An Introduction to Group Theory


Description: This course is intended as an introduction to the group theory for physics students. .

Over the last century the ideas and methods of group theory become very important tool in many different areas of physics, e.g., quantum mechanics, solid state physics, high energy physics, quantum field theory etc.

This course aims at giving a general introduction to the description of symmetry properties of physical systems. It provides a set of simple examples of the physical concepts and mathematical ideas which where developed in that context.

Syllabus:

  1. Symmetry operations, Group axioms, subgroup, Lagrange’s theorem, Direct product, multiplication table, homomorphism and isomorphism, finite groups, permutation group.
  2. Equivalence class, Invariant subgroup, Simple group, Coset, Quotient group.
  3. Homomorphism and isomorphism, theorem of homomorphism.
  4. Representation theory, Schur’s lemma, Orthogonality theorem, Conjugacy class, Character table, Direct sum, Tensor product.
  5. Continuous groups, Lie algebra, Jacobi identity, Representation theory of Lie alge- bra, Fundamental representation, Adjoint representation.
  6. Rotational group, SO(2) and SO(3).
  7. SU(2) group, Pauli matrices, Ladder operators, Casimir invariants. Local isomorphism between SU(2) and SO(3)
  8. SU(3) Lie algebra, Gell-Mann matrices, Structure constants.
  9. Integration over continuous groups, Topology, Group manifold.
  10. Cartan-Weyl basis, Classification of simple Lie algebras, Roots and weights, Dynkin diagrams.
  11. Lorentz and Poincare groups. Conformal symmetry

Lectures