Instanton
In quantum field theory, an instanton is a topologically nontrivial field configuration in four-dimensional Euclidean space (considered as the Wick rotation of Minkowski spacetime). Specifically, it refers to a Yang-Mills gauge field A which locally approaches pure gauge at spatial infinity. This means the field strength defined by A,
- <math>\bold{F}=d\bold{A}+\bold{A}\wedge\bold{A}<math>
vanishes at infinity. The name instanton derives from the fact that these fields are localized in space and (Euclidean) time - in other words, at a specific instant.
The Yang-Mills energy is given by
- <math>\frac{1}{2}\int_{\mathbb{R}^4} \operatorname{Tr}[*\bold{F}\wedge \bold{F}]<math>
where * is the Hodge dual. If we insist that the solutions to the Yang-Mills equations have finite energy, then the curvature of the solution at infinity (taken as a limit) has to be zero. This means that the Chern-Simons invariant can be defined at the 3-space boundary. This is equivalent, via Stokes' theorem, to taking the integral
- <math>\int_{\mathbb{R}^4}\operatorname{Tr}[\bold{F}\wedge\bold{F}]<math>.
This is a homotopy invariant and it tells us which homotopy class the instanton belongs to.
Since the integral of a nonnegative integrand is always nonnegative,
- <math>0\leq\frac{1}{2}\int_{\mathbb{R}^4}\operatorname{Tr}[(*\bold{F}+e^{-i\theta}\bold{F})\wedge(\bold{F}+e^{i\theta}*\bold{F})]
=\int_{\mathbb{R}^4}\operatorname{Tr}[*\bold{F}\wedge\bold{F}+2\cos\theta \bold{F}\wedge\bold{F}]<math>
for all real θ. So, this means
- <math>\frac{1}{2}\int_{\mathbb{R}^4}\operatorname{Tr}[*\bold{F}\wedge\bold{F}]\geq\frac{1}{2}\left|\int_{\mathbb{R}^4}\operatorname{Tr}[\bold{F}\wedge\bold{F}]\right|<math>
If this bound is saturated, then the solution is a BPS state. For such states, either *F = F or *F = − F depending on the sign of the homotopy invariant.
