Group and Gauges
4.3 Formal Bundle Atlases
4.4 Frame Bundles
4.5 Vector Bundles
4.5.1 Definitions and Basic Concepts
4.5.2 Linear Algebra Constructions for Vector Bundles
4.6 The Clutching Construction
4.7 Associated Vector Bundles
4.7.1 Basic Concepts
4.7.2 Adapted Bundle Atlases for Associated Vector Bundles
4.7.3 Bundle Metrics on Associated Vector Bundles
4.7.4 Examples
4.8 Exercises for Chap. 4
5 Connections and Curvature
5.1 Distributions and Connections
5.1.1 The Vertical Tangent Bundle
5.1.2 Ehresmann Connections
5.2 Connection 1-Forms
5.2.1 Basic Definitions
5.2.2 A Connection 1-Form on the Hopf Bundle S3→S2
5.3 Gauge Transformations
5.3.1 Bundle Automorphisms as G-Valued Maps on P
5.3.2 Physical Gauge Transformations
5.3.3 The Action of Bundle Automorphisms on Associated Vector Bundles
5.4 Local Connection 1-Forms and Gauge Transformations
5.5 Curvature
5.5.1 Curvature 2-Forms
5.5.2 The Structure Equation
5.5.3 The Bianchi Identity
5.6 Local Curvature 2-Forms
5.6.1 The Curvature 2-Form of the Connection on the Hopf Bundle S3→S2
5.7 Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4
5.8 Parallel Transport
5.9 The Covariant Derivative on Associated Vector Bundles
5.10 Parallel Transport and Path-Ordered Exponentials
5.10.1 Path-Ordered Exponentials
5.10.2 Explicit Formula for Parallel Transport
5.11 Holonomy and Wilson Loops
5.12 The Exterior Covariant Derivative
5.13 Forms with Values in Ad(P)
5.14 A Second and Third Version of the Bianchi Identity
5.15 Exercises for Chap. 5
6 Spinors
6.1 The Pseudo-Orthogonal Group O(s,t) of IndefiniteScalar Products
6.2 Clifford Algebras
6.2.1 Existence and Uniqueness of Clifford Algebras
6.2.2 Clifford Algebras and Exterior Algebras
6.3 The Clifford Algebras for the Standard SymmetricBilinear Forms
6.3.1 Gamma Matrices
6.3.2 The Chirality Operator in Even Dimensions
6.3.3 Raising Indices of Gamma Matrices
6.3.4 Examples of Clifford Algebras in Low Dimensions
6.3.5 The Structure of the Standard Clifford Algebras
6.4 The Spinor Representation
6.5 The Spin Groups
6.5.1 The Pin and Spin Groups
6.5.2 The Spinor Representation of the Orthochronous Spin Group
6.5.3 The Lie Algebra of the Spin Group
6.6 Majorana Spinors
6.7 Spin Invariant Scalar Products
6.7.1 Majorana Forms
6.7.2 Dirac Forms
6.7.3 Relation Between Invariant Forms and Majorana Spinors
6.8 Explicit Formulas for Minkowski Spacetime of Dimension 4
6.8.1 The Lorentz Clifford Algebra
6.8.2 The Orthochronous Lorentz Spin Group
6.9 Spin Structures and Spinor Bundles
6.9.1 Spin Structures
6.9.2 Spinor Bundles
6.9.3 Structures on Spinor Bundles
6.10 The Spin Covariant Derivative
6.10.1 Spin Connection
6.10.2 Spin Covariant Derivative
6.10.3 Dirac Operator
6.11 Twisted Spinor Bundles
6.12 Twisted Chiral Spinors
6.13 Exercises for Chap. 6
Part II The Standard Model of Elementary Particle Physics
7 The Classical Lagrangians of Gauge Theories
7.1 Restrictions on the Set of Lagrangians
7.1.1 Existence of Symmetries
7.1.2 The Quantum Field Theory Should Be Renormalizable
7.1.3 The Quantum Field Theory Should Be Free of Gauge Anomalies
7.1.4 The Lagrangian of the Standard Model
7.2 The Hodge Star and the Codifferential
7.2.1 Scalar Products on Forms and the Hodge Star Operator
7.2.2 The Codifferential
7.3 The Yang–Mills Lagrangian
7.3.1 The Yang–Mills Lagrangian
7.3.2 The Yang–Mills Equation
7.3.3 Massive Gauge Bosons
7.4 Mathematical and Physical Conventions for Gauge Theories
7.5 The Klein–Gordon and Higgs Lagrangians
7.5.1 The Pure Scalar Field
7.5.2 The Scalar Field Coupled to a Gauge Field
7.6 The Dirac Lagrangian
7.6.1 The Fermion Field Coupled to a Gauge Field
7.6.2 Lagrangians for Chiral Fermions
7.7 Yukawa Couplings
7.8 Dirac and Majorana Mass Terms
7.9 Exercises for Chap. 7
8 The Higgs Mechanism and the Standard Model
8.1 The Higgs Field and Symmetry Breaking
8.1.1 The Yang–Mills–Higgs Lagrangian
8.1.2 Spontaneously Broken Gauge Theories
8.1.3 The Hessian of the Higgs Potential
8.1.4 The Nambu–Goldstone and Higgs Bosons
8.1.5 Unitary Gauge and the Nambu–Goldstone Bosons
8.2 Mass Generation for Gauge Bosons
8.2.1 Broken and Unbroken Gauge Bosons
8.2.2 The Combined Lagrangian
8.2.3 Simplifying the Lagrangian
