520024 VU VDSP Summer Academy 2025 "Symmetries, Particles and Fields" (2025S)

This course offers a modern and concise exploration of symmetries in both classical and quantum physics, with a particular focus on their applications in particle physics. It is organized into three distinct yet interconnected lecture series, outlined as follows.

1) 5 lectures on “What is spin?” by Prof. Nils Carqueville:

Spin is a key feature of quantum physics. For example, spin plays an important role in understanding electrons and quarks, where "rotations by 360 degrees act non-trivially". These lectures aim to answer the titular question from a perspective of symmetry and the representation theory of groups. We introduce spin groups and discuss their relation to rotation groups SO(n) as well as Lorentz groups. To talk about spin as a symmetry, we then consider the action of spin groups on Hilbert spaces. These representations are naturally described in terms of Clifford algebras, such as those obtained from Pauli matrices or the gamma-matrices featuring in the Dirac equation. The goal of this lecture series is to describe all this in one coherent story that does not assume much more mathematics than linear algebra.

2) 5 lectures on “Symmetries in Particle Physics: From Relativistic Fields to Phenomenology” by Prof. Josef Pradler:

These five lectures explore how symmetries shape modern particle physics, starting with Lorentz-invariant wave equations and moving through Noether's theorem, gauge invariance, discrete transformations, spinor-helicity amplitude techniques, and spontaneous symmetry breaking. We examine how mass generation emerges from local gauge symmetries and see how spinor-helicity efficiently handles newly introduced redundancies in the symmetry prescriptions. The series concludes by linking these foundational ideas to phenomenological questions such as the nature and stability of dark matter.

3) 5 lectures on “Effective Theory Method in classical and quantum physics” by Prof. André Hoang:

The descriptions of all physical systems can be formulated in terms of a Lagrangian function. Interestingly, assuming that there is a "Theory of Everything", which using the same logic should also have a Lagrangian formulation, there should be a way to derive the Lagrangian of a particular system, may it be quantum or classical, from the Lagrangian of that Theory of Everything. Likewise it is also possible to derive the Lagrangian of a classical system from quantum theory, or of a non-relativistic system from the relativistic formulation.
The Effective Theory Method describes how to derive an "effective Lagrangian" from a more general Lagrangian. In the series of lectures we discuss the Effective Theory Method in general and focus on a few practical applications such as the classical inverted pendulum problem and the derivation of relativistic corrections to the hydrogen atom. We also discuss the important role of symmetries in the construction of effective Lagrangians.

Throughout the two-week course, lectures will be held in the mornings, followed by an afternoon tutorial each day to give students the opportunity to practice and apply the concepts and methods covered in the lectures.

Knowledge of classical mechanics, electrodynamics, quantum mechanics and linear algebra is an essential prerequisite.