Instanton - crowlogic/arb4j GitHub Wiki

In the field of mathematical physics, an instanton is a finite-energy solution to the Yang-Mills equations, which describe non-Abelian gauge fields. More precisely, instantons are solutions to the anti-self-dual Yang-Mills equations over Euclidean 4-space, which can be understood as solutions to the Yang-Mills equations in Euclidean spacetime that are localized both in space and time. These solutions are important in the study of quantum field theory, especially quantum chromodynamics (QCD), and the quantum behavior of the gauge field.

The formal definition involves differential geometry and gauge theory. Let's use the basic setup of a principal $G$-bundle $P$ over a four-dimensional manifold $M$ (usually taken to be the Euclidean space $\mathbb{R}^4$ or the four-sphere $S^4$), where $G$ is a Lie group (often the group $SU(2)$ or $SU(3)$ in physical applications).

The Yang-Mills field, or the connection on this bundle, can be understood as a Lie-algebra valued 1-form $A$ on $M$. The curvature of this connection is a 2-form $F_A$ which is given by the exterior derivative of $A$ plus a term involving the Lie bracket: $F_A = dA + [A \wedge A]$. The Yang-Mills equations in four dimensions can be written as the vanishing of the self-dual part of the curvature: $F_A^+ = 0$. Here $F_A^+$ denotes the self-dual part of $F_A$ with respect to the Hodge star operator on $M$.

An instanton is a solution $A$ to this equation such that the corresponding curvature $F_A$ has finite action, defined by the Yang-Mills functional: $YM(A) = \int_M |F_A|^2 dv < \infty$. Here $|F_A|^2$ is the pointwise norm of the curvature with respect to a metric on $M$ and the Lie algebra, $dv$ is the volume form, and the integral is over the entire manifold $M$.

This definition abstracts the properties of solutions to the Yang-Mills equations that have important physical interpretations, in terms of tunnelling effects between different vacuum states in quantum field theory. The degree or winding number of an instanton, defined as the integral of the normalized trace of $F_A \wedge F_A$ over $M$, measures the topological charge of the field configuration, and corresponds to the number of quantum transitions induced by the field.