8.3 Massive Gauge Bosons in the SU(2) X U(1)-Theory of the Electroweak Interaction
8.3.1 The Lie Algebra su(2)L X u(1)Y
8.3.2 The Gauge Bosons
8.3.3 The Physics Notation
8.3.4 Experimental Values
8.3.5 Charges
8.4 The SU(3)-Theory of the Strong Interaction (QCD)
8.4.1 Basis for su(3)C
8.5 The Particle Content of the Standard Model
8.5.1 Fermions
8.5.2 Antiparticles
8.5.3 Chirality of the Standard Model
8.5.4 Higgs Field
8.5.5 Gauge Fields
8.5.6 The Total Particle Content of the Standard Model
8.5.7 Hypercharges: Constraints from Group Theory
A Z6-Subgroup of the Standard Model Group
Charge Quantization
8.5.8 Hypercharges: Constraints from Vanishing of Anomalies
8.6 Interactions Between Fermions and Gauge Bosons
8.6.1 The Electroweak Interaction Vertex
8.6.2 The Strong Interaction Vertex
8.6.3 The Dirac Lagrangian for Fermions
8.7 Interactions Between Higgs Bosons and Gauge Bosons
8.7.1 The Higgs Lagrangian
8.7.2 The Yang–Mills Lagrangian
8.8 Mass Generation for Fermions in the Standard Model
8.8.1 Yukawa Couplings for Leptons
8.8.2 Yukawa Couplings for Quarks and Quark Mixing Across Generations
8.8.3 Experimental Values for the CKM Matrix and Fermion Masses
8.8.4 The Yukawa Lagrangian for Fermions
8.9 The Complete Lagrangian of the Standard Model
8.10 Lepton and Baryon Numbers
8.11 Exercises for Chap. 8
9 Modern Developments and Topics Beyond the Standard Model
9.1 Flavour and Chiral Symmetry
9.1.1 Further Reading
9.2 Massive Neutrinos
9.2.1 Dirac Mass Terms
9.2.2 Neutrino Oscillations
9.2.3 Experimental Values for the PMNS Matrix and Neutrino Masses
9.2.4 Majorana Mass Terms
9.2.5 Dirac–Majorana Mass Terms and the Seesaw Mechanism
9.2.6 Further Reading
9.3 C, P and CP Violation
9.3.1 The CKM Matrix and the Jarlskog Invariant
9.3.2 C and P Transformations
9.3.3 CP Violation in the Standard Model
9.3.4 Further Reading
9.4 Vacuum Polarization and Running Coupling Constants
9.4.1 Casimir Operators
9.4.2 Running Coupling for Gauge Theories with Fermions
9.4.3 Experimental Values for Coupling Constants
9.4.4 Further Reading
9.5 Grand Unified Theories
9.5.1 Group Theoretic Preliminaries
9.5.2 Embeddings of the Standard Model Gauge Group GSM/Z6 into the Simple Lie Groups SU(5) and Spin(10)
9.5.3 Normalized Hypercharge and Unification of Coupling Constants
9.5.4 The Fermions in the SU(5) Grand Unified Theory
9.5.5 The Fermions in the Spin(10) Grand Unified Theory
9.5.6 The Fermions in the E6 Grand Unified Theory
9.5.7 Gauge Anomalies
9.5.8 The Gauge Bosons in the SU(5) Grand Unified Theory
9.5.9 Symmetry Breaking and the Higgs Mechanism in the SU(5) Grand Unified Theory
9.5.10 Further Reading
9.6 A Short Introduction to the Minimal Supersymmetric Standard Model (MSSM)
9.6.1 Graded Lie Algebras and the Supersymmetry Algebra
9.6.2 Supersymmetric Field Theories
9.6.3 Further Reading
9.7 Exercises for Chap. 9
Part III Appendix
A Background on Differentiable Manifolds
A.1 Manifolds
A.1.1 Topological Manifolds
A.1.2 Differentiable Structures and Atlases
A.1.3 Differentiable Mappings
A.1.4 Products of Manifolds
A.1.5 Tangent Space
A.1.6 Differential of a Smooth Map
A.1.7 Immersed and Embedded Submanifolds
A.1.8 Vector Fields
A.1.9 Integral Curves
A.1.10 The Commutator of Vector Fields
A.1.11 Vector Fields Related by a Smooth Map
A.1.12 Distributions and Foliations
A.2 Tensors and Forms
A.2.1 Tensors and Exterior Algebra of Vector Spaces
A.2.2 Tensors and Differential Forms on Manifolds
A.2.3 Scalar Products and Metrics on Manifolds
A.2.4 The Levi-Civita Connection
A.2.5 Coordinate Representations
A.2.6 The Pullback of Forms on Manifolds
A.2.7 The Differential of Forms on Manifolds
B Background on Special Relativity and Quantum Field Theory
B.1 Basics of Special Relativity
B.2 A Short Introduction to Quantum Field Theory
B.2.1 Quantum Field Theory and Quantum Mechanics
B.2.2 Free Quantum Field Theory on 0-Dimensional Space
B.2.3 Free Quantum Field Theory on d-Dimensional Space
Canonical Quantization
B.2.4 Unitary Representation of the Poincaré Group
B.2.5 Interacting Quantum Field Theories
B.2.6 Path Integrals
B.2.7 Series Expansions
Perturbation Theory
Semi-Classical Approximation
Non-Perturbative Quantum Field Theories
B.2.8 Renormalization
B.2.9 Further Reading
References
Index