Classical Field Theory:
( pdf file
), updated: 10/10/2003
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
), updated: 10/10/2003
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
), updated: 10/2/2003
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 ), updated: 10/16/2003
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
), updated: 10/23/2003
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
), updated: 10/31/2003
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
), updated: 12/9/2003
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
), updated: 12/9/2003
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
), updated: 12/10/2003
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.
Perturbation Theory and Feynman Rules ( pdf file
), updated: 11/13/2003
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.
Applications of Perturbation Theory
electron-positron anihilation and scattering processes in QED.
Hartree Approximation in many-body systems.
Mean Field Theory in Classical Statistical Mechanics.
Connected and Disconnected Green's functions.
Exponentiation of connected diagrams. Reducible and Irreducible
Diagrams.
One particle Irreducible (1 PI) Vertex Functions. Physical content.
Self Energy. Vacuum Polarization.
Vertex Part.
The generating functional of 1PI vertex functions. Theory of the
effective potential. Spontaneous and explicit symmetry breaking.
Ward Identities.
Fluctuations and Radiative Corrections.
One-loop radiative corrections in QED. Gauge Invariance.
Divergencies, regularization and renormalization. Effective charge.
Fluctuations in the Landau Theory of phase
transitions; mass and coupling constant renormalization (to 1 loop).
Hartree, Hartree-Fock and Random Phase Approximations in the Fermi gas: self
energy, vacuum polarization, and vertex part. Dialectric Constant and
Screening.
Canonical Transformations in Field Theory.
Bogoliubov Transformations for Fermi and Bose Systems. Ground state
instabilities. Examples, BCS superconductors (baby version).
Jordan-Wigner transformation and quantum spin chains. Diagonalization and
spectra.
Finite Temperature Field Theory and Quantum Statistical mechanics
Fermions and Bosons. Boundary Conditions. Matsubara sums. Thermal
Green's function. Feynman rules. Response functions at finite temperature and
frequency.
Path Integrals: connection between Classical Statistical Mechanics and
Quantum Field Theory
Path Integrals and the
connection between Classical Statistical Mechanics and Quantum Field Theory.
The Transfer Matrix. The Ising Model as a Quantum Field Theory.