University of Illinois at Urbana-Champaign · Department of Physics

Physics 582

General Field Theory

Fall Semester 2021

Instructor: Professor Eduardo Fradkin


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

Time: 2:30 pm-3:50 pm (US Central Time) Monday-Wednesday
Place: Room 158 Loomis Lab (and by zoom if needed)
Office Hours: Tuesdays 4:00 am-5:00 pm (US Central Time), by zoom (zoom link supplied)
Credit: 1 unit.
CRN: 30717

TA: Marcus Rosales
Office Hours: Wednesdays 11 am - 12:00 noon Central Time (by zoom)
E-mail: marcusr2@illinois.edu

TA: Matthew O'Brien
Office Hours: Thursdays at 4:30-5:30 pm Central Time (by zoom)
E-mail: mco5@illinois.edu





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 582 is the first half of a two-semester sequence of courses in Quantum Field Theory. The second half, Physics 583, will be taught in the Spring Semester, 2022. 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 graduate level Quantum Mechanics (or equivalent). having successfully passed the graduate level sequence in Quantum Mechanics is a prerequisite for the Quantum Field Theory sequence. You should not register in this class if you did not pass this prerequisite.
In Physics 582 we will study the basic conceptual and computational tools of quantum field theory. We will discuss 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 Physics 583 we will discuss advanced topics including Gauge Theories, the Renormalization Group in Field Theory and in Statistical Physics, non-perturbative methods in QFT (solitons and instantons), elementary Conformal Field Theory and its applications to String Theory and Critical Phenomena, and QFT, Topology and quantum Hall physics.
Below you will find a detailed Course Plan (or Syllabus) for Physics 582. It is divided in items and there you will find links to my class notes. I will post them as they become available.

As you are aware this Fall 2021 this course will be taught in person (and possibly by zoom). If you cannot attend in person it is important that you have access to a computer and good internet access for this to work. My zoom lectures from the Fall 2020 were recorded and subtitled and the recordings are available in MediaSpace (Recorded Lectures). Access to the recorded lectures is available only to registered students.

I strongly recommend that you read the material that will be covered in class from my book. I also want to encourage you to come with questions and to send me the questions ahead of time by email.

You will also find links to the homework sets and to their solutions. There will be a total of six homework sets. The homework problem sets 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 homework sets that you will need to download with a clearly stated deadline (all times are in US Central Time). You will either have to prepare your solutions using LaTeX (much preferred) or by scanning a clear handwritten version of your solutions. You will have to upload your solutions to a website indicated below. It is fine to upload a scanned file of a handwritten solution provided it is clearly legible. This means that your solutions must not contain anything crossed out and written with dark pen with large enough script on doubly spaced paper with the equations presented (and not written as part of a line text) that it can be easily read. You should not email your solutions! Unreadable solutions will not be graded. The website where you will be able to upload your solutions (and where you will also be able to download the graded set) is accessed through this link. 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), both TAs will host each an office hour also by zoom. The TAs will send the invitations to each registered student to their office hours zoom meetings by email.

There will not be a midterm exam but there will be a Take Home Final Exam. You may prepare your solution set electronically, and you have the choice of either preparing the solution in LateX (strongly preferred) and to send me the pdf file by email no later than Friday December 17 at 5:00 pm (CST). Alternatively you may prepare a handwritten solution. In that case you must write with dark ink and be clearly legible. You should send me the pdf file of your solution by email. You should not put your solution neither in the Physics 582 homework box nor in my mail box as I will be out of town. Please recall that the Final Exam is mandatory and I will have to give an F to any student who does not turn a solution in time. There will be no extensions as I need to turn the grades in promptly.

The Final grade will be determined by the six homework sets and the Take Home Final Exam. The Take Home Final Exam will carry the same weight as one of the six homework sets. 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 582 Gradebook here


Announcements


Homeworks:

Please find below the HW sets for Physics 582. Please notice the clearly shown deadlines. To turn in the pdf file of your work we will follow the following procedure. Upon clicking on this link you will access a website where you can upload your pdf file. To access this secure website you will need to enter your University of Illinois netid and password. Then you will see a menu with the list of homeworks for Physics 582 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 : Monday August 23, 2021 Due: Friday September 10, 2021, 5 pm (US Central Time)

Solutions to homework set 1; Posted: Wednesday September 22, 2021.

Homework Set No. 2 Posted: Sunday September 12, 2021 Due: Friday September 24, 2021, 9:00 pm (US Central Time)

Solutions to homework set 2; Posted: Friday October 1, 2021.

Homework Set No. 3 Posted: Friday September 24, 2021Due: Friday October 8, 2021, 9:00 pm (US Central Time).

Solutions to homework set 3; Posted: Wednesday October 13, 2021.

Homework Set No. 4; Posted: Friday October 9, 2021 Due: Friday October 22, 2021, 9:00 pm (US Central Time) New deadline: Monday October 25, 2021, 9:00 pm US Central Time

Solutions to homework set 4; Posted: Wednesday October 27, 2021.

