Department for General and Applied Physics, Moscow Institute of Physics and Technology

Chair for "Problems in theoretical physics"

based at Landau Institute for theoretical physics


Introduction to the quantum field theory

Ya.P. Pugay

The lectures consists of two basic parts. The first part begins with the introduction to the classical Yang-Mills field theory. Next, the non-abelian gauge theories quantization is explained, and basic notions of the perturbation theory including renormalization theory are introduced. The asymptotic freedom and confinement phenomena are discussed. The second part of the lectures is devoted to explanation of basic notions of the eletro-weak theory.

Prerequisite Knowledge: The knowledge of basic notions of quantum electrodynamics is assumed (Volume 1 of ref. [1]). The course (except for lectures on BRST quantization and quantum anomalies) is designed for theoretical-physics students, who do not specialize in the QFT and mathematical physics areas.

Examinations and Grading: For A grade it is required to hand in 80% of the exercise sheets. An overall grade depends on the problems solved by the student and his final-exam grade.



  1. Introduction. Strong and weak interactions as gauge ones
  2. Basics from Lie algebras and its representation theory
  3. Classical Yang-Mills theory
    • Gauge fields, gauge transform. Local and non local observables
    • Action
    • Equations of motion
    • Conserved current
    • Poisson brackets
    • Geometrical interpretation of gauge fields
  4. Gauge theory quantization in hamiltonian approach. Quantization of fields with the first order constraints and gauge fixing. Finite dimensional model example
  5. Choosing a gauge. Background field gauge. Faddeev-Popov ghosts. Gribovs ambiguity
  6. Feynman rules
  7. BRST quantization
    • BRST operator and its cohomologies
    • Geometrical interpretation of BRST
    • Ward identities (Slavnov-Taylor)
    • Unitarity
  8. Renormalization theory. Regularizations and subtraction schemes
    • Review on different regularizations
    • Divergency index for a diagram
    • Z factors
    • Gauge theory renormalization
  9. Asymptotic freedom
    • One loop beta function
    • Effective charge
    • Anomalous dimensions
    • Dimensional transmutation
    • Dilatational anomaly
  10. Quantum Chromodynamics
    • Quarks and hadrons
    • Confinement
    • Large N limit
  11. Weak interactions
    • Leptons
    • Weak current structure
    • V-A theory
    • Meson decay. C- P- violation
  12. Spontaneous local and global symmetry breaking
    • Goldstone theorem
    • Higgs mechanism
    • Quantization of fields with spontaneous gauge symmetry breaking
    • tHooft gauge
  13. Standard model
    • Gauge sector
    • Higgs sector
    • Quarks and leptons
    • CKM matrix
  14. Anomalies in gauge theories
    • Anamalous symmetries
    • Classification of anomalies
    • Vergeles-Fujikawa method
    • Chiral anomalies cancellation in Standard model

Exercise classes:

  1. Repetition. Basic notion on free fields theory
  2. Repetition. Basic notions of quantum elctrodynamics
  3. Repetition. Fermionic functional integral. Effective poteniial. Renormalization group, beta function (Gross-Nevue model)
  4. Yang-Mills Lagrangian. Gauge symmetry. Lie algebras. Representations of SU(2) and SU(3)
  5. Yang-Mills theory. Feynman rules. Ghosts. Gauge fixing
  6. Effective action. Background field method. One loop computations. Functional determinants
  7. computation
  8. Renormalization. Beta function, anomalous dimensions. Renormalization group
  9. Callan-Symanchik equation. Asymptotic freedom. Dimensional transmutation, logarithmic corrections to scaling, spontaneous mass generation
  10. Renormalization group. Double logarithms
  11. Spontaneous symmetry breaking (Coleman-Wienberg model)
  12. Higgs mechanism
  13. Electro weak theory. CP violation and simple scatetings
  14. Chiral anomalies
  15. Discussion on remaining homework problems

Recommended texts

  1. Claude Itzykson, Jean-Bernard Zuber, Quantum Field Theory. McGraw-Hill, New York.
  2. Steven Weinberg, "The quantum theory of fields" Vol. 1-3, Cambridge university press.
  3. M. Peskin & D. Schroeder, "An introduction to quantum field theory", Westview press.
  4. . , " ", ., , 2009.
  5. A.A. Slavnov, L.D. Faddeev, Gauge Fields: Introduction to Quantum Theory (Frontiers in Physics S.)
  6. L. B. Okun, Leptons and Quarks, Elsevier Science.
  7. . . , . . -, " ", ., , 1984.
  8. . . , " ", 150, 337-416 (1986).