Syllabus
Relativistic quantum mechanics: Klein - Gordon equation. Continuity equation and probability density.
Dirac equation. Continuity equation.
Spin and total angular momentum. Non-relativistic limit and Pauli equation.
Spin magnetic moment of electron. Covariant form of Dirac equation.
Algebra of Dirac gamma matrices. Equivalence of representations.
Standard representation. Trace identities.
Invariance of Dirac equation under proper Lorentz transformations. Spinor representations of Lorentz group.
Space inversion. Covariant bilinear forms.
Solutions of Dirac equation for free particle. States with positive and negative energy.
Bispinor plane-wave amplitudes u and v. Charge conjugation.
Time reversal. Spin states.
Spin four-vector. Helicity.
Projection operators for energy and spin. Gordon decomposition.
Massless particles. Chirality.
Weyl equation and its invariance properties: proper Lorentz transformations, P, C, CP. Dirac equation for particle in external spherically symetric field.
Stationary states. Commuting observables.
Spinor harmonics (spherical spinors). Separation of angular and radial variables.
Solution of radial equations in the case of Coulomb potential. Energy spectrum of hydrogen-like atom.
Degree of degeneracy and fine structure of energy levels. Difficulties of one-particle interpretation of Dirac equation.
Proca equation. Plane waves and properties of polarization vectors.
Lagrange formalism for relativistic classical fields: Variational (stationary action) principle and Euler - Lagrange equations. Lagrangian density for Klein - Gordon, Dirac, Maxwell and Proca (massive vector) fields.
Symmetries and conservation laws. Noether's theorem.
Consequences of invariance under Poincaré group: energy-momentum tensor, angular momentum. Internal symmetry.
Invariance under phase transformations and conservation of vector current (charge). Local gauge transformations.
Quantizatrion of free fields and particle interpretation: Real and complex Klein - Gordon field. Canonical quantization and commutation relations for creation and annihilation operators.
Energy, momentum and charge of the quantized field. Fock space.
Vacuum and normal ordering. Dirac field.
Positivity of energy and anticommutation relations. Bosons and fermions - spin and statistics.
Antiparticles. Quantization of massive vector (Proca) field.
Relativistic covariance of canonical quantization. Interactions of quantized fields: Examples - Yukawa interaction, interaction of fermions with vector field (electrodynamics), direct four-fermion interaction.
Interaction (Dirac) representation in description of time evolution. Dyson perturbation expansion of evolution operator.
Chronological product (time-ordering). S-matrix.
Relativistically invariant transition amplitude. Decay probability per unit time (decay rate).
Cross section of a two-particle collision. Kinematics of binary processes: Mandelstam variables s, t, u.
Examples of some processes in the first order of perturbation expansion - decay of scalar and vector boson into a fermion-antifermion pair. Neutrino - electron scattering.
Representation of the corresponding transition amplitudes by means of Feynman diagrams.
Annotation
The equations of relativistic quantum mechanics. The Lagrange formalism at the classical field theory. The canonical quantization of free fields. The interaction of quantized fields. The perturbative expansion of S-matrix.
Feynman diagrams.