Groups and Symmetries

This course unit is intended primarily for Mathematical Physics 4 students but it may be attended - either seriously or on a `sitting in' basis - by Astrophysics 4 and Physics 4 students, MPhys students, physics graduate students, and indeed by anyone else who may be interested. Students are warned that there will be no printed lecture notes: come to the lectures and make your own!

• Synopsis

  This course provides an introduction to the theory of groups and their applications in physics. No previous knowledge of groups is required, but the applications build on classical and quantum mechanics. The topics covered are: finite group theory; representations of finite groups; an introduction to Lie groups and their representations; applications in classical and quantum physics.

• Syllabus
  • Finite groups. Definition of a group; multiplication table; cyclic groups; subgroups; cosets; Lagrange's theorem; permutation groups; isomorphism and homomorphism; conjugate elements and classes; normal subgroups and factor groups; direct product.
  • Representations. Definition and elementary properties; equivalence; invariant sub-spaces and reducibility; unitary representations; irreducible representations; Schur's lemmas; orthogonality theorems; characters; reduction of reducible representations; the regular representation; character tables; reduction of direct products of representations.
  • Applications in classical and quantum mechanics. Vibrational problems; the displacement representation and its reduction to normal modes; projection operators. Symmetry in quantum mechanics; symmetry of the Hamiltonian and degeneracy; symmetry breaking; selection rules; the Wigner-Eckart theorem; wave functions and induced transformations; Bloch's theorem.
  • Continuous groups. Abelian groups: translations, SO(2), U(1), momentum and energy in quantum mechanics. Compact nonabelian Lie groups and algebras: SO(3) and SU(2), Weyl homomorphism, quantum theory of angular momentum, applications to selection rules. Isospin.

• Recommended Textbooks

  • H.F. Jones, Groups, Representations and Physics, IoP (1998).
  • J.P. Elliot & P.G. Dawber, Symmetry in Physics, Vol 1, Macmillan (1984).
  • Wu-Ki Tung, Group Theory in Physics, World Scientic, 1985.

  • Group Theory and Quantum Mechanics} M. Tinkham, McGraw-Hill, 1964
  • Group Theory in Physics: An Introduction} J.F. Cornwell, Academic Press (1997)
  • Introduction to Group Theory}, W. Lederman (Longman Scientific and Technical)

• Examples

  Problem solving is the best way to learn and understand mathematical physics. For this course there will be six problems classes at which solutions to the more difficult problems will be presented on the blackboard as required. I will also be available in my office to answer any remaining questions about the lectures and the associated problem sheets.

  Students are warned that no written solutions will be provided to the problems. It is thus essential that each student intending to take the course seriously makes a determined attempt to construct their own solutions to the problems BEFORE the appropriate problems class, so that any difficulties he or she finds during the course of solving the problems may be resolved during the class.

  • Problem set 1 : Group Axioms
  • Problem set 2 : Classes and Cosets
  • Problem set 3 : Irreducible Representations
  • Problem set 4 : Applications: Molecular Vibrations
  • Problem set 5 : Applications: Degeneracy and Selection Rules
  • Problem set 6 : Translations and Rotations

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R.D. Ball
October 2003