Quantum Anomalies as Projective Phases
2022-04-21
Abstract
This is a lecture note originally prepared for “26th APCTP Winter School on Fundamental Physics”, Feb. 14-18, 2022. Later it is expanded for the lecture at KEK (online), Apr. 19-21, 2022. (The expansion is planned to be done before/through the lectures.)
In this lecture we will study the symmetry and its anomaly in low-dimensional, i.e. 0+1d and 1+1d, quantum field theories. In 0+1-dimensional quantum field theory, a.k.a quantum mechanics, the Wigner’s theorem tells that a global symmetry forms a group and acts on the Hilbert (state) space as a projective representation. We will see example with non-trivial projective phases and how it can be related to symmetry protected topological phases in 1+1-dimensions. We then see how the story are generalized/changed in 1+1-dimensional (relativistic) quantum field theory, where the locality of the theory plays an important role. This lecture aims to formalize quantum anomalies from Hamiltonian perspective, while a conventional approach usually heavily relies on path-integral perspectives.
Chapter 1 Introduction
Symmetry is a guiding principle in physics. In many case, given a system, you first analyze its symmetry. Or, to model a given phenomena, the symmetry is often be the first clue. Therefore, there have been numerous research on the topic. What is surprising is that, still, in 2022, it is a hot area of research and there are many things to be understood. Maybe the main goal of this lecture is to give the sense of unfinished-ness.