# References for classical Yang-Mills theory

I am looking for a reference to study classical (i.e., not quantized) Yang-Mills theory.

Most of the sources I find focus on mathematical aspects of the theory, like Bleecker's book Gauge theory and variational principles, or Baez & Muniain's Gauge fields, knots and gravity.

But I am more interested something similar to the standard development of electromagnetism, as can be found, for example, in Landau & Lifchitz's course on theoretical physics. To be more exact, I would like to learn about the field equations (Yang-Mills and Einstein equations), but also about the corresponding Lorentz Law, the energy of the Yang-Mills field,... and the analogous concepts of what is made for the electromagnetism.

That is, I look for a rigorous exposition where, at the same time, I could learn whether it is possible to prove, at the classical level, the quick decrease of the strong interaction, and things like that.

• If by the "quick decrease of the strong interaction" you mean asymptotic freedom, you are not going to find a proof of this at the classical level because it is an intrinsically quantum effect. Jan 10, 2012 at 19:00
• Mmm.. No, I meant the fact that the strong force is much stronger than electromagnetism or gravitation at the atomic scale, but it becomes negligible at large scale (I guess this is not the same of "asymptotic freedom"). To see this, you would only need a kind of analogous of "Coulomb's law" for the strong force... Jan 11, 2012 at 10:11
• The strong force is negligible at large distances because the "charge" it couples to is confined. I don't think you are going to understand confinement from classical gauge theory either. Jan 12, 2012 at 6:43

This is underrepresented in the literature. I have Nakahara and have looked at Frenkel (both listed in other answers) as well as many other "standard" references. The best book reference for classical YM theory that I found was Rubakov's Classical Theory of Gauge Fields.

• Thanks, Steve. I didn't know that book, but it seems precisely the kind of reference I'm looking for. And I totally agree this topic is underrepresented in the literature. Jan 11, 2012 at 10:03
• Full text of Rubakov's book is available here: gen.lib.rus.ec/… Jan 12, 2012 at 10:39
• Re last comment, related meta discussion: meta.mathoverflow.net/questions/3953/… Mar 23, 2019 at 23:12

I would add: Atiyah, Michael F. (1979), Geometry of Yang–Mills fields and then the book of the same author about gauge theories: Atiyah, Michael F. (1988e), Collected works. Vol. 5 Gauge theories

• Thank you, Jon. But I think these references are "too sophisticated" (and too mathematically oriented) for my purposes. Jan 11, 2012 at 10:01

Have you tried the book "the Geometry of physics" by Th. Frankel?

• Thanks for the reference, Liviu. I remembered having looked at this book, but I didn't try it seriously. I'll take it again. Jan 11, 2012 at 9:57

Maybe you can have a look to Nakahara's Geometry, Topology and Physics, or is it too elementary for your purposes?

• I didn't knew of this book, but it seems nice. Definitely, another petition for the library of the Department... Jan 11, 2012 at 9:59

My personal suggestion is 'Differential Geometry, Gauge Theories, and Gravity' by M. Gockeler and T. Schucker. However, it assumes a fairly high degree of mathematical sophistication (it's one of the texts in the 'Cambridge Monographs on Mathematical Physics). If you do get it, it is really worth the effort to master the topics inside, and the range of topics covered is fascinating.