MATH 6100 — Introduction to Mathematical Physics

Learning Seminar on

Classical Gauge Field Theory

Spring 2025 — Iowa State University

Contents

Elements of classical field theory, including tensor calculus, Lagrangians, Noether’s theorem, electromagnetism, gravity, gauge symmetry, Dirac equation.

Description

This learning seminar will provide an opportunity for those with mathematical background to become acquainted with certain powerful key concepts in fundamental physics. It may also be of relevance to physics students that are interested in a mathematical perspective on these topics.

The focus this semester will be on classical field theory, which is the study of vector fields (and, more generally, tensor fields), defined on Minkowski spacetime or other manifolds, and that are subject to equations of motion. We reserve the topic of quantum field theory for a possible future learning seminar. (This means that any physics we discuss will be, at best, low-energy approximations of the real world; although in the case of gravity, no quantum theory is known.)

The emphasis in the seminar is on the principles and ideas that (it turns out) give rise to physically relevant equations, rather than the techniques and methods for actually solving those equations and comparing with experiments.

Prerequisites

There are no formal prerequisites besides basic calculus and linear algebra, but a certain “mathematical maturity”, or otherwise adventurous spirit, may be useful. For those interested in geometry, you can also take Manifolds, Tensors and Differential Geometry (MATH 6240, also runs Spring 2025). If you are more interested in algebra, the course Representation Theory (MATH 6180, Spring 2025) is recommended. Lastly, if you are interested in analysis, the course Partial Differential Equations II (MATH 6560, Spring 2025) is a good supplement to the seminar.

Instructor

• jthartwig.net

Time and Place

• Lectures: Tuesdays 2:10pm–3:00pm in Carver 0232.

Literature

In addition to lecture notes (to be posted), I provide here a list of recommended books, in order of more mathematical to more physical:

• Mathematical Gauge Theory, by Mark J.D. Hamilton (Springer)

This is the best book I have found that is written for mathematicians, but still tries to explain a lot of physics notation, conventions and terminology. It has careful treatments of manifolds and Lie groups, vector bundles and connections, and covers the key classical field theory topics needed to begin to understand the standard model of particle physics up to the most common attempts at grand unified theories. The downside is that general relativity is not discussed at all.

• Tensors, Differential Forms, and Variational Principles, by Lovelock and Rund (Dover)

This is my favorite book on differential and pseudo-Riemannian geometry. Its final chapter contains an interesting treatment of variational principles for vector and metric lagrangians, which specialize to Maxwell and Einstein’s theories. It is mathematical, but also uses the tensor index notation commonly seen in physics.

• Classical Field Theory, Florian Scheck (Springer)

A solid book that could be used as the main text on a course in classical field theory.

• Classical Field Theory, Davison E. Soper (Dover)

A lovely book that contains a lot of nice results, including the derivation (as it were) of the Einstein-Hilbert lagrangian, and other topics.

• Gauge Field Theories — An Introduction, J. Leite Lopes (Pergamon Press, 1980)

An older book that covers a lot of examples of theories that are helpful to understand general phenomena such as Higgs mechanism, symmetry breaking etc. It also touches on supersymmetry. The downside is it assumes certain basic familiarity with the subject, so might be best for physics students.

• Classical Theory of Gauge Fields, Valery Rubakov (Princeton University Press, 2002)

Part I contains the basics of gauge field theory up to electroweak symmetry breaking via Higgs. Part II contains more advanced topological considerations (that go beyond the scope of this seminar), while Part III deals with fermions. Again, sadly, gravitation is not mentioned.

• Fields, Symmetries, Quarks, by Ulrich Mosel (Springer)

This wonderful book cleverly guides the reader from quantum electrodynamics through beta decay and early gauge theories, to electroweak unification, Higgs mechanism, and ultimately the complete standard model of particle physics. Though a physics book, it requires only a minimum of quantum field theory to follow. The downside (from the perspective of using it as our main text) is that it lacks any treatment of gravity.

There are many other text books that have an overlap with the planned course topics, for example Gauge Fields, Knots, and Gravity, by Baez and Muniain (World Scientific, 1994) but it is less detailed in certain aspects.

Examination

The seminar is pass/fail. To receive a passing grade, you need to be active in the class, which means handing in at least 50% of the homework or equivalent effort.

Homework sets will be posted here.

Preliminary Lecture Plan (subject to change)

The following is a rough current outline. The material may move around and some topics might be moved to homework problems or self-study.

1. Numerical Tensors: Linear Change of Coordinates, Vectors and Covectors, Sum and Product, Tensors of rank \((r,s)\), Contraction and trace, Kronecker tensor, metric, raising and lowering. Lorentz Tensors, Levi-Civita symbol.
2. General Tensors: the gradient of a scalar field, general metric tensor, oriented change of coordinates. Examples: Klein-Gordon Equation, \(\partial_\mu\phi\partial^\mu\phi\), \(\partial_\mu A_\nu-\partial_\nu A_\mu\), Newton-Laplace Equation.
3. Relative tensors, scalar densities, Levi-Civita tensor, differential \(n\)–form, \(d^nx\) and \(\sqrt{-g}\).
4. Integration and the Divergence Theorem
5. Principle of Least Action; Equations of Motion (Euler-Lagrange). Example: (non-)interacting scalar fields.
6. Noether’s First Theorem; momentum, angular momentum, energy. Lorentz invariance, Energy-momentum tensor.
7. Classical Electromagnetism: The Maxwell Lagrangian. Gauge transformation of the 4-potential.
8. Covariant Differentiation: Affine connections, compatible metrics, Fundamental Theorem of Riemannian Geometry. Points where connection vanishes. Klein-Gordon in curved spacetime.
9. Curvature and torsion, the curvature scalar, metric theories. (Derivation of Einstein-Hilbert Lagrangian) and its equations of motion: Einstein’s Field Equations.
10. Gravitational waves, reduction to Newton-Laplace for low intensity fields, black holes
11. Revisiting Klein-Gordon and Maxwell in presence of gravity. Gravitational lensing
12. Fermions and the Dirac Equation: Dirac gamma matrices. Chirality and Weyl fermions. Global \(U(1)\)–symmetry, charge.
13. Local Symmetry: Noether’s Second Theorem. The need for a gauge field \(A_\mu\). The QED Lagrangian, Dirac-Maxwell equations.
14. Yang-Mills. SU(2) and weak interaction. Apparent problems.
15. Higgs Mechanism: The complex Higgs doublet, spontaneous symmetry breaking, the Standard Model Lagrangian.

Statement on Free Expression

_{Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.}

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