Department of Physics
University of Illinois at UrbanaChampaign
Room 2119 ESB, MC704,
1110 West Green Street, Urbana, IL 618013080
Phone: 2173334409
Fax: 2172447704
Email efradkin@illinois.edu
Eduardo Fradkin's Homepage
Time: 2:30 pm3:50 pm MondayWednesday
Place: Rm. 158 Loomis
Office Hours: Tuesdays 4:00 pm5:00 pm, Rm 2119 ESB
Credit: 1 unit.
CRN: 30717
TA: Mr. ZeMin Huang, Rm 390K, Loomis Lab
Office Hours: Thursdays 4:00 pm to 5:00 pm
Email: zeminh2@illinois.edu
TA: Mr. Marcus Rosales
Office Hours: Mondays 4:00 pm to 5:00 pm, Third Floor Commons Area, ICMT (ESB)
Email: marcusr2@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 twosemester sequence of courses
in Quantum Field Theory. The second half, Physics 583, will be taught in the
Spring Semester, 2020. 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, nonperturbative 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 LLPMRL Interpass, 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 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 Thursday December 19 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
Homeworks:
Homework Set No. 1
Posted : Sunday August 25, 2019 Due: Friday September 13, 2019
Solutions to homework set 1;
Posted: Thursday September 19, 2019.
Homework Set No. 2
Posted: Sunday September 15, Due: Friday September 27, 5:00 pm
Solutions to homework set 2;
Posted:
Homework Set No. 3
Posted: Due:
Solutions to homework set 3;
Posted:
Homework Set No. 4 pdf file;
Posted: Due:
Solutions to homework set 4;
Posted:
Homework Set No. 5 pdf file;
Posted: Due: ,
Solutions to homework set 5;
Posted:
Homework Set No. 6
Posted: Due:
Solutions to homework set 6;
Posted:
Homework Set No. 7/ Final Exam
Posted: Due: Thursday December 19, 2019,
at 5:00 pm CST. A clearly readable pdf file (composed in LaTeX) 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. SpaceTime 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 KleinGordon 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.
NonAbelian Gauge Invariance. Minimal Coupling.
The role of topology: the AharonovBohm effect.
SpaceTime Symmetries and the EnergyMomentum Tensor.
The EnergyMomentum tensor and the geometry of
spacetime.
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 Smatrix.
Propagators and path integrals. Propagator for a Relativistic Real and
Complex Scalar Fields. PathIntegral
representation of the Smatrix and Green functions. Imaginary time. Minkowski
space and Euclidean space
NonRelativistic Field Theory
Review of Second Quantization for ManyParticle Systems.
ManyBody Systems as a Field Theory.
NonRelativistic Fermions at zero temperature: ground
state, spectrum of lowlying excitations.
Propagator for the
NonRelativistic 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 spinstatistics
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
Pathintegral 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 YangMills nonAbelian Gauge theories.
Gauge fixing, covariant gauges and the FaddeevPopov construction.
Ghosts. BRST invariance.
Physical Observables and Propagators .
The Propagator in NonRelativistic Quantum Mechanics:
retarded, advanced and Feynman propagators.
Green Functions in Classical Electrodynamics.
Propagators, TimeOrdered Products and Green
Functions in Quantum Field Theory.
Smatrix elements and Green functions. Analytic
properties. Lehman representation. Spectrum.
Crosssections and and the Smatrix.
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,
nonrelativistic Fermi gas at zero
temperature, and Landau Theory of Phase transitions.
