BPST instanton

In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU(2) Yang–Mills theory in Euclidean space-time (i.e. after Wick rotation), meaning it describes a transition between two different vacua of the theory. It was originally hoped to open the path to solving the problem of confinement, especially since Polyakov had proven in 1987 that instantons are the cause of confinement in three-dimensional compact-QED. This hope was not realized, however.

The BPST instanton is an essentially non-perturbative classical solution of the Yang–Mills field equations. It is found when minimizing the Yang–Mills SU(2) Lagrangian density:

with F_μν^a = ∂_μA_ν^a – ∂_νA_μ^a + gε^abcA_μ^bA_ν^c the field strength. The instanton is a solution with finite action, so that F_μν must go to zero at space-time infinity, meaning that A_μ goes to a pure gauge configuration. Space-time infinity of our four-dimensional world is S³. The gauge group SU(2) has exactly the same structure, so the solutions with A_μ pure gauge at infinity are mappings from S³ onto itself. These mappings can be labelled by an integer number q, the Pontryagin index (or winding number). Instantons have q = 1 and thus correspond (at infinity) to gauge transformations which cannot be continuously deformed to unity. The BPST solution is thus topologically stable.

It can be shown that self-dual configurations obeying the relation F_μν^a = ± ½ ε^μναβF_αβ^a minimize the action. Solutions with a plus sign are called instantons, those with the minus sign are anti-instantons.

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