Introduction to Renormalization 1102-6`IRen

The purpose of the course is to discuss the rules of construction of Hamiltonians in quantum theories. Hamiltonians of basic theories include interactions which occur at distances so small that the phenomena observed in experiments are of a much greater size than the range of the basic interaction. When the ratio formally tends to infinity, naive candidates for the basic theory produce diverging predictions. The problem is solved by a procedure called renormalization. This procedure renders an effective theory that describes observables at the experimentally accessible scales without producing divergences and eo ipso defines what is meant by the underlying basic theory. The principles and meaning of the renormalization procedure will be explained using simple model Hamiltonians and should be understandable to the students and graduate students who are familiar with quantum mechanics. The lecture and exercises will provide the participants with opportunities for hands-on experience of applying in practice the principles that make the renormalization group procedure a natural element of construction of a quantum theory. The program foresees discussion of
1. Examples of theories that require renormalization,
2. Renormalization of the Dirac delta-potential according to Jackiw,
3. Wilson's renormalization group for Hamiltonians,
4. Asymptotic freedom, triviality and fixed point,
5. Limit cycle and the search for fundamental theory,
6. Similarity renormalization group procedure,
7. Wegner's equation - from femtouniverse to condensed matter,
8. Theory of effective particles,
and may evolve as a result of questions in class. In particular, the course should be useful for students interested in analysis beyond perturbation theory. Students work in small teams and become familiar with the subject matter by solving and discussing problems that arise due to the divergences in model examples. The result of students' work in a previous edition of the course I2R is published as an article "Renormalization group procedure for potential −g/r^2" in Physics Letters B 777, 260-264 (2017). Methods discussed with students during the course can be used by them for carrying original research on basic theoretical issues. For a most recent example, see https://arxiv.org/abs/2012.11947.

Description by Stanisław Głazek, November 2021.

Time estimate:
Lecture = 30 hours
Exercises = 15 hours
Homework = 30 hours
Exam preparation = 30 hours
Total of about 105 hours