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Dec 11, 2006
RENORMALIZATION THEORY
Texts
Leo P. Kadanoff, Statistical Physics: Statics, Dynamics and Renormalization (World Scientific, Singapore 2000);
badly mangled GaTech library
"ebook" version
Physics Today review
All chapter and exercise
numbers refer to this book, unless stated otherwise.
P. Cvitanović, Field theory webbook
P. Cvitanović Quantum Field Theory lecture notes
PROBLEM SETS:
I would like to receive solutions to problem sets by Tuesday, at the
lecture, or in my mailbox, School of Physics.
Course Schedule

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 selfsimilar
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 selfsimilar 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 perioddoubling 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 perioddoubling 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, 2d 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 1d and 2d 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)

10/1617 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;
FokkerPlanck 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 DysonSchwinger 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: Selforganized criticality


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

Phase transitions in meanfield 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

11/2324 thanksgiving

tuesday
11/28
Real space renormalization

This is the part of the course that is "not in another textbooks"
(says Leo):
the Kadanoff spinblocking and the
KadanoffMigdal construction

Reading:

L.P. Kadanoff,
Statistical Physics

Section 13.8: 2d 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 BogoliubovShirkov.
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
QFT topics this course did not cover

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
Reading:
P. Cvitanović Lattice Field Theory notes. They are not finished yet, so you might not want to print them on paper
yet.
P. Cvitanovicć Field theory, chapter 3, sections 3.B, 3.C,
Problems:
3.B.1 on page 36.
Lecture 10 12:30
 2:00 thursday, Sep 29, 2006 in Howey W505
Asymptotic expansions
Saddle point expansions, phi4 perturbative corrections.
Reading:
Intro to Dingle monograph Asymptotics 
A Behavioural Survey pp. 14.
Problems:
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:
Section 2.F purports to relate full and connected Green functions in
a few lines. Many books work do this via painful combinatoric proofs which
seem pointless and not very bright to me,
so let me know if there is a hole in my derivation.
Exercise 2.D.1 defines Quantum Electro Dynamics notation that I forgot to
explain in exercise 3.F.3. I gave you this exercise to treat diagrams
that one runs into always in condensed matter physics (particlehole
exciations, etc).
Problems:
do the
current set
from the Lattice Field Theory notes.
Lecture 16 12:30
 2:00 thursday, Oct 20, 2006 in Howey W505
Renormalization in 3.99 dimensions
We focus on Wilson's momentum space renormalization in 3.99 dimensions.
Reading:
C Jayaprakash's
4d dimension renormalization on three sheets of paper
Predrag
Cvitanović