# Introduction to Classical Field Theory

- A tour of the fundamental interactions

- Jarrett L. Lancaster

- September 2018

### Description

This book is a short introduction to classical field theory, and is most suitable for undergraduate students who have completed at least intermediate-level courses in electromagnetism and classical mechanics. The main theme of the book is showcasing the role of fields in mediating action-at-a-distance interactions. Suitable technical machinery is developed to explore at least some aspect of each of the four known fundamental forces in nature. Beginning with the physically-motivated introduction to field theory, the text covers the relativistic formulation of electromagnetism in great detail so that aspects of gravity and the nuclear interaction not usually encountered at the undergraduate level can be covered by using analogies with familiar electromagnetism. Special topics such as the behavior of gravity in extra compactified dimensions, magnetic monopoles and electromagnetic duality, and the Higgs mechanism are also briefly considered.### About Editors

Jarrett L Lancaster obtained a BS in physics and applied mathematics from the University of North Carolina at Greensboro and his PhD in physics from New York University. His research focuses on dynamics of low-dimensional quantum systems and emergent phenomena. He worked previously as a postdoctoral researcher at the Joint School of Nanoscience and Nanoengineering and as a visiting assistant professor of physics at Roanoke College. He is currently an assistant professor of physics at High Point University.### Table of Contents

Table of Contents

1 Motivation and Introduction

1.1 The four fundamental interactions

1.1.1 Electromagnetism

1.1.2 Strong and weak nuclear forces

1.1.3 Gravity

1.2 Particle exchange and force mediation

1.3 Examining a simple model

1.4 Relativity emerges

1.5 The necessity of fields and a conundrum

1.6 Exercises

2 Basics of Scalar Field Theory

2.1 From oscillators to fields

2.2 Lagrange and Hamilton

2.3 Hamiltonian with sources

2.4 The attractive Yukawa potential

2.5 Some relativistic technology

2.6 Relativistic field theories

2.7 Exercises

3 Electromagnetism

3.1 Maxwell's equations

3.2 Lagrangian formulation

3.3 Why like charges repel: the Coulomb potential

3.4 Resolution of a conundrum and magnetic energy

3.5 The electric field in arbitrary spatial dimensions

3.6 Propagation of interactions

3.7 Electromagnetic duality and magnetic monopoles

3.8 Gauge invariance

3.9 Exercises

4 Yang-Mills Theory 78

4.1 From Maxwell to Yang-Mills

4.2 Nonabelian gauge theory formalism

4.3 The static potential

4.4 The strong nuclear interaction

4.5 Classical color charge dynamics

4.6 Effective static quark-antiquark potential

4.7 Electroweak unification and Higgs mechanism

4.8 Exercises

5 Gravity as a Field Theory

5.1 The trouble with Newtonian gravity

5.2 Constructing an appropriate field theory

5.3 Emergence of Newton's law of gravity

5.4 Interactions of light and matter

5.5 A glimpse at general relativity

5.6 Gravity with extra, compactified dimensions

5.7 Exercises

A Mathematical Results

A.1 Taylor expansions

A.2 Gaussian integrals

A.3 Gamma function

A.4 Dirac delta function

A.5 Fourier transforms

A.6 Curvilinear coordinates

A.7 Levi-Civita symbol

A.8 Contour integration

A.9 Summation of series

### Bibliographic

Paperback ISBN: 9780750329378

Ebook ISBN: 9781643270838

DOI: 10.1088/978-1-64327-084-5

Publisher: Morgan & Claypool Publishers