Homework Set No. 5; Posted: Monday October 25, 2021 Due: Friday November 12, 2021, 9:00 pm (US Central Time) , replaced with an edited version on Wednesday October 27, 2021

Solutions to homework set 5; Posted: Thursday November 18, 2021

Homework Set No. 6 Posted: Friday November 12, 2021 Due: Saturday December 4, 2021, 9:00 pm (US Central Time) (new deadline)

Solutions to homework set 6; Posted: Tuesday December 7, 2021.

Homework Set No. 7/ Final Exam Posted: Friday Saturday December 4, 2021, Due: Friday December 17, 2021, 9:00 pm (US Central Time). A clearly readable pdf file (composed in LaTeX or a clearly legible handwritten scanned version) should be uploaded to the same website you used for the other problem sets. This last homework set will be the Take Home Final Exam for Physics 582. No late solutions will be accepted.

Course Plan

Introduction to Quantum Field Theory

Classical Field Theory
Fields, Lagrangians and Hamiltonians. The action. Real and complex fields. Space-Time and Internal symmetries. The Least Action Principle. Field Equations. Minkowski and Euclidean spaces.
The free massive relativistic scalar field. The Klein Gordon Equation, its solutions and their physical interpretation. Relativistic Covariance.
Statistical Mechanics as a Field Theory. Coarse graining and hydrodynamic picture. The Landau Theory of Phase Transitions and Landau functionals. Symmetries. Analogy with the Klein-Gordon field.
Field Theory and the Dirac Equation. The Dirac Equation: The Dirac and the Klein Gordon operators. Spinors. The Dirac Algebra. Relativistic Covariance. Solutions and their physical interpretation. Symmetries. Holes. Massless particles and chirality.
Maxwell's Electrodynamics as a Field Theory. Maxwell's Equations. Gauge invariance. Solutions and gauge fixing. Helicity.
Classical Field Theory in the Canonical Formalism. Analytic Continuation to imaginary time and the connection between Quantum Field Theory and Classical Statistical Mechanics.

Symmetries and Conservation Laws

Continuous Symmetries, Conservation Laws and Noether's Theorem.
Internal Symmetries. Global Symmetries and Group Representations. Local Symmetries and Gauge Invariance. Non-Abelian Gauge Invariance. Minimal Coupling.
The role of topology: the Aharonov-Bohm effect.
Space-Time Symmetries and the Energy-Momentum Tensor. The Energy-Momentum tensor and the geometry of space-time.

Canonical Quantization

Elementary Quantum Mechanics.
Canonical Quantization in Field Theory.
A simple example: Quantized elastic waves.
Quantization of the Free Scalar Field Theory.
Symmetries of the Quantum Theory: the case of the free charged scalar field.

Path Integral Quantization in Quantum Mechanics and in Quantum Field Theory

Path Integrals and Quantum Mechanics. Density matrix.
Evaluating Path Integrals in Quantum Mechanics
Path Integral quantization of the Scalar Field Theory. Schrödinger, Heisenberg and interaction representations. The Evolution operator and the S-matrix.
Propagators and path integrals. Propagator for a Relativistic Real and Complex Scalar Fields. Path-Integral representation of the S-matrix and Green functions. Imaginary time. Minkowski space and Euclidean space, and finite temperature; erratum

Non-Relativistic Field Theory

Review of Second Quantization for Many-Particle Systems. Many-Body Systems as a Field Theory. Non-Relativistic Fermions at zero temperature: ground state, spectrum of low-lying excitations.
Propagator for the Non-Relativistic Fermi Gas. Holes, particles and the analytic properties of the propagator.

Quantization of the Dirac Theory

Quantization of the Dirac Theory: ground state, spectrum, quantum numbers of excitations, causality and spin-statistics theorem.
Propagator for the Dirac Field Theory.

Coherent State Path Integrals

Coherent State path integral quantization of bosonic and fermionic systems.
Path integrals for spin.
Grassmann variables. Path integral quantization of the Dirac theory.
Fermion and Boson determinants. Zeta function regularization.

Quantization of Gauge Theories

Path-integral quantization of the Maxwell Abelian gauge theory; quantization and gauge fixing. Propagator for the free electromagnetic field. The Wilson loop operator. Path Integral quantization of Yang-Mills non-Abelian Gauge theories. Gauge fixing, covariant gauges and the Faddeev-Popov construction. Ghosts. BRST invariance.

Physical Observables and Propagators .

The Propagator in Non-Relativistic Quantum Mechanics: retarded, advanced and Feynman propagators. Green Functions in Classical Electrodynamics.
Propagators, Time-Ordered Products and Green Functions in Quantum Field Theory. S-matrix elements and Green functions. Analytic properties. Lehman representation. Spectrum. Cross-sections and and the S-matrix.
Linear Response Theory. Measurements and correlation functions. Application to the electromagnetic response of a metal. Sum rules. (See my additional notes from Physics 561 on Linear Response Theory)

Perturbation Theory and Feynman Rules

Wick's Theorem, generating functional and perturbation theory. Perturbation expansion for vacuum amplitudes and Green functions. Feynman Diagrams. Examples of Feynman Rules: φ^4 theory and QED, non-relativistic Fermi gas at zero temperature, and Landau Theory of Phase transitions.

Bibliography


Last updated 12/7/2021