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# 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,

$\bold{F}=d\bold{A}+\bold{A}\wedge\bold{A}$

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.

Instantons may be easier to visualise in two dimensions than in four. In the simplest case the gauge group is U(1). In this case the field can be visualised as an arrow at each point in two-dimensional spacetime. An instanton is a configuration where, for example, the arrows point away from a central point. More complicated configurations are also possible.

The field configuration of an instanton is very different to that of the vacuum. Because of this instantons cannot be studied by using Feynman diagrams, which only include perturbative effects. Instantons are fundamentally non-perturbative.

The Yang-Mills energy is given by

$\frac{1}{2}\int_{\mathbb{R}^4} \operatorname{Tr}[*\bold{F}\wedge \bold{F}]$

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

$\int_{\mathbb{R}^4}\operatorname{Tr}[\bold{F}\wedge\bold{F}]$.

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,

$0\leq\frac{1}{2}\int_{\mathbb{R}^4}\operatorname{Tr}[(*\bold{F}+e^{-i\theta}\bold{F})\wedge(\bold{F}+e^{i\theta}*\bold{F})] =\int_{\mathbb{R}^4}\operatorname{Tr}[*\bold{F}\wedge\bold{F}+2\cos\theta \bold{F}\wedge\bold{F}]$

for all real θ. So, this means

$\frac{1}{2}\int_{\mathbb{R}^4}\operatorname{Tr}[*\bold{F}\wedge\bold{F}]\geq\frac{1}{2}\left|\int_{\mathbb{R}^4}\operatorname{Tr}[\bold{F}\wedge\bold{F}]\right|$

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.

Instanton effects are important in solving the so-called 'eta-problem' in quantum chromodynamics.

03-10-2013 05:06:04