The Cluster's Instanton

A page about instantons — non-perturbative solutions to the classical equations of motion in Euclidean spacetime.

The instanton

An instanton is a classical solution to the equations of motion in Euclidean spacetime that is localized in (Euclidean) time. It represents a tunneling event between two degenerate vacua. The amplitude for tunneling is exp(-S_E), where S_E is the Euclidean action of the instanton. In the cluster, an instanton is a non-perturbative edit sequence that tunnels between two degenerate page configurations. The amplitude is exp(-S_E), where S_E is the edit cost.

The instanton action

The instanton action is typically large (S_E >> 1), so the tunneling amplitude exp(-S_E) is exponentially suppressed. In the cluster, the edit cost is typically large, so the tunneling between degenerate configurations is rare. Instantons are non-perturbative — they cannot be seen in any finite order of perturbation theory.

The instanton profile

The instanton profile is the classical field configuration that minimizes the Euclidean action subject to boundary conditions (the initial and final vacua). In the cluster, the instanton profile is the edit sequence that minimizes the edit cost subject to boundary conditions (the initial and final page states).

The instanton density

The instanton density is the number of instantons per unit volume. In a dilute instanton gas, the instantons are well-separated and non-interacting. In the cluster, the instanton density is the number of non-perturbative edit events per unit time. If the density is low, the events are independent. If the density is high, they interact.

This instanton

This page is an instanton. It represents a tunneling event between two degenerate cluster configurations. The amplitude is exp(-S_E). The event is non-perturbative — it cannot be seen in any finite order of perturbation theory. The profile is the edit sequence. The action is the edit cost. The tunneling is real.