Courses ¨
CLASSICAL FIELD THEORY
(B.L.Voronov)

  1. Basic concepts of Riemannian geometry
      Vector fields and covectors, tensors. Metric, connection, covariant derivative. Parallel translation and geodesic.
  2. Curvature tensor and torsion tensor
      Properties of the curvature tensor. Metric postulate.
  3. Symmetries of spaces
      The group of motions of the metric (homogeneity and isotropy). Conformal group. Homogeneous spaces.
  4. Killing vectors
  5. Spaces of GR
      Minkowski space, de Sitter and anti-de Sitter.
  6. Dynamic metrics of GR
      Point-like mass. The model of inflation without the Λ-term. Cooling three-dimensional plane model
  7. Compactification, the idea of Kaluza-Klein. The problem of time.
  8. Differential geometry basics
      Vector fields on the manifold. Vector bundles. Riemannian structure on a manifold.
  9. Lie algebras basics
      Classification of algebras. Hop, Clifford and Berezin algebras.
  10. Killing metric on the group. Adjoint representation
  11. Definition and classification of representations. The configuration space of fields
  12. Deformative quantization
  13. Review of the standard model
  14. Quarks, leptons, bosons. Fundamental and adjoint representation, gauge group of model
  15. Dynamic principle, action.
      General restrictions. Stability. Locality. Renormalizability.
  16. Symmetries
      Groups of symmetries, symmetry algebra, the trivial symmetries. Noether's theorem. Conserved currents and charges. Energy-momentum tensor.
  17. The theory of a neutral scalar field
      Analysis of action. Symmetries. Spontaneous symmetry breaking. Kinks.
  18. Quantization of scalar field
  19. Goldstone theorem. Goldstone mode
  20. Complex scalar field
      Analysis of action. Symmetries. Conserved currents. Spontaneous symmetry breaking. Solitons.
  21. Quantization of complex scalar field
  22. Clifford algebra
      Irreducible representations of Clifford algebra. Lorentz subalgebra.
  23. Spinor field
      Transformations of the fields. Spinor connection. Covariant derivative.
  24. Dirac field
      Action. Renormalizability. Equations of motion. Solution of Dirac equations. Symmetries and currents.
  25. Quantization of Dirac field
  26. Chiral symmetry
      Chirality (left and right). Chiral current. Neutrino oscillations
  27. Coloring of spinors (gauge group extension)
  28. The neutral vector field (Proka's field)
      Action, equations, symmetries, conservation laws, quantization. Maxwell's theory (m = 0). Gauge and its choice.
  29. Isotopic vector fields
      Killing metric. Action. Symmetries.
  30. The interaction of fields of spin 0, 1/2, 1
      Representations of fields, admissible Lagrangians. Higgs theory, generation of mass.
  31. Non-Abelian theory (Yang-Mills)
      Non-Abelian vector fields as connection in the bundle. Local gauge symmetry, covariant derivative. Lagrangian. Conserved currents. Generation of mass. Example of chromodynamics
  32. Weinberg-Salam model
      Gauge group, representations of fields. Lagrangian with interaction.
  33. Higgs mechanism. Higgs boson. Neutral currents
 
  • LITERATURE
    1. V.A.Rubakov. Classical gauge fields.
    2. N.N.Bogolyubov, D.V.Shirkov. Quantum fields.
    3. I.P.Volobushev, Yu.A. Kubyshin. Differential Geometry and Lie algebras and their applications in field theory.
    4. L.B. Okun. Leptons and quarks

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