Principal Bundles pp 173-193 | Cite as

# Yang–Mills Theory

## Abstract

In this chapter, we try to present the theory as Yang and Mills saw it in [54] some 60 years ago. Rather than develop gauge theory in all its generality and then remark that the Yang–Mills theory is just a special case, we want to show the ideas and equations that these physicists worked with, and see later how the generalization came about. So we will use much of the notation of [54], including a lot of indices, as is the tradition in the physics literature to this day. This will require the reader to understand the operations of raising and lowering indices, sometimes not too lovingly known as “index mechanics.” Even though [54] is only five pages long, we only discuss in detail some of its topics. Nonetheless, our approach is somewhat anachronistic since it is our current understanding of gauge theory that illuminates the present reading of [54]. Also, we use a bit of the language of differential forms to prepare the reader for the next chapter. However, this language is inessential for the strict purpose of this chapter.

## Bibliography

- 2.V.I. Arnold, V.V. Kozlov, and A.I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 2nd edition, Springer, 1997.Google Scholar
- 7.E.B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, 1995.Google Scholar
- 12.L. Faddeev, Advent of the Yang–Mills field, in: Highlights of Mathematical Physics, Eds. A. Fokas et al., Am. Math. Soc., 2002.Google Scholar
- 13.R.P. Feynman, Quantum theory of gravitation, Acta Phys. Polonica
**24**(1963) 697–722.MathSciNetGoogle Scholar - 17.W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, vol.
**129**, Springer, 1991.Google Scholar - 18.I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, 1963.Google Scholar
- 19.S.J. Gustafson and I.M. Sigal, Mathematical Concepts of Quantum Mechanics, Springer, 2003.Google Scholar
- 20.B.C. Hall, Lie Groups, Lie Algebras, and Representations, Springer, 2003.Google Scholar
- 22.P. Higgs, Spontaneous symmetry breakdown without massless bosons, Phys. Rev.
**145**(1966) 1156–1163.MathSciNetCrossRefGoogle Scholar - 28.J.D. Jackson, Classical Electrodynamics, John Wiley \(\&\) Sons, 2nd edition, 1975.Google Scholar
- 36.W. Miller, Jr., Symmetry Groups and Their Applications, Academic Press, 1972.Google Scholar
- 49.V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Graduate Texts in Mathematics, Vol.
**102**, Springer, 1984.Google Scholar - 54.C.N. Yang and R.L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev.
**96**(1954) 191–195.MathSciNetCrossRefGoogle Scholar - 55.D.Z. Zhang, C.N. Yang and contemporary mathematics, Math. Intelligencer
**15**, no. 4, (1993) 13–21.Google Scholar