Topology in Statistical Physics - Lecture 1

Explore the fundamental concepts of topology in statistical physics through this comprehensive lecture, part of the Bangalore School on Statistical Physics XI. Delve into the mathematical structures underlying physical phenomena, focusing on topological quantum numbers and their significance. Examine Maxwell's equations as a classical effective field theory, and investigate the incorporation of magnetic monopoles. Learn about ultraviolet regulation, symmetries, and the Dirac string. Gain insights into different types of quantum numbers and the path integral formulation of quantum mechanics. Engage with a detailed exploration of these advanced topics, designed to bridge the gap between master's-level courses and cutting-edge research in statistical physics.

Topology in statistical physics - 1
BSSP 2020 : Introduction to Topology in statistical physics
Contents
Lecture -1: Topology in statistical physics: What and why?
Our course work in mathematical physics should tell us that a mathematical structures have two characteristic features
Assume that you are familiar
With this, I intent develop the course as follows
The aim would be to try and tell you the story in a way you can start thinking for yourself.
Example of topological quantum number and associated ideas
In this sense Maxwell's equation describe a classical EFFECTIVE field theory
Ultra-violet regulation : Define a small length, a and do all the calcula- tions.
Sometimes it is easy to see what is a natural regular - Symmetries
Now we modify the Maxwell's equation to incorporate the magnetic monopole This may come from more high energy theories which we would not discuss,
Magnetic Monopole
In the magnetostatic limit with the monopole at the origin, i.e. on = mor,
Line singularity Dirac String : This singularity is not in a physical observable.
General Forecast : Two types of Quantum Numbers.
Term String
Path Integral formulation of Quantum Mechanics
Integrate out P
Q&A