1.1 Lecture guide
1.1.1 Usage of this note
This note is provided primarily as a website. The usage is self-explaining but you might find a useful tips if you click the “i” mark in the top navigation bar. Also from the navigation bar one can download the pdf version. (If you want a epub file, KO can easily generate it too, so let me know.) However, the equation number does not much currently between the html and pdf format.
The parts having * at the tail of its title will probably be skipped in the lecture.
1.1.2 Objective
This lecture aims to be an introduction to the field of symmetry and its anomaly in quantum field theory (QFT). However this lecture turns out to be idiosyncratic. KO still think this is the best way of understanding, but admittedly unconventional (KO am not going to talk about topological operators, background fields, or even fermions.) The aim here is to stress that the anomaly is the projectiveness. The linear algebra we love so much is redundant when connected to the physics by the Born rule, and anomaly or the projectiveness is one reflection of that fact. For more conventional introductions see the references below.
In the first part of the lecture topological aspects of symmetry in quantum mechanics are reviewed, then in the latter part of the lecture we proceed to symmetry in 1+1-dimensional quantum field theory.
1.1.3 Prerequisite
Proficiency in the undergraduate level quantum mechanics and some basic knowledge about quantum field theory (at least knowing how to quantize the free scalar field) and group theory (e.g. what are \(SO(3)\), \(SU(2)\), \(\mathbb{Z}_2\), and so on) are assumed, but (hopefully) not much more. Especially, the first half will focus on quantum mechanics so it is hopefully understandable to even advanced undergraduates.
1.1.4 Useful references
(Random order)
[1]: Yuji Tachikawa’s full semester lecture note. The first half of Yuji’s lecture is about the big framework the most of researchers assume (but not necessarily proven), which KO will mostly omit. The second half of Yuji’s might serve as an advanced version of this lecture.
[2]: Clay Córdova’s lecture at Cern (video and hand-written notes). You can find other versions of his video on the same topic via googling. This might be regarded as more conventional version of this.
[3]: Older lecture by myself (video and hand-written notes). This one was more normal.
[4], [5]: E. Witten’s reviews/articles about fermions. While there are not so much overlap between this lecture by me and these lecture note and paper by E. Witten, and Witten’s is a bit more advanced, they are undoubtedly ones of the best entry points to the field.