University of Illinois at Urbana-Champaign · Department of Physics

Physics 583

Advanced Field Theory

Academic Year 2016/2017

Spring Semester 2017

Instructor: Professor Eduardo Fradkin

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


Time: 2:00 pm -3:20 pm Tuesday-Thursday
Place: 136 Loomis Laboratory
CRN: 36786
Credit: 1 unit.
Office Hours: Wednesdays 2:30-3:30 pm, Rm 2119 ESB
TA: Mr. Hart Goldman
Office Hours: Thursdays 3:30 pm - 4:30 pm, Third Floor, Commons Area, ICMT
E-mail: hjg2@illinois.edu




Announcements

updated on 5/11/2017


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 583 is the second half of a two-semester sequence of courses in Quantum Field Theory. The first half, Physics 582, was taught by me in the Fall Semester 2016. 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 580/581 sequence on Quantum Mechanics (or equivalent), and that they have taken Physics 582 this Fall 2016 with me where 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. Using this link you can find my Physics 582 Lecture Notes.

In this semester Spring 2017, in Physics 583 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). 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 583 Gradebook . There will not be a midterm exam but there will be a Final Exam in the form of a Term paper. Important:. The problem sets have a due date for a good reason. Solution sets turned in during within 2 days after the due date will be accepted with a 20% penalty. Unless there is an overwhelming reason, and my prior approval, no solution sets will be accepted past the two day extension period.

Homeworks


Homework set No. 1 pdf file
posted Tuesday January 30, 2017 , Due Friday February 24, 2017, 9:00 pm ,

Solutions to Homework Set No.1


Homework set No. 2 pdf file
posted Sunday February 26, 2017; Due Friday March 31, 2017

Solutions to Homework Set No.2


Homework set No. 3 pdf file
posted Friday March 31, 20017; Due Friday April 28, 2017

Solutions to Homework Set No.3;


Term Paper/Final Exam


List of Suggested Term Papers: Please click here to see a list of suggested Term Papers. The list of suggested Term papers will be posted on February 26. Please select three topics of your choice and send me an e-mail with your selections (ranked ordered) no later than Friday March 31, 5:00 pm.

The Term paper will be due at 9:00 pm CDT on TBA. You will have to send me the pdf file of your Term paper by e-mail to my e-mail address: efradkin@illinois.edu. Your e-mail must be posted before 9:00 pm CDT. I will not accept Term Papers after that time.

There will also be an oral presentation of the Term Papers on the official date of the Final Exam, TBA. We will meet in Room 190 ESB (notice the change of room) from 9:00 am through 1:00 pm. Each student will have 15 minutes for the presentation. The presentations will have a "workshop" style: each presenter will send me the pdf or powerpoint file (keynote is also fine) on TBA no later than 9:00 pm. I will put it on my MacBook and we will use use it for the presentations. my computer Note: the Term Paper and the Presentation will be the Final Exam for this course.

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. It 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 examples of papers in LaTeX: latex file, pdf file (using bibtex for the bibliography) or latex file, pdf file (using a more straightforward way of entering the bibliography)


Course Plan

Quantization of Gauge Field Theories. Please see my Physics 582 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. Latex version (pdf file)
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 Low Energy Effective Action.

Regularization and Renormalization

The Loop Expansion.. 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.

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.

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. Conformal Invariance and renormalization. Upper and lower critical dimensions. Scaling behavior and corrections to scaling.
Two case studies: non-linear sigma models and non-abelian gauge theories.
Renormalized perturbation 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 number 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. Topological field theory.

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).

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

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. Dzyaloshinski. ``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", 2nd edition, Cambridge University Press, 2013.

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

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 5/11/2017