Marcin Kisielowski (Warsaw) "Introduction to Yang-Mills theory"

Marcin Kisielowski

Introduction to Yang-Mills theory

Sundays, room 311 (PDMI), 10:00

Yang-Mills theory is a theory of three (out of four) interactions: electromagnetic, weak and strong, and shares a geometrical language with the fourth interaction, the gravity. Although its predictions have been tested by many experiments, its full mathematical foundation is still not known. It stays a big challenge for modern mathematics and theoretical physics to prove an existence of non-trivial Yang-Mills theory in four dimensions -- in fact, a person who proves it will win a one-million dollar prize. The lecture will be an introduction to the Yang-Mills theory covering also an introduction to path integral quantization. A tentative plan of the lectures is the following.

  1. Geometrical foundations: elements of a theory of manifolds and fibre bundles
    1. Manifolds, tangent vectors, differential forms
    2. Groups, Lie groups, Lie algebras
    3. Vector bundles, principal bundles
    4. Connection on principal bundle
    5. Forms of type rho
    6. Curvature
  2. Classical Yang-Mills theory
    1. Electromagnetism as a simplest example of Yang-Mills theory
    2. Lagrangian of Yang-Mills theory
    3. Hamiltonian formalism of Yang-Mills theory
    4. Gauge fixing, Gribov ambiguities
  3. Path integral formulation of quantum mechanics
    1. The path integral over phase space
    2. Wick’s theorem
    3. Feynman’s diagrams
    4. Loop expansion
    5. 1-particle irreducible diagrams and effective action
    6. Wick rotation
  4. Path integral quantization of Yang-Mills theory
    1. Faddeev-Popov procedure
    2. Feynman rules for Yang-Mills theory
    3. Background field method and Yang-Mills effective action

  1. Millennium Prize Problems - Clay Mathematics Institute,
  2. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience Publishers
  3. L. D. Faddeev, A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory, Westview Press
  4. P. Ramond, Field Theory: A Modern Primer, Westview Press
  5. P. Etingof, 18.238 Geometry and Quantum Field Theory, Fall 2002. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed 14 Jun, 2012). License: Creative Commons BY-NC-SA
  6. L. D. Faddeev, Quantum Field Theory (notes by Lisa Jeffrey), Quantum Field Theory program at IAS: Fall Term,
  7. L. F. Abbott, Introduction to the Background Field Method, Acta Physica Polonica vol. B13 (1982)