Yang-Mills Theory(Non-Abelian gauge theory)

introduction

• This is not a quantum theory.
• This can be regarded as a generalization of theory of electromagetisms., i.e. bundle + connections
• looks like the coordinate invariance of gravity theory
• Gauge theory
• Usually, non-abelian gauge theory is called the YM theory.
  • QCD is one example.

basic concepts

• connection
• curvature

original Yang-Mills model

• three kinds of photon
  • one ordinary photon
  • two electrically charged photons with spin 1 which is physically impossible to exist
• massless gauge fields
  • for example, electromagnetic field(the only example at that time)

recipe

• prepare Dirac fields
• start with the free Dirac Lagrangian
• we demand the Lagrangian to be invariant under the SU(N) local gauge transformations
• structure constants are needed
• self-interaction of gauge fields starts to appear

Yang-Mills potential

• dual role
  • a field in space-time
  • operator in the isotopic-spin space

quantization of Yang-Mills theory

• We want to quantize this theory.
• standard model is a quantized version of a Yang-Mills theory of classical fields
• mass gap in Yang-Mills theory

books

• Baez and Muniain (“Knots, Gauge Fields and Gravity”, World Scientific 1994
• M. Nakahara “Geometry, Topology and Physics”.
• Fifty years of Yang-Mills theory http://books.google.com/books?id=WV57GpYjYREC&pg=PA102&lpg=PA102&dq=weakly+interacting+boson+yang-mills&source=bl&ots=ztMiD8nOgz&sig=nhFiCtCQw2lAodU1zacqg3wueUQ&hl=ko&ei=W1jZTtuZCIqRiALR3c26BA&sa=X&oi=book_result&ct=result&resnum=10&ved=0CHEQ6AEwCQ#v=onepage&q&f=false

expository

• Rivière, Tristan. “The Variations of Yang-Mills Lagrangian.” arXiv:1506.04554 [math], June 15, 2015. http://arxiv.org/abs/1506.04554.
• Slavnov, A. A. ‘New Approach to the Quantization of the Yang-Mills Field’. arXiv:1503.03380 [hep-Ph, Physics:hep-Th], 11 March 2015. http://arxiv.org/abs/1503.03380.
• Gauge Theory in Two Dimensions: Topological, Geometric and Probabilistic Aspects Ambar N. Sengupta
• Introduction to Yang-Mills theories
• http://michaelnielsen.org/blog/yang_mills.pdf
• History of Yang-Mills theory and wishful thinking The Reference Frame
• Electromagnetic duality for children
• What Is Geometry? Shiing-Shen Chern, The American Mathematical Monthly, Vol. 97, No. 8, Special Geometry Issue (Oct., 1990), pp. 679-686
• Introduction to gauge theories and the Standard Model CERN video lectures

articles

• Huang, Teng. “Yang-Mills Connections on \(G_{2}\)-Manifolds and Calabi-Yau 3-Folds.” arXiv:1502.02090 [math-Ph], February 6, 2015. http://arxiv.org/abs/1502.02090.
• Yang-Mills Theory and Geometry, Donaldson
• C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 - 195 (1954)

encyclopedia

• http://en.wikipedia.org/wiki/Yang-Mills_theory
• http://en.wikipedia.org/wiki/Gauge_theory
• http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/Yang-Mills_theory

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