Group Theory and Gauge Symmetries in Physics

Group Theory & Gauge Symmetries: Lorentz & Poincaré, Local Gauge Invariance, SU(3), QCD, Spherical Harmonics, and more

What you'll learn
- Understand the role of group theory and Lie algebras in describing spacetime and internal symmetries of physical systems.
- Derive and apply Lorentz and Poincaré transformations to vector and spinor fields in special relativity and quantum mechanics.
- Explain and implement the concept of local gauge invariance and its consequences for conserved currents, QED and QCD.
- Use spherical harmonics, Legendre polynomials, Wigner matrices and harmonic analysis on groups to solve angular problems and study representations.
- Analyze the structure of SU(2), SO(3), SU(3) and other unitary groups SU(N) and their connection to quarks, gluons and non-Abelian gauge theories.
- Gain an intuitive introduction to path integrals and Faddeev–Popov ghosts, understanding their role in gauge field quantization.
- Understand the Rodrigues' rotation formula, and how it provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3)

Requirements
- A solid understanding of undergraduate-level physics, including classical mechanics, quantum mechanics, and special relativity.
- Familiarity with calculus, linear algebra, and differential equations.
- Exposure to Lagrangian mechanics and tensor notation is highly recommended.
- General Relativity is not mandatory, but recommended to fully grasp some of the lectures

Description
Group Theory and Gauge Symmetries in Physics

Modern physics is built on symmetry. From the Lorentz invariance of relativity to the local gauge symmetries of the Standard Model, group theory provides the language that unifies spacetime transformations, conservation laws, and fundamental interactions.

This course offers a comprehensive journey throughgroup theory, Lie algebras, gauge invariance, and their applications in theoretical physics, blending rigorous mathematics with physical insight, explained as intuitively as possible. You will learn how the symmetry principles of classical and quantum physics lead naturally to the gauge theories underlying electromagnetism, QCD, and beyond.

Topics covered

The role ofLorentz and Poincaré transformationsin special relativity and field theory.

HowLie groups and Lie algebrasdescribe angular momentum, spin, and spacetime symmetries.

Vector and spinor field transformations, the generators of these transformations, and commutation relations.

The mathematics behind theDirac equation, conserved currents, and charge conservation.

Local gauge invariance: why it’s needed, how it arises, and its implications for the Standard Model.

The structure ofSU(2), SO(3), SU(3), and unitary groups SU(N), and their connection to quarks and gluons in QCD.

Spherical harmonics, Legendre polynomials, and Wigner D-matrices: the angular part of quantum wavefunctions and their group-theoretic meaning.

Harmonic analysis on groupsand the deep relation between SO(3) and SU(2).

A heuristic introduction topath integrals, Faddeev–Popov ghosts, and gauge consistency. This will be an "extra", only if you are interested in path integrals (which are covered extensively in other courses of the author).

Course Structure

The course is organized in progressive modules, starting from the fundamentals of spacetime symmetry and building up to advanced topics like non-Abelian gauge theories and harmonic analysis on groups. Each lecture includes derivations, step-by-step explanations, and mathematical appendices to strengthen your skills.

If you manage to work your way through the course, you will definitely have apowerful conceptual and computational toolkitfor working with symmetries, Lie groups, and gauge theories—the essential backbone of modern theoretical physics.

Who this course is for:
- Undergraduate or graduate physics students who want to understand the mathematical structure behind symmetries in mechanics, quantum theory, and field theory.
- Researchers or postgraduates in theoretical physics seeking a deeper grasp of Lie groups, Lie algebras, and gauge invariance.
- Mathematicians or engineers with a background in linear algebra and calculus who want to apply group theory to physical problems.
- Learners of quantum mechanics looking to connect angular momentum, spin, and field transformations to underlying symmetry principles.
- Students preparing for advanced studies or research in particle physics, quantum field theory, or gauge theories, who need a mathematical but "intuitive" explanation of the relevant mathematics.
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