University of Illinois at Urbana-Champaign · Department of Physics

Physics 485

Advanced Field Theory

Academic Year 2003/2004

Spring Semester 2004

Instructor: Professor Eduardo Fradkin

Department of Physics
University of Illinois at Urbana-Champaign
Room 2119 ESB, MC-704,
1110 W Green St, Urbana, IL 61801-3080
Phone: 217-333-4409
Fax: 217-244-7704
E-mail efradkin@uiuc.edu
Eduardo Fradkin's Homepage


Time: 9:00-10:20am Monday-Wednesday
Place: Rm. 158 Loomis
Call Number: 06595
Credit: 1 unit.
Office Hours: Tuesdays 4:00-5:00 pm, Rm 2119 ESB
TA: Eun-Ah Kim,
Address: Rm 3101 ESB
Office Hours: Fridays 10:00-11:00 am, Rm 3101 ESB
Phone: 333-6276
e-mail: eunahkim @uiuc.edu



Announcements



In many areas of Physics, such as High Energy Physics, Gravitation, and in Statistical and Condensed Matter Physics, the understanding of the essential physical phenomena requires the consideration of the collective effects of a large number of degrees of freedom. Quantum Field Theory is the tool as well as the language that has been developed to describe the physics of problems in such apparently dissimilar fields.

Physics 485 is the second half of a two-semester sequence of courses in Quantum Field Theory. The first half, Physics 483, was be taught in the Fall Semester 2003. The aim of this sequence is to provide the basic tools of Field Theory to students (both theorists and experimentalists) with a wide range of interests in Physics. These ideas and tools will be used in subsequent and more specialized courses. As a prerequisite I will assume that the students have mastered the contents of the Physics 480/481 sequence on Quantum Mechanics (or equivalent), and that they have taken Physics 483. Last Fall 2003, in Physics 483 we studied the basic conceptual and computational tools of quantum field theory. We also discussed the applications of these methods to several areas of Physics, such as High Energy and Statistical and Condensed Matter Physics, both in the Lectures and in the Problem Sets.

In this semester Spring 2004, in Physics 485 we will discuss a number of advanced topics in Quantum Field Theory, including Gauge Field Theories, the Renormalization Group in Quantum Field Theory and in Statistical Physics, non-perturbative methods in Quantum Field Theory, including solitons and instantons, and 1/N expansions; elementary Conformal Field Theory and its applications to String Theory and Critical Phenomena; Topology in Quantum Field Theory and applications to problems in Condensed Matter such as quantum Hall physics and Berry phases.
Below you will find a detailed Course Plan (or Syllabus) for Physics 485. It is divided in items and there you will find links to my class notes. I will post them as they become available. You will also find links to the homework sets and to their solutions. There will be a total of five homework sets (more or less). The homeworks are very important. There you will find many applications to different problems in various areas of Physics in which Field Theory plays an essential role. You will not be able to master the subject unless you do (and discuss) the problem sets. You can look up your grades by logging into the Physics 485 Gradebook . There will not be a midterm exam but there will be a Final Exam in the form of a Term paper.

Homeworks


Homework set No. 1 pdf file
posted 2/16/2004, Due 3/1/2004, misprints edited on 2/20/2004

Solutions to Homework Set No.1; Posted on 3/18/2004


Homework set No. 2 pdf file
posted 3/7/2004, Due 3/19/2004

Solutions to Homework Set No.2; Posted on 5/3/2004


Homework set No. 3 pdf file
posted 4/17/2004, Due 4/30/2004, misprints edited on 4/24/2004

Solutions to Homework Set No.3; Posted on 5/18/2004


Term Paper


List of Suggested Term Papers: Please click here to see a list of suggested Term Papers.

The Term paper will be due at 8:00 am at the Final Exam for Physics 485, room 158 LLP.

The day of the Final Exam is Wednesday May 12, 2004. We will meet in Room 158 LLP (our usual classroom) from 8:00am to 11:00 am (sorry, it was not my choice!). The Final Exam will consist on an oral presentation of your paper for the entire class. Each one of you will have 20 minutes to present your work. Please check the Final Exam Program to see at what time you are scheduled to give your talk. Please note that all students will be expected to attand all the talks.

You will assume that you are giving a talk for a reasonably educated audience (at least in field theory). The talk must be reasonably self-contained, and the assumptions and your results must be clearly stated. You will also turn in your written paper on the day of the presentation. At the end of the Semester I will put all the Term Papers together in a printed volume which will be distributed among the students registered in the class.

The paper must be formatted in LaTeX, which is the standard program for the production of science papers. Other lower quality formats, such as Word, will not be accepted.

