Entry requirements

120 credits with Mathematical Methods of Physics. Participation in Symmetry and Group theory or the course Algebraic structures is required. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• define and provide several explicit examples of a smooth manifold;
• perform basic calculations involving differential forms, exterior derivatives, define integration of differential forms, formulate and use Stokes'  theorem;
• define Riemannian manifolds, complex manifolds, and symplectic manifolds;
• derive basic properties and provide simple concrete examples of connection and curvature on Riemannian manifolds;
• define Lie groups both abstractly and concretely, work with the matrix groups, explain such concepts as left-invariant vector fields and one-forms;
• define abstractly and concretely Lie algebras, give basic examples of matrix Lie algebras, provide examples of Lie algebras of vector fields, in particular of Hamiltonian vector fields;
• explain basic concepts of abstract Lie algebra theory, such as roots and weights, and construct Lie algebras using Cartan matrix, state the classification theorem of simple Lie algebras over complex field;
• use basic properties of representation theory for some examples of matrix Lie algebras, perform tensor product and decompositions of representations and apply them to explain certain phenomenon in hadron physics.

Content

Basic point set topology, smooth atlas, coordinate charts, transition functions. Vector fields, differential forms. Integration of differential forms, Stokes theorem. Complex structure, symplectic structure, metric, connection and curvature on Riemannian manifolds. Application of symplectic geometry in Hamiltonian mechanics. Application of differential geometry in general relativity. Abstract Lie groups, matrix Lie groups, group action, left- and right-invariant vector fields and one-forms. Abstract Lie algebras and their matrix realisation, exponentiation. Roots, weights, classification of simple Lie algebras. Highest-weight representations. Unitary representations of SU(2), SO(3), SU(n), SO(n) and their applications in physics.  

Instruction

Lectures and lessons.

Assessment

Hand-in problems. Oral exam.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.