# Aspects of Renormalization Group (PHY771, Spring 2020)

## Schedule

The class timings are

- Thursday: 5pm-6pm
- Friday 3:15pm-4:15pm

## Lecture Notes

## Announcements

### Jan 10, 2020

More details are given below. To summarize:

- We will do roughly 5 weeks (~10 lectures) on ideas of renormalization group (RG) in field theory.
- We will start the course in the week of Feb 24 and go on through March. I will coordinate with the registered students before that to figure out a good meeting time.
- Quantum field theory (QFT) is not required as a prerequisite, but will certainly help.
- There will be (optional) problem sets, but no exams.
- You are welcome to just sit in without registering, just send me an email to get included in the mailing list.

Please let me know if you have any other questions, topic suggestions or any other comments!

## Course Description

The ideas of renormalization group (RG) are a fundamental aspect of our modern understanding of field theories. Besides providing a intuitive explanation for the otherwise mysterious tricks for removing infinities in quantum field theory (QFT), RG also provides a unifying framework for condensed matter physics and particle physics, and organizes several concepts in field theory.

In this course, we will go over the broad conceptual ideas and look at some concrete applications of RG to physical phenomenon, both in particle physics and condensed matter physics. We will introduce RG and use it to motivate (or “define”) field theory. On the condensed matter side, we will see how field theory arises from simple lattice models, explore how RG organizes ideas of critical phenomenon in CM, and helps us understand “universality.” On the particle physics side, it will clarify the meaning of renormalizabilty, tell us why the quark mass depends on the energy scale, etc. While we will also discuss some differences between the condensed matter and high energy point of view, most of our discussion will not depend on whether you live in real time (quantum field theory) or imaginary time (statistical field theory).

### Prerequisites

Quantum mechanics is necessary. QFT is not required, but will help. In fact, this might be a good place to learn QFT if you have never studied it, because RG can be used to define and motivate QFT. However, the course will be ideal if you have seen at least a semester of quantum field theory.

### Topics

The topics to be covered are flexible and we can tailor it to the class interest. The sort of things I will cover might be:

- Simple examples from quantum mechanics to illutrate the need for renormalization, even without field theory.
- How do field theories arise as low energy effective theories of lattice models close to critical points? What is the meaning of ‘universality’?
- How is the perturbative renormalization (counterterms, etc.) procedure related to the broader renormalization group?
- What do people mean when they say that the “Higgs boson has a naturalness problem”? Why does the quark masss “run” as a function of energy scale?
- How do sigma models arise as effective field theories of Goldstone modes? What is the role of symmetry?
- Perturbative renormalization: computation of the beta function (“running” of the coupling) and anomalous dimensions using Feynman diagrams.
- Non-perturbative RG with lattice field theory: concrete lattice field theory computations of phi^4 theory to illustrate how non-perturbative renormalization works.

### Homework

I will prepare some (optional) problem sets. They will not be graded, but will supplement the material covered in lectures.

### Exams

There will be no exams!

### Auditing

If you don’t wish to register, but are interested in just sitting in for the lectures, please send me an email and I can include you in the mailing list.

## References

- Zinn Justin’s books Phase Transitions and Renormalisation Group and Quantum Field Theory and Critical Phenomena
- Nigel Goldenfeld’s book Lectures On Phase Transitions And The Renormalization Group
- John Mcgreevy’s lecture notes on Renormalization Group
- David Tong’s lecture notes on Statistical Field Theory