Symmetry and the Standard Model: Mathematics and Particle Physics (Hardcover)

<內容簡介>

。Provides the mathematical and physical groundwork of particle physics
。Introduces the standard model in an insightful and elementary way
。Contains clear, intuitive explanations and plenty of examples

While elementary particle physics is an extraordinarily fascinating field, the huge amount of knowledge necessary to perform cutting-edge research poses a formidable challenge for students. The leap from the material contained in the standard graduate course sequence to the frontiers of M-theory, for example, is tremendous. To make substantial contributions to the field, students must first confront a long reading list of texts on quantum field theory, general relativity, gauge theory, particle interactions, conformal field theory, and string theory. Moreover, waves of new mathematics are required at each stage, spanning a broad set of topics including algebra, geometry, topology, and analysis.

Symmetry and the Standard Model: Mathematics and Particle Physics, by Matthew Robinson, is the first volume of a series intended to teach math in a way that is catered to physicists. Following a brief review of classical physics at the undergraduate level and a preview of particle physics from an experimentalist's perspective, the text systematically lays the mathematical groundwork for an algebraic understanding of the Standard Model of Particle Physics. It then concludes with an overview of the extensions of the previous ideas to physics beyond the Standard Model. The text is geared toward advanced undergraduate students and first-year graduate students.

