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 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 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 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 Final Exam. It will be a take home exam. You will
have to prepare your solution set electronically. 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 19 at 9:00 pm. Alternatively you may prepare a
handwriten solution. In that case you must write with dark ink, so that it can be
scanned and be clearly legible. You the must scan your solution and
send me the pdf file by email. You should not put your solution neiher in the Physics
582 homework box nor in my mail box as I will be out of town. In order for your Final
to count you must send me your solution by email.
Classical Field Theory:
( pdf file
)
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 ( pdf file
)
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 ( pdf file
)
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
( pdf file )
Path Integrals and Quantum Mechanics. Density matrix.
Evauating 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 ( pdf file
)
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 ( pdf file
)
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 ( pdf file
)
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 ( pdf file
)
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 . ( pdf file
)
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; pdf file)
Perturbation Theory and Feynman Rules ( pdf file
)
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.