University of Illinois at Urbana-Champaign · Department of Physics

Physics 583

Advanced Field Theory

Academic Year 2023/2024

Spring Semester 2024

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
Eduardo Fradkin's Homepage

Time: 2:00 pm -3:20 pm Tuesday-Thursday
Place: Rm 158 Loomis Lab
Rubric: Phys583
CRN: 36786
Credit: 1 unit.
Office Hours: Mondays 4:00-5:00 pm US Central Standard Time, Rm 2119 ESB (occasionally on zoom)
TA: Matthew O'Brien
Office Hours: Wednesdays 3-4pm and Fridays 4-5 pm, Rm 3110 ESB


updated on 4/4/2024

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 2023. 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 2023 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 2024, 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. I strongly recommend that you read the material that will be covered in class from my lecture notes that you can find in this webpage.

You will also find links to the homework sets and to their solutions. There will be a total of three homework sets (possibly one more). 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.

All homework sets are due on 9:00 pm (US Central Time) of the assigned due date. Below you will find the HW sets that you will need to upload with a clearly stated deadline (all times are in US Central Time). You will have to upload your solutions to the my.physics space for Physics 583, from where you will be able to download the graded sets. You must upload a clearly legible pdf file so that the TA can read it. You should not email your solutions! I strongly recommend that you write your solutions using LaTeX and send the pdf file thus created. If for some reason you need to send a handwritten solution it should be written with dark pen on a double spaced paper with the equations presented (not a written in the line text). Unreadable solutions will not be graded. No late solution sets will be accepted unless you prearrange this beforehand with the TA and with me. There will be a 20 percent grade penalty for late solution sets.

In addition to my office hours (see above), the TA will host each an office hour.

There will not be a midterm exam but there will be a Final Exam in the form of a Term paper and and Final Oral Presentation. See below for details.

The Final grade will be determined by the three homework sets, the Term Paper and the Final Oral Presentation. Each will carry the same weight. The letter grades will be determined as follows: an A+ will require at least 95% of the grade, an A at least 90 % of the grade, an A- at least 85% of the grade, a B+ at least 80% of the grade, a B at least 75% of the grade, a B- at least 70% of the grade, a C+ at least 65% of the grade, and so on.

You can access the Physics 583 Gradebook here


Please find below the HW sets and the list of suggested Term Papers for Physics 583. Please notice the clearly shown deadlines. To upload the pdf file of your work you will go to the Physics 583 space in my.physics. There you will find the link "Course uploads" and clicking on it you will be able to upload your solutions and download the graded solutions. There you will see a menu with the list of homeworks for Physics 583 and you will choose which homework solution you wish to upload. After the homework is graded by the TA you will be able to access and download the graded work by using this same link.

Homework set No. 1
posted Sunday January 21, 2022 , Due Friday, February 9, 2024, at 9:00 pm CST, edited on 1/24/2024

Homework set No. 2
posted Sunday February 11, 2024; Due Sunday March 17, 2024, 9:00 pm CST

Homework set No. 3
posted Sunday March 17, 2024 ; Due Tuesday April 16, 2024, 9:00 pm CDT (extended deadline)

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 March 17, 2024. Please select three topics of your choice and send me an e-mail with your selections (ranked ordered) no later than Friday April 5, 2024, 9:00 pm, US Central Time.

The Term paper will be due at 9:00 pm US CDT on Wednesday May 8, 2024. You will have to send me the pdf file of your Term paper by e-mail to my e-mail address: 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 date of the Final Exam on Thursday May 9, 2024, in Rm. 3110 ESB beginning at 9:00 am CDT. 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 Wednesday May 8, 2024 no later than 9:00 pm US Central Time. 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 4-2), 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.

Course Plan

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 (1PI) 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.

The Perturbative Renormalization Group
Renormalized perturbation theory. Upper and lower critical dimensions. Scaling behavior and corrections to scaling. Callan-Symanzik equations and scaling behavior; dimensional regularization with 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.

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.

Anomalies in Quantum Field Theory.

The chiral anomaly in 1+1 and 3+1 dimensions. Non-perturbative behavior in 1+1 dimensions. Abelian and non-Abelian bosonization. Fractional charge. Parity anomaly in 2+1 dimensions. Anomaly inflow. Theta vacua.

Scale and Conformal Invariance in Field Theory.
Conformal Field Theory in Quantum Field Theory,
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.

Topological Field Theory.

Gauge theories and topology; discrete gauge theories; Chern-Simons gauge theory and knots.


We will use my textbook "Quantum Field theory: An Integrated Approach", Princeton University Press.

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", Princeton University Press.

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

Last updated 4/4/2024