Table Of Contents

1 Review of Classical Physics
1.1 Hamilton’s Principle
1.2 Noether’s Theorem
1.3 Conservation of Energy
1.4 Special Relativity
1.4.1 Dot Products and Metrics
1.4.2 The Theory of Special Relativity
1.4.3 Lorentz Transformations Revisited
1.4.4 Special Relativity and Lagrangians
1.4.5 Relativistic Energy-Momentum Relationship
1.4.6 Physically Allowable Transformations
1.5 Classical Fields
1.6 Classical Electrodynamics
1.7 Classical Electrodynamics Lagrangian
1.8 Gauge Transformations
1.9 References and Further Reading
2 A Preview of Particle Physics: The Experimentalist’s Perspective
2.1 The Ultimate “Atoms”
2.2 Quarks and Leptons
2.3 The Fundamental Interactions
2.3.1 Gravitation
2.3.2 Electromagnetism
2.3.3 The Strong Interaction
2.3.4 The Weak Interaction
2.3.5 Summary
2.4 Categorizing Particles
2.4.1 Fermions and Bosons
2.4.2 Baryons and Mesons
2.4.3 Visualizing the Particle Hierarchy
2.5 Relativistic Quantum Field Theories of the Standard Model
2.5.1 Quantum Electrodynamics (QED)
2.5.2 The Unified Electroweak Theory
2.5.3 Quantum Chromodynamics (QCD)
2.6 The Higgs Boson
2.7 References and Further Reading
3 Algebraic Foundations
3.1 Introduction to Group Theory
3.1.1 What is a Group?
3.1.2 Definition of a Group
3.1.3 Finite Discrete Groups and Their Organization
3.1.4 Group Actions
3.1.5 Representations
3.1.6 Reducibility and Irreducibility: A Preview
3.1.7 Algebraic Definitions
3.1.8 Reducibility Revisited
3.2 Introduction to Lie Groups
3.2.1 Classification of Lie Groups
3.2.2 Generators
3.2.3 Lie Algebras
3.2.4 The Adjoint Representation
3.2.5 SO.2/
3.2.6 SO.3/
3.2.7 SU.2/
3.2.8 SU.2/ and Physical States
3.2.9 SU.2/ for j D 12
3.2.10 SU.2/ for j D 1
3.2.11 SU.2/ for Arbitrary j
3.2.12 Root Space
3.2.13 Adjoint Representation of SU.2/
3.2.14 SU.2/ for Arbitrary j : : : Again
3.2.15 SU.3/
3.2.16 What is the Point of All of This?
3.3 The Lorentz Group
3.3.1 The Lorentz Algebra
3.3.2 The Underlying Structure of the Lorentz Group
3.3.3 Representations of the Lorentz Group
3.3.4 The Vector Representation in Arbitrary Dimension
3.3.5 Spinor Indices
3.3.6 Clifford Algebras
3.4 References and Further Reading
4 First Principles of Particle Physics and the Standard Model
4.1 Quantum Fields
4.2 Spin-0 Fields
4.2.1 Equation of Motion for Scalar Fields
4.2.2 Lagrangian for Scalar Fields
4.2.3 Solutions to the Klein-Gordon Equation
4.3 Spin-1=2 Fields
4.3.1 A Brief Review of Spin
4.3.2 A Geometric Picture of Spin
4.3.3 Spin-1=2 Fields
4.3.4 Solutions to the Clifford Algebra
4.3.5 The Action for a Spin-1=2 Field
4.3.6 Parity and Handedness
4.3.7 Weyl Spinors in the Chiral Representation
4.3.8 Weyl Spinors in Any Representation
4.3.9 Solutions to the Dirac Equation
4.3.10 The Dirac Sea Interpretation of Antiparticles
4.3.11 The QFT Interpretation of Antiparticles
4.3.12 Dirac and Majorana Fields
4.3.13 Summary of Spin-1=2 Fields
4.4 Spin-1 Fields
4.4.1 Building a Lagrangian for Vector Fields
4.4.2 Vector Fields in the Massless Limit
4.5 Gauge Theory
4.5.1 Conserved Currents
4.5.2 The Dirac Equation with an Electromagnetic Field
4.5.3 Gauging the Symmetry
4.5.4 A Final Comment: Charge Conjugation
4.6 Quantization
4.6.1 Review of What Quantization Means
4.6.2 Canonical Quantization of Scalar Fields
4.6.3 The Spin-Statistics Theorem
4.6.4 Canonical Quantization of Fermions
4.6.5 Symmetries in Quantum Mechanics
4.6.6 Insufficiencies of Canonical Quantization
4.6.7 Path Integrals and Path Integral Quantization
4.6.8 Interpretation of the Path Integral
4.6.9 Expectation Values
4.6.10 Path Integrals with Fields
4.6.11 Interacting Scalar Fields and Feynman Diagrams
4.6.12 Interacting Fermion Fields
4.6.13 A Brief Glance at Renormalization
4.7 Final Ingredients
4.7.1 Spontaneous Symmetry Breaking
4.7.2 Breaking Local Symmetries
4.7.3 Non-Abelian Gauge Theory
4.7.4 Representations of Gauge Groups
4.7.5 Symmetry Breaking Revisited
4.7.6 Simple Examples of Symmetry Breaking
4.7.7 A More Complicated Example of Symmetry Breaking
4.8 The Standard Model
4.8.1 Helpful Background
4.8.2 The Outline
4.8.3 A Short-Range Force: The Gauge and Higgs Sector
4.8.4 The Gauge Bosons and Their Coupling to the Higgs Boson
4.8.5 The Lepton Sector: The Origin of Mass
4.8.6 The Quark Sector
4.8.7 Yukawa Couplings Among Generations
4.9 References and Further Reading
5 Beyond the Standard Model of Particle Physics
5.1 Overview of Physics Beyond the Standard Model
5.2 Grand Unified Theories
5.2.1 Unification of the Coupling Constants
5.2.2 The Basic SU.5/
5.2.3 Supersymmetry
5.3 Higher-Rank GUT Unification
5.3.1 A GUT Implication
5.3.2 GUT Summary
5.4 Alternate Directions and Quantum Gravity
5.4.1 Extra Dimensions
5.4.2 What About (Quantum) Gravity?
5.4.3 String Theory
5.4.4 Loop Quantum Gravity
5.4.5 Causal Dynamical Triangulation
5.4.6 Causal Sets
5.4.7 Non-commutative Geometries
5.4.8 Twistor Theory
5.4.9 Hoˇrava-Lifshitz Gravity
5.4.10 Quantum Gravity Summary
5.5 References and Further Reading
References
Index