Students will explore the crucial role of continuous symmetries in physics through Wigner’s classification of “elementary particles” and learn how rest mass, spin and chirality emerge. They will also learn how to construct a relativistic field theory using the rules for constructing the Lagrangian and the variational principle. Finally, they will be introduced to gauge theories through scalar electrodynamics, and to antimatter through the Dirac equation.
The course aims to introduce the principles that underpin all of modern physics: spacetime and the concept of a field; continuous and discrete symmetries in physics; the classification of representations of symmetry groups; the Lagrangian approach and the variational principle; and scalar and spinor fields. These tools are used throughout physics, from general relativity to elementary particle physics.
The course begins with a review of special relativity and the symmetries of Minkowski spacetime.
Lie group theory is then applied to the classification of reducible and irreducible representations and to Wigner’s classification of “elementary particles”. It is shown how spin, mass and chirality arise from symmetries.
The course subsequently introduces massive scalar field theory and the construction of a relativistic field theory using a variational approach.
Students are also required to undertake an independent self-study assignment on the Dirac equation, based on the notes provided.
Minkowski spacetime and its symmetries
Review of special relativity; Lorentz transformations; the Poincaré group; discrete symmetries: parity, time reversal and charge conjugation.
Classification of fields
The concept of a field; Lie group theory and the classification of representations; application to the rotation group and the concept of spin; representations of the Lorentz group and chirality; infinite-dimensional representations of the Poincaré group: mass and orbital angular momentum.
Free massive scalar field theory
The Lagrange chain and the continuum limit leading to the scalar field; construction of a field theory using the variational principle; Noether’s theorem; the Klein–Gordon equation; complex scalar fields and charge conjugation; local gauge invariance of the complex scalar field.
In addition, students are required to undertake an independent self-study assignment on the Dirac equation, based on the notes provided.
Spinor field theory
Chiral (Weyl) spinors and Dirac bispinors; the Dirac equation; the variational principle; propagation of free fermions and particular solutions of the Dirac equation; Dirac and Weyl bases.
The exercises provide practical reinforcement of selected topics covered in the course (SU(2) and SO(3), the Lorentz group, etc.).
Lectures are delivered at the blackboard. Tutorial sessions complement the lectures. Students are required to complete an independent assignment on the Dirac equation.
The oral examination is based on a list of questions provided in advance. The independent assignment on the Dirac equation replaces the traditional examination of the tutorial material. Questions about the independent assignment are included in the oral examination.
| Training | Block | Credits | Mandatory |
|---|---|---|---|
| Bachelor in Physics | 3 | 3 | Yes |