Introduction to the quantum field theory
The lectures consists of two basic parts. The first part begins with the
introduction to the classical YangMills field theory. Next, the nonabelian
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 eletroweak 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
theoreticalphysics 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 finalexam grade.
Syllabus
Lectures
 Introduction. Strong and weak interactions as gauge ones
 Basics from Lie algebras and its representation theory
 Classical YangMills theory
 Gauge fields, gauge transform. Local and non local observables
 Action
 Equations of motion
 Conserved current
 Poisson brackets
 Geometrical interpretation of gauge fields
 Gauge theory quantization in hamiltonian approach.
Quantization of fields with the first order constraints and gauge fixing. Finite dimensional model example
 Choosing a gauge. Background field gauge. FaddeevPopov ghosts. Gribov’s ambiguity
 Feynman rules
 BRST quantization
 BRST operator and its cohomologies
 Geometrical interpretation of BRST
 Ward identities (SlavnovTaylor)
 Unitarity
 Renormalization theory. Regularizations and subtraction schemes
 Review on different regularizations
 Divergency index for a diagram
 Z factors
 Gauge theory renormalization
 Asymptotic freedom
 One loop beta function
 Effective charge
 Anomalous dimensions
 Dimensional transmutation
 Dilatational anomaly
 Quantum Chromodynamics
 Quarks and hadrons
 Confinement
 Large N limit
 Weak interactions
 Leptons
 Weak current structure
 VA theory
 Meson decay. C P violation
 Spontaneous local and global symmetry breaking
 Goldstone theorem
 Higgs mechanism
 Quantization of fields with spontaneous gauge symmetry breaking
 t’Hooft gauge
 Standard model
 Gauge sector
 Higgs sector
 Quarks and leptons
 CKM matrix
 Anomalies in gauge theories
 Anamalous symmetries
 Classification of anomalies
 VergelesFujikawa method
 Chiral anomalies cancellation in Standard model
Exercise classes:
 Repetition. Basic notion on free fields theory
 Repetition. Basic notions of quantum elctrodynamics
 Repetition. Fermionic functional integral. Effective poteniial. Renormalization group, beta function (GrossNevue model)
 YangMills Lagrangian. Gauge symmetry. Lie algebras. Representations of SU(2) and SU(3)
 YangMills theory. Feynman rules. Ghosts. Gauge fixing
 Effective action. Background field method. One loop computations. Functional determinants
 computation
 Renormalization. Beta function, anomalous dimensions. Renormalization group
 CallanSymanchik equation. Asymptotic freedom. Dimensional transmutation, logarithmic corrections to scaling, spontaneous mass generation
 Renormalization group. Double logarithms
 Spontaneous symmetry breaking (ColemanWienberg model)
 Higgs mechanism
 Electro weak theory. CP violation and simple scatetings
 Chiral anomalies
 Discussion on remaining homework problems
Recommended texts
 Claude Itzykson, JeanBernard Zuber, Quantum Field Theory. McGrawHill, New York.
 Steven Weinberg, "The quantum theory of fields" Vol. 13, Cambridge university press.
 M. Peskin & D. Schroeder, "An introduction to quantum field theory", Westview press.
 Ý. Çè, "Êâàíòîâàÿ òåîðèÿ ïîëÿ â äâóõ ñëîâàõ", Ì., ÐÕÄ, 2009.
 A.A. Slavnov, L.D. Faddeev, Gauge Fields: Introduction to Quantum Theory (Frontiers in Physics S.)
 L. B. Okun, Leptons and Quarks, Elsevier Science.
 Ì. Á. Âîëîøèí, Ê. À. ÒåðÌàðòèðîñÿí, "Òåîðèÿ êàëèáðîâî÷íûõ âçàèìîäåéñòâèé ýëåìåíòàðíûõ ÷àñòèö", Ì., ÝÀ, 1984.
 À. Þ. Ìîðîçîâ, "Àíîìàëèè â êàëèáðîâî÷íûõ òåîðèÿõ", ÓÔÍ 150, 337416 (1986).
