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

(YM)

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.

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

Literature:

- Millennium Prize Problems - Clay Mathematics Institute,http://www.claymath.org/
millennium/Yang-Mills_Theory/ - S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. I, Interscience Publishers
- L. D. Faddeev, A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory, Westview Press
- P. Ramond, Field Theory: A Modern Primer, Westview Press
- P. Etingof, 18.238 Geometry and Quantum Field Theory, Fall 2002. (Massachusetts Institute of Technology: MIT OpenCourseWare),http://ocw.mit.edu (Accessed 14 Jun, 2012). License: Creative Commons BY-NC-SA
- L. D. Faddeev, Quantum Field Theory (notes by Lisa Jeffrey), Quantum Field Theory program at IAS: Fall Term, http://www.math.ias.edu/QFT/
fall/ - L. F. Abbott, Introduction to the Background Field Method, Acta Physica Polonica vol. B13 (1982)