PH-701 Group Theory

  1. Finite Groups: groups and representations, the regular representations, irreducible representations, transformation groups and applications, Schur’s lemma, orthogonality relations, characters, eigenstates, tensor products.
  2. Lie groups: generators, Lie algebras, Jacobi identity, the adjoint representation, simple algebras and groups, states and operators.
  3. SU(2): eigenstates of J3, raising and lowering operators, tensor products.
  4. Tensor operators: orbital angular momentum, Wigner Eckart theorem and examples, product of tensor operators.
  5. Isospin: charge independence, creation operator, number operators, isospin generators, symmetry of tensor products, the deuteron, superselection rules.
  6. Roots , weights and SU(3): Gellmann matrices, weights and roots of su(3), positive weights, simple roots, constructing the algebra, Dynkin diagrams and examples, the Cartan matrix, the trace of generator, fundamental representation of SU(3), constructing the states, the Weyl group, complex conjugation and example of other representation.
  7. Tensor method: lower and upper indices, tensor components and wave functions, irreducible representation and symmetry, invariant tensor, Clebsch-Gordon decomposition, triality, matrix elements and operators, normalization, tensor operators.
  8. Hypercharge and strangeness: the eight-fold way, the Gellmann-Okubo formula, hadron resonances, quarks.
  9. Young tableaux and SU(n): raising and lowering indices, Clebsch-Gordan decomposition, U(1), generalization of gell-mann matrices, SU(N) tensors, dimensions, complex representations
  10. The Lorentz and Poincare groups and space-time symmetries: generators and the Lie algebra, irreducible representation of the proper Lorentz group, unitary irreducible representation of the Poincare group, relation between representation of the Lorentz and Poincare groups, relativistic wave functions, fields and wave equations.


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