Symmetries Graduate School 2023

This lecture is devoted to Noether’s theorems and the study of the interplay between symmetries and conservation laws, from ordinary mechanics to general relativity. In order to start on a common ground and interest a broad audience, we will begin with a review of Noether’s (first) theorem in ordinary non-relativistic mechanics. This will enable us to settle some subtleties, agree on conventions, and especially explore some curious and lesser-known symmetry features of familiar models (such as particles and celestial mechanics). We will then move on to field theory, and discuss the construction of conserved currents and energy-momentum tensors. This will include a discussion of conserved quantities in general relativity. Finally, we will turn to the core of the topic, which is Noether’s (second) theorem for gauge symmetries. After recalling the basic properties of gauge theories in Lagrangian and Hamiltonian form, we will derive the consequences of gauge symmetry for the construction of conserved charges. For this, we will introduce the so-called covariant phase space formalism, which enables to construct symmetry charges and algebras, and derive (non) conservation laws. This will be illustrated in Maxwell’s theory and in general relativity. In particular, we will focus in depth on the example of three-dimensional gravity as an exactly soluble model in which all aspects of symmetries can be understood. We will end with an outlook towards the notion of asymptotic symmetries and their use in classical and quantum gravity. Ideally, the audience should be familiar with: Hamiltonian mechanics differential forms basic features of general relativity

Interacting quantum particles can form non-trivial states of matter characterized by topological order, which features several unconventional properties such as topological degeneracy and fractionalized quasiparticles. In addition, it also provides a promising platform for realizing quantum computing in a robust manner. In this series of lectures, I will introduce the basics of topological order and its connection to quantum computing from various aspects involving lattice models, symmetry, and entanglement structure. Several frontier topics such as fracton topological phases, self-correcting quantum memory, state preparation, and quantum LDPC codes will be briefly discussed.

These lectures will cover the concepts and techniques of effective field theory. I will try to introduce several of the useful techniques which do not usually get covered in the standard QFT courses and books. We will start with the effective field theory aspects of QED, and end with the treatment of general relativity as a quantum field theory using effective field theory techniques.