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 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.
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
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.
to imaginary time and the connection between Quantum Field Theory
and Classical Statistical
Symmetries and Conservation Laws
Continuous Symmetries, Conservation Laws and
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
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
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
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
(See my additional notes from Physics 561 on
Linear Response Theory)
Perturbation Theory and Feynman Rules
Wick's Theorem, generating functional and perturbation
Perturbation expansion for vacuum amplitudes and Green functions. Feynman
Examples of Feynman Rules: φ^4 theory and QED,
non-relativistic Fermi gas at zero
temperature, and Landau Theory of Phase transitions.