University of Illinois at Urbana-Champaign · Department of Physics

Physics 582

General Field Theory

Fall Semester 2016

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 Monday-Wednesday
Place: Rm. 158 Loomis
Office Hours: Thursdays 4:00 pm-5:00 pm, Rm 2119 ESB
Credit: 1 unit.
CRN: 30717

TA: Chang-Tse Hsieh
Office Hours: Tuesdays 4 pm to 5 pm, Rm 4139 ESB
E-mail: chsieh9@illinois.edu

TA: Han-Yi Chou
Office Hours: Fridays 10:00 - 11:00 am, Rm 4015 ESB
E-mail: hchou10@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, 2017. 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.

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 of the assigned due date. you must put your solution sets in the Physics 582 drop box (located near the Loomis end of the LLP-MRL Inter-pass, 2nd floor). No late solution sets will be accepted unless you prearrange that with the TA and with me. There will be a penalty for late solution sets.

There will not be a midterm exam but there will be a Final Exam. It will be a take home 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 Sunday December 15 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.


You can access the Physics 582 Gradebook here


Announcements


Homeworks:

Homework Set No. 1 Posted : Monday August 22, 2016 Due: Sunday September 18, 2016

Solutions to homework set 1; Posted: Thursday October 6, 2016

Homework Set No. 2 Posted: Monday September 19, 2016 Due: Monday October 3, 2016, 5 pm

Solutions to homework set 2; Posted: Thursday October 20, 2016.

Homework Set No. 3 Posted: Wednesday October 5, 2016Due: Wednesday October 19, 2016

Solutions to homework set 3; Posted: Thursday October 27, 2016

Homework Set No. 4 pdf file; Posted: Wednesday October 19, 2016 Due: Sunday October 30, 2016, 5:00 pm

Solutions to homework set 4; Posted: Posted December 4, 2016

Homework Set No. 5 Posted: Monday October 31, 2016 Due: Monday November 14, 2016, 9:00 pm,

Solutions to homework set 5; Posted: December 7, 2016.

Homework Set No. 6 Posted: Sunday November 13, 2016 Due: Friday December 2, 2016, 9:00 pm

Solutions to homework set 6; Posted: Sunday, December 11, 2016.

Homework Set No. 7/ Final Exam Posted: Saturday December 3, 2016 Due: Friday December 16 at 5:00 pm CST. A clearly readable pdf file (composed in LaTeX or clearly written) should be sent to me (not to the TA's) by email. This homework will be the Take Home Final Exam for Physics 582

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. Schrodinger, 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's functions. Imaginary time. Minkowski space and Euclidean space

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's Functions in Classical Electrodynamics.
Propagators, Time-Ordered Products and Green's Functions in Quantum Field Theory. S-matrix elements and Green's 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's functions. Feynman Diagrams.
Feynman Rules for scalar fields and QED.
Feynman Rules for a non-relativistic Fermi-Gas at zero temperature.
Feynman Rules for the Landau Theory of Phase transitions.

Bibliography


Last updated 12/11/2016