# The Schwinger model

The Schwinger model is the 2d QED with massless fermions. An important result about it (which I would like to understand) is that this is a gauge invariant theory which contains a free massive vector particle.

The original article by Schwinger Gauge invariance and mass, II, Phys. Review, 128, number 5 (1962), is too concise for me.

QUESTION: Is there a more detailed/modern exposition of the above result?

## 5 Answers

You can take a look at Zinn-Justin, "Quantum field theory and critical phenomena", section 31.4 in 3rd edition.

• It's appendix A31.2 in the old edition if that's what your library has. Mar 19, 2013 at 1:59
• It's section 32.4 in the 4th edition. Mar 18, 2016 at 20:02

There is a best pedagogical book! "Selected topics in Gauge theories" by Walter Dittrich. Page 135. Best explanation!

• Welcome to Physics SE. Please provide a concise summary of the citation. Mar 18, 2013 at 20:36
• @Dibakar: yah, it is awesome! Alhamdulillah! Jan 26, 2014 at 15:03

Schwinger model is a solvable case in 2D QED. It is the special example for a gauge field to obtain its mass without a scalar involved, which is called the dynamical symmetry breaking, caused by the broken chiral symmetry.

A book written by Ashok Das, , includes this topic, see the Sec. 13.2

To understand Schwinger model better, you may know something about chiral anomaly, all related knowledge are be given in Sec. 13 of this book.

The book Non-Perturbative Methods in 2 Dimensional Quantum Field Theory by Abdalla, Abdalla, and Rothe discusses 2d QED (and a lot of other stuff) in detail.

I would recommend getting this from a library, or maybe reading Abdalla's lecture notes if you can't. The book is helpful, but it is not worth the price it is being sold at. (Google lists the ebook for sale at \$281!)

You can also try Shifman's "Advanced Topics in Quantum Field Theory" which is fairly new and up to date with material on non-perturbative things like solitons and instantons, SUSY and gauge theory. Includes a discussion of the Schwinger model.