Syllabus
Quantization of free fields (continued from QFT I):
Propagator of a free quantized field (scalar, spinor and massive vector fields). Propagator as the Green´s function of relativistic wave equation. Pauli - Jordan commutator functions for scalar field. Quantization of free electromagnetic (Maxwell) field. Non-covariant physical gauge ("radiation gauge"). Propagator. Covariant quantization. Feynman gauge. Gupta - Bleuler method. Indefinite metric. Lorentz condition and physical states. Propagator in a class of covariant gauges (a-gauges, with Feynman gauge corresponding to a = 1).
Basics of Feynman diagram techniques:
Wick theorems. Normal form of S-matrix. Quantum electrodynamics (QED). Normal ordering of the current in interaction Lagrangian (elimination of "tadpole" diagrams). Summary of basic rules for Feynman diagrams (vertex, internal and external lines). 2nd order processes in QED. Production of muon pair in electron-positron annihilation. Compton scattering (Klein - Nishina formula). Two-photon annihilation of electron-positron pair. Elastic scattering of electron and positron (Bhabha) and scattering of electrons (Möller). Scattering in external Coulomb field: Mott formula for the cross section. Non-relativistic and ultrarelativistic limits. Polarization phenomena (degree of longitudinal polarization as a function of energy and scattering angle).
Closed-loop diagrams:
Examples of fourth-order contributions to the S-matrix for a specific scattering process (e - m scattering). Diagrams involving closed loops of internal lines. Ultraviolet divergences. Practical calculation of one-loop Feynman graphs. Dimensional regularization. Feynman parametrization. Algebra of Dirac matrices. Symmetric integration. Computation of n-dimensional scalar integrals. Pauli - Villars regularization. Photon self-energy (vacuum polarization). Transverzality. Calculation of the scalar formfactor P. Imaginary part of the P and dispersion relation. Electron self-energy. Infrared divergence. Vertex correction. Ward identity. Takahashi identity and current conservation. Renormalization in QED. Bare and renormalized quantities. Counterterms. Renormalization of field operators, mass and coupling constant (charge). Equality of renormalization constants for vertex and electron field as a consequence of the Ward identity. Effect of renormalization for external lines. Triangle loops of fermion lines. Furry´s theorem. Box diagram (square loop of fermion lines): absence of ultraviolet divergences. Divergence index (superficial degree of divergence) of a general one-particle irreducible graph and criterion for renormalizability in field theory models.
Examples of physical effects of closed-loop diagrams:
Photon - photon scattering: low-energy limit and estimation of the cross section by means of effective Lagrangian of dimension 8. QED correction to the Coulomb potential due to vacuum polarization (Uehling correction, dependence of effective charge on distance). Anomalous magnetic moment of electron (Schwinger correction). Summation of corrections to photon propagator and "running coupling constant". Basic ideas of renormalization group. Homogeneous Callan - Symanzik equation and its solution. Beta function for renormalization of coupling constant. Anomalous dimensions.