Dec 11, 2006

RENORMALIZATION THEORY

Texts

PROBLEM SETS: I would like to receive solutions to problem sets by Tuesday, at the lecture, or in my mailbox, School of Physics.

tuesday 08/22
Overture

There are 3 strains of renormalization theory; inseparability of charged particles from accomagning fields, scaling of fluctuations close to a phase transitions, and renormalization on dynamically generated fractals. We start by discussing the self-similar structure of Julai/Mandelbrot sets, by iteration of the Fatou complex quadratic polynomial.

Optional reading:

Teaching the renormalization group H.J. Maris and L.P. Kadanoff, American Journal of Physics , 46, 652 (1978)

thursday 08/24
tuesday 08/29
Complex renormalization

Renormalization on dynamically generated fractals: the self-similar structure of Julia/Mandelbrot sets.

Reading:

Complex universality

(boyscout version version11.8, Sep 7 2006)
Very preliminary draft - please return your copy to Predrag with your edits and comments, by tuesday 09/12

Playing:

E. Demidov,

Anatomy of Mandelbrot and Julia sets
Play with it, renormalize by clicking

thursday 08/31
tuesday 09/05
Period doubling route to chaos

Renormalization of period-doubling bifurcation sequences observed in a wide range of physical systems

Reading:

Universality in transitions to chaos

(boyscout version version11.8, Sep 7 2006)
Very preliminary draft - please return your copy to Predrag with your edits and comments,
by tuesday 09/12

Optional reading:

A simpler derivation of Feigenbaum's renormalization group equation for the period-doubling bifurcation sequence S. N. Coppersmith, Am. J. Phys., 67, 52 (1999). (does not look ``simpler" to Predrag)

Exercises:

(optional: 26.1 Period tripling - complex universality)

27.1 Period doubling (updated Fri Sep 15)
(optional: 27.2 Period doubling in a nonlinear oscillator)
due tuesday 09/19

Solutions:

Period doubling solutions by Jorge and Ed (Tue Sep 19)

thursday 09/07
tuesday 09/12
thursday 09/14
Gaussian distributions

Probabilistic behavior of many variables, lattice Green function

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 3: Gaussian distributions

tuesday 09/19
Gaussian lattice Green function

Translation invariance diagonalizes the Green function in Fourier space. Fourier decomposition as the simplest example of representation theory for finte groups.

Reading:

P. Cvitanović, Quantum Field Theory lecture notes

Section 3.7: Propagator in the space representation
Section 3.8: Periodic lattices

thursday 09/21
tuesday 09/26
thursday 09/28
1- Ising and Gaussian models, duality

Transfer operators along a space direction = QM in imaginary, periodic time

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 4: Quantum Mechanics and Lattices

tuesday 10/03
Operators, correlations, 2-d Ising model

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 4: Quantum Mechanics and Lattices

thursday 10/05
tuesday 10/10
Duality

Exact duality strong/weak coupling for 1-d and 2-d lattices

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 15: Duality
Sections 15.1 to 15.5 (skip 15.6 and beyond)

Wikipedia, Poisson resummation

thursday 10/12
Random Dynamics

From hopping and stopping to propagators

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 5: Diffusion and Hopping
Sections 5.1 to 5.5 (skip security prices)

10/16-17 Fall 2006 Recess Break

thursday 10/19
tuesday 10/24
thursday 10/26
Statistical mechanics and diffusion

Einstein theory of Brownian motion

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 6: From Hops to Statistical Mechanics
Sections 6.1 to 6.7: Random walk in momentum; evolution of probability densities; Fokker-Planck equation

Optional reading:

Transporting densities

(boyscout version version11.8, Sep 7 2006)
Sects. 9.5 and 9.6 (in particular) cover the same material as Kadanoff
Noise

(boyscout version version11.8, Sep 7 2006)
Sects. 35.1 to 35.3 cover the same material as Kadanoff
Sects. 35.4 and 35.5 are preliminary - if you spot some errors, please return your copy to Predrag with your edits and comments

tuesday 10/31
thursday 11/02
Field theory in a few easy pieces

all about partition functions, free energies, Gibbs free energies and Dyson-Schwinger equations

Reading:

Generating functionals

P. Cvitanović, Field theory: chapter 2

tuesday 11/07
NOTE: class starts at 10:05 (Predrag fingerprinted again for FBI's delectation)
Generating functionals

partion function, free energy, Gibbs free energy for innocent, summed up

NOTE: guest lecture at 11:05
Kurt Wiesefeld: Self-organized criticality

thursday 11/09
tuesday 11/14
thursday 11/16
Mean field theory of critical behavior

Phase transitions in mean-field Ising and infinite range models; critical exponents, scaling functions, correlations at criticality

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 10: Phase transions
Chapter 11: Mean field theory
[skipped Chapter 12: Continuous phase transitions; you might want to read it anyway, for the history of the subject]

tuesday 11/21
Renormalization, simple exact examples

Decimation, renormalization flows

Reading:

L.P. Kadanoff, Statistical Physics

Chapter 13: Renormalization in 1 dimension

11/23-24 thanksgiving

tuesday 11/28
Real space renormalization

This is the part of the course that is "not in another textbooks" (says Leo): the Kadanoff spin-blocking and the Kadanoff-Migdal construction

Reading:

L.P. Kadanoff, Statistical Physics

Section 13.8: 2-d Ising model
Chapter 14: Real space renormalization techniques

thursday 11/30
tuesday 12/05
thursday 12/07
Renormalization group

How come the same renormalization technique works for the highbrow QFT (all graphs divergent, with relativity and QM apparently playing key roles) and for the lowly stat mech of phase transitions? The whole structure of renomalizability comes from demanding that a theory is fully specified with a finite number of coupling (or correlation) strength parameters; this condition separates the good from the bad among theories. Too bad it relegates general relativity to the bad.

Reading:

B. Delamotte, A hint of renormalization, Am. J. Phys 72, 170 (2004).

Technical parts are a pedagogical presentation of Bogoliubov-Shirkov. The summary gives a good overview of the current understaning of renormalization in QFT (where it is unavoidable), and in statistical mechanics (where it is of importance only in an unaturally small neighborhood of a phase transition).

Friday 12/08:    classes end

Monday 12/18 grades deadline   

Optional reading:

Hans Bethe and Quantum Electrodynamics, Freeman Dyson, Physics Today (Oct 2005)
Evolution of the Bogoluibov Renormalization Group, D.V. Shirkov (1999): a rambling overview, of no pedagogical use as it cannot be understood unless one already knows the history of QFT renomrmalization.
Renormalization group Wikipedia

Lecture 9          12:30 - 2:00 tuesday, Sep 27, 2006 in Howey W505
Perturbative lattice field theory

Lecture 10          12:30 - 2:00 thursday, Sep 29, 2006 in Howey W505
Asymptotic expansions

Lecture 11           12:30 - 2:00 tuesday, Oct 4, 2006 in Howey W505
Feynman diagrams

Reading:
P. Cvitanović Lattice Field Theory notes. Current version is still not final (!), but read it anyway, do the problems (in particular, the one on phi^4 asymptotics).

Lecture 12          12:30 - 2:00 thursday, Oct 6, 2006 in Howey W505
Functional integrals in Fourier space

Reading:
P. Cvitanović Field theory, chapter 2. I do not mean that you should really work through this - it is too much field theory for our goals in this course, but there are a few point that might be of interest:

Lecture 16          12:30 - 2:00 thursday, Oct 20, 2006 in Howey W505
Renormalization in 3.99 dimensions