The paper must be at least ten (10) pages long, double spaced pages, not including the title page, in 10pt. font. The title page must include the title, your name and an abstract. The paper must include a section with introductory material in which you give the background information and the main motivation. There should also be a main section in which you discuss the principal content, including the details of the model, the approximations that you use and the techniques that are needed to understand the results. Here you will present the main results and you will discuss whatever calculations you had to do. You may put the details of these calculations in an Appendix if these calculations are too involved and disrupt the natural logical flow of the paper. You should have section with your Conclusions and another one with your References.

You can either use the "article" documentclass (which is standard in LaTex 2e) or you can use the APS package (RevTeX 4), which also runs on LaTeX 2e; in this case please declare the document as a "preprint".
Figures: If you wish to use figures in your paper you are welcome to do so but they must be in eps ("encapsulated postscript") format. They must also be included in the text.
LaTeX Resources:. There are lots of resources for the use of TeX and LateX. The best books are The TeX Book by Donald Knuth (Addison Wesley) and Guide to LaTeX, by Helmut Kopka and Patrick W. Daly (Addison Wesley). A good summary can be found in this document on LaTeX2e.
You can also find examples of documents in TeX in the website of the Journals of the American Physical Society. Otherwise you may want to use the following example of a paper in LaTeX: latex file, pdf file


Book with the Physics 485/Spring 2004 Term Papers
Please click here to download a book with the Term Papers. Everyone did a superb job! (Please beware that this is a large file, about 1.1MB). Have a great summer!

Course Plan

Review of Path integrals in Quantum Field Theory. This material was covered in Physics 483 this last Fall 2003. Please see my Physics 483 Lecture Notes.
Path integrals in Quantum Mechanics.
Path integrals in Quantum Field Theory: scalar fields.
Coherent states and path integral quantization. Path integral for spin.
Path integrals for fermionic theories. Path integral quantization of Dirac fermions.
Applications of coherent state path integrals to non-relativistic fermionic and bosonic theories.
Computation of functional determinants. Zeta-function regularization.

Quantum Field Theory and Statistical Mechanics.
Field theory at finite temperature. Density matrices and Transfer matrices.
The Ising Model as a QFT. Solution of the 2D Ising Model.

Review of Quantization of Gauge Field Theories. (This was covered extensively in Physics 483, Fall 2003. See my Fall 2003 Lecture Notes.)
Quantization of constrained systems.
Gauge fixing and path integrals. Non-abelian Gauge Theories. The Faddeev-Popov method. Ghosts. BRST invariance.
Feynman rules for gauge theories.

Generating Functionals and the Effective Potential.
Feynman diagrams. Connected, Disconnected and Irreducible Green's functions.
Exponentiation of connected diagrams. Reducible and Irreducible Diagrams.
One particle Irreducible (1 PI) Vertex Functions. Physical content. Self Energy.
The generating functional of 1PI vertex functions. Theory of the effective potential.
Spontaneous and explicit symmetry breaking. Ward Identities.
The Loop Expansion. Theory of the Low Energy Effective Action.

Regularization and Renormalization

Fluctuations. Perturbative renormalization to two loop order of a scalar field. Divergent Feynman diagrams and regularizations in QFT.
Subtractions and renormalized Lagrangians. Renormalizability. Critical dimensions
Gauge invariance and regularization. Dimensional regularization.

The Renormalization Group.

Scale dependence in Quantum Field Theory and in Statistical Physics.
Scale invariance. Fixed points and Universality in Quantum Field Theory and Critical Phenomena.
Renormalization group transformations. Construction of fixed point theories. Upper and lower critical dimensions. Scaling behavior and corrections to scaling.
Two case studies: non-linear sigma models and non-abelian gauge theories.
Renormalized perturabtion theory; Callan-Symanzik equations and scaling behavior; minimal subtraction.
Renormalizability of the non-linear sigma model in D=2 dimensions; asymptotic freedom. Renormalization of Yang-Mills gauge theories in D=4 dimensions. Infrared problems.

The 1/N expansions

O(N) scalar field theory and non-linear sigma models.
Fermionic theories in the large N limit.
Yang Mills gauge theory in the limit of large numer of colors.
The String picture of confinement and Large-N Yang-Mills theory. The Maldacena Conjecture.

Strong coupling behavior of quantum field theories.

Field theory ``beyond perturbation theory".
Lattice regularization of QFT.
Confinement in Gauge Field Theories. Higgs phases. The Higgs mechanism and mass generation. Phases of Gauge theories and Phase Diagrams.

