nLab
Yang-Mills equation

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Chern-Weil theory

Contents

Idea

The Yang-Mills equations are the equations of motion/Euler-Lagrange equations of Yang-Mills theory. They generalize Maxwell's equations.

References

(For full list of references see at Yang-Mills theory)

General

Solutions

General

Wu and Yang (1968) found a static solution to the sourceless SU(2)SU(2) Yang-Mills equations. Recent references include

  • J. A. O. Marinho, O. Oliveira, B. V. Carlson, T. Frederico, Revisiting the Wu-Yang Monopole: classical solutions and conformal invariance

There is an old review,

  • Alfred Actor, Classical solutions of SU(2)SU(2) Yang—Mills theories, Rev. Mod. Phys. 51, 461–525 (1979),

that provides some of the known solutions of SU(2)SU(2) gauge theory in Minkowski (monopoles, plane waves, etc) and Euclidean space (instantons and their cousins). For general gauge groups one can get solutions by embedding SU(2)SU(2)‘s.

Instantons and monopoles

For Yang-Mills instantons the most general solution is known, first worked out by

for the classical groups SU, SO , Sp, and then by

  • C. Bernard, N. Christ, A. Guth, E. Weinberg, Pseudoparticle Parameters for Arbitrary Gauge Groups, Phys. Rev. D16, 2977 (1977)

for exceptional Lie groups. The latest twist on the Yang-Mills instanton story is the construction of solutions with non-trivial holonomy:

  • Thomas C. Kraan, Pierre van Baal, Periodic instantons with nontrivial holonomy, Nucl.Phys. B533 (1998) 627-659, hep-th/9805168

There is a nice set of lecture notes

on topological solutions with different co-dimension (Yang-Mills instantons, Yang-Mills monopoles, vortices, domain walls). Note, however, that except for instantons these solutions typically require extra scalars and broken U(1)‘s, as one may find in super Yang-Mills theories.

Some of the material used here has been taken from

Another model featuring Yang-Mills fields has been proposed by Curci and Ferrari, see Curci-Ferrari model.

Last revised on November 17, 2019 at 01:55:38. See the history of this page for a list of all contributions to it.