**PHY512 Renormalization**

**S. G. Rajeev**

**Tuesdays and Thursdays 11: 05am to 12:20pm at Bausch and Lomb 269 (Physics Building)**

** First Class: Thursday, Jan 15 2014**

**Office Hours: **Tuesdays /Thursdays 10:15-11:05 am

**I will be using Blackboard to post Notes and Problems.**

**Synopsis**

In several areas of physics,the calculation of physical quantities lead to answers that are infinite. The first example was the energy of a point charge in classical electromagnetism. Dirac showed that this could be absorbed into a redefinition of the mass of the charge but with a left-over effect: the radiation emitted by the particle brakes its acceleration.

### Later, similar divergences were encountered in quantum electrodynamics. Heroic work by Schwinger and Feynman showed that again the infinities could be removed with experimental consequences that could be tested with high precision (magnetic moment of the electron). These methods were extended and used by Wilson to explain century old puzzles in a completely different area of physics: phase transitions in classical statistical physics. It is the purpose of this course to introduce you to these revolutionary ideas in physics.

### The procedures used cannot be justified by existing mathematics: a new integral calculus of functions of an infinite number of variables is needed. Instead we will study examples from several area of physics and applied mathematics. See Syllabus below for a list. Some of the more ambitious parts of the syllabus will be covered by individual student projects rather than in lectures. Potential applications to turbulence and a glimpse of particle physics beyond the standard model will be mentioned only in passing.

**Pre-requisites**

### Knowledge of Electrodynamics (PHY218/ 415), Quantum Mechanics (e.g., PHY237/408), Statistical Mechanics (PHY227/418). Also, you should have access to Mathematica and some ability to code in C.** **

**References**

### There is no one textbook that covers this ground. A mix of books and review articles will be used instead, along with my own notes.

**Grades**

### There will be no exams. Instead, there will be a homework set of three problems about every other week. In addition, each student will work on a special topic involving renormalization and submit a term paper of about 10 pages. This can be a summary of research papers or (modest) original research.

**Syllabus**

**1 Historical Introduction **

## Self-energy of a Point Charge. Mass renormalization. Lorentz-Dirac Equation of motion. Landau-Lifshitz equation for radiating particle. Critical (center) manifold theory for Ordinary Differential Equations. Casimir Energy. Zero-point energy of harmonic oscillators. The Electromagnetic Field as a collection of oscillators. Casimir energy. Zeta function regularization. Euler Constant. Euler manipulated divergent series with abandon to get many useful formulas. The methods he used are a precursor to what we call renormalization. They were established rigorously a hundred or so years later using modern analysis by Weierstrass and others.

**2 Ising Model**

## Hamiltonian. Free energy. One dimensional Transfer matrix . Block Spin Transformation. Renormalization semi-group. Monotonicity. Migdal-Kadanoff approximate recursion relations in 2D. Ising Model on Hierarchical Lattices. Fixed point. Phase transition. M Kaufman, RB Griffiths PRB 1981-1984 Generalizations: Potts/Clock models. O(n)-models. Metric spaces.

**3 Singular Potentials in Quantum Mechanics**

## Delta potential in two dimensions. Coupling constant renormalization. Asymptotic Freedom. Ground state energy. Scattering amplitude. Bound state in three dimensional quantum mechanics with short range interactions. Scattering length. Interaction between cold atoms. Gross-Pitaevskii model.

**4 Feigenbaum**

## Iteration of bimodal functions. Period doubling. Renormalization map. Fixed point. Feigenvalues. Feigenbaum-Cvitanovic function.

**5 Percolation**

## Bond percolation. Clusters. Real space renormalization in percolation theory. Exact solution on Bethe Lattices. Approximate renormalization transformation on 2d triangular lattice. Fractal dimension of clusters. Comparison with Monte-Carlo simulations. Relation to Ising/Potts models.

**6 Scalar Quantum Field Theory**

## Gaussian integrals. Action. Green's Functions (Correlation Functions). Tree diagrams. Feynman diagrams. Loop expansion. Self-energy. Hierarchy problem in the Standard Model of elementary particles.

**7 Critical Phenomenon**

## Correlation length. Universality. Critical exponents. Renormalization semi-group transformations. Calculation of critical exponents of the Wilson-Fisher fixed point. Critical equation of state.

**8 Lattice Field Theory**

## Lattice regularization. Continuum limit. Correlation functions. Monte-Carlo simulation of the two dimensional Ising model. Metropolis and Wolff algorithms. Relation to percolation.