While crystal symmetries rely on discrete groups, the fundamental forces of nature rely on continuous symmetries (Lie groups). Tung provides a masterful introduction to:
Clebsch–Gordan decomposition (example): For two spin-1/2: 1/2 ⊗ 1/2 = 1 ⊕ 0 (triplet + singlet). Triplet symmetric, singlet antisymmetric.
: Chapters 8–9 (Lorentz group). This is the hardest part. Spend two weeks just understanding the difference between SO(3,1) and SL(2,C). Do the spinor algebra until it becomes intuitive. Wu-ki Tung Group Theory In Physics Pdf
: Essential for understanding rotations and spin in quantum mechanics. Lorentz and Poincaré Groups
Unlike many pure math treatments, Tung's book is highly regarded for its physics-first approach — covering finite groups, Lie groups, and their representations with clear connections to angular momentum, particle classification, and scattering theory. It sits nicely between the rigor of Hamermesh and the more applied style of Georgi. While crystal symmetries rely on discrete groups, the
Moving beyond discrete symmetries, Tung introduces —groups whose elements form a continuous differentiable manifold. Instead of studying the infinite elements of a Lie group directly, physicists study its Lie algebra (the infinitesimal generators of the group). Tung provides clear mathematical derivations of roots, weights, and the classification of semi-simple Lie algebras. 4. The Lorentz and Poincaré Groups
Constantly remind yourself of the physical meaning of the math. For example, recognize that the Casimir operators of the Poincaré group correspond exactly to physical mass and spin. : Chapters 8–9 (Lorentz group)
A: For problems and computational practice, "Lie Groups for Pedestrians" by Lipkin (old but gold). For modern QFT applications, "Quantum Field Theory" by Schwartz has excellent group theory appendices that complement Tung.
Despite being written decades ago, Tung’s book remains a staple on the syllabi of advanced physics courses globally for several reasons:
For modern high-energy physics, discrete symmetry is not enough. Tung provides a masterful introduction to Lie groups and Lie algebras. He explains how infinitesimal transformations lead to conserved quantities, linking directly to . 5. The Lorentz and Poincaré Groups
Crucial tools used to reduce complex representations into irreducible ones (irreps).