Wu Ki Tung - Group theory in Physics
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Learning Objectives
Knowledge acquired:
See contents of the course.
Competence acquired :
How to deal with symmetries in physics in a rigorous and effective fashion. A new understanding of Quantum Mechanics.
Skills acquired (at the end of the course):
A method to solve problems and to determine the consequences of the symmetries in physics problems.
Prerequisites
Degree in Physics
Teaching Methods
6 CFU
Lectures hours: 48 hours and 6 more hours for exercises
Type of Assessment
Oral test.
The test lasts about an hour. The student will be asked to solve a problem among those purposely assigned by the teacher. Thereafter, two questions on relevant topics.
Course program
Basics of group theory. Classes and invariant subgroups. Cosets and quotient group.
Permutation group.
Theory of group representations.
Irreducible and inequivalent representations.
Orthogonality and completeness relations.
Product and reduction.
Wigner projector.
Wigner-Eckart theorem.
Representation of the permutation group and applications. Young tableaux.
Decomposition of tensors.
Continuous groups. Lie groups and Lie algebras.
Invariant measure and its determination. Integrals over the group.
Rotation group and SU(2).
Parametrizations of SU(2) and SO(3): Euler angles and axis-angle. Wigner D matrices of the irreducible representations.
Lorentz group SO(1,3) and its algebra. Finite dimensional representations of the Lorentz group.
The group SL(2,C).
Martices of the irreducible representations.
Poincarè group and its algebra. Unitary representations of the Poincarè group, massive and massless.
Applications to quantum field theory.