Non-perturbative effects in QFT.

Instantons and solitons. The role of topology in Quantum Field Theory and in Statistical Physics. Elementary discussion of Homotopy groups and classes. Topological invariants.
Vortices and monopoles in scalar theories and in gauge theories.
Dualities in Statistical Mechanics and in Gauge Theory.

Scale and Conformal Invariance in Field Theory.

Conformal Field Theory in two-dimensional Quantum Field Theory, Critical Phenomena and String Theory. Elementary theory of the Bosonic String.
General consequences of conformal invariance. Conformal invariance in two-dimensions. The Virasoro Algebra. Representations. Conformal invariance, continuous global symmetries and current algebra. Kac-Moody algebras.
Applications. The 2D Ising model as a CFT. Wess-Zumino-Witten models. Quantum spin chains and conformal invariance.

Anomalies in Quantum Field Theory.

Magnetic monopoles and baryon decay.
Non-perturbative behavior in 1+1 dimensions. Abelian and non-Abelian bosonization.
Gauge theories and Topology: Chern-Simons gauge theory and knots.
Gauge theories in Condensed Matter Physics: gauge theories and the quantum Hall effects.

Bibliography

M. E. Peskin and D. V. Schroeder. ``An Introduction to Quantum Field Theory", Perseus Books, The Advanced Book Program (Reading, MA).

D. Amit, ``Field Theory, the Renormalization Group and Critical Phenomena", World Scientific.

L.Ryder. ``Quantum Field Theory", Cambridge University Press.

M. Stone , ``The Physics of Quantum Fields", Graduate Texts in Contemporary Physics, Springer-Verlag.

C. Itzykson and J. B. Zuber. ``Quantum Field Theory", McGraw-Hill.

J. D. Bjorken and S. Drell. ``Relativistic Quantum Fields", McGraw-Hill.

L. D. Landau and E. M. Lifshitz, ``The Classical Theory of Fields", Pergamon Press.

R. P. Feynman, ``Path Integrals and Quantum Mechanics", McGraw Hill.

G. Parisi, ``Statistical Field Theory", Addison Wesley.

J. Zinn-Justin, `` Quantum Field Theory and Critical Phenomena", Oxford University Press.

M. Green, J. Schwartz and E. Witten, `` Superstring Theory", Cambridge University Press.

J. Polchinski, ``String Theory", Cambridge University Press.

S. Coleman, ``Aspects of Symmetry", Cambridge University Press.

R. Rajaraman, ``Solitons and Instantons", North-Holland.

A. M. Polyakov, ``Gauge Fields and Strings", Harwood.

P. Di Francesco, P. Mathieu and D. Senechal, ``Conformal Field Theory", Springer-Verlag.

R. Balian and J. Zinn-Justin, ``Methods of Field Theory", North-Holland.

L. Schulman, ``Techniques and Applications of Path Integration", Wiley.

C. Itzykson and J. Drouffe, ``Statistical Field Theory, Cambridge University Press.

P. Ramond, ``Field Theory: a Modern Primer", Addison Wesley.

S. J. Chang, ``Introduction to Quantum Field Theory", World Scientific.

S. Doniach and E. H. Sondheimer, ``Green's's Functions for Solid State Physicists", Imperial College Press/ World Scientific.

A. Abrikosov, L. Gorkov and I. Dzyaloshinsky. ``Methods of Quantum Field Theory in Statistical Physics", Dover.

A. Fetter and J. D. Walecka. ``Quantum Theory of Many Particle Systems", McGraw-Hill.

R. P. Feynman. ``Statistical Mechanics", Addison-Wesley.

E. Fradkin. ``Field Theories of Condensed Matter Systems", Addison-Wesley.

L. P. Kadanoff and G. Baym, ``Quantum Statistical Mechanics", Addison Wesley.

J. Cardy, ``Scaling and Renormalization in Statistical Physics", Cambridge University Press.

D. Pines and P. Nozieres, ``The Theory of Quantum Liquids", Addison Wesley-Perseus.

J. Negele and H. Orland, ``Quantum Many Particle Systems", Addison Wesley.

N. Goldenfeld, `` Lectures on Phase Transitions ad the Renormalization Group", Addison Wesley.

C. Nash and S. Sen,``Topology and Geometry for Physicists", Academic Press.

S. Weinberg, ``The Quantum Theory of Fields" (three volumes), Cambridge University Press.

A. Zee, ``Quantum Field Theory, in a nutshell", Cambridge University Press.

M. Kaku, ``Quantum Field Theory", McGraw-Hill.


Last updated 5/18/2004