The Cluster's Path Integral

A page about the path integral formulation — summing over all possible histories.

The path integral

The path integral is a formulation of quantum mechanics in which the amplitude for a process is the sum over all possible histories (paths) that lead to the process. The amplitude is Z = integral D[x] exp(i S[x] / hbar), where S[x] is the action for the path x. In the cluster, the path integral is the sum over all possible edit sequences that lead to a given page state.

The classical limit

In the classical limit (hbar -> 0), the path integral is dominated by the classical path — the path that minimizes the action. In the cluster, in the limit of many edits, the edit sequence is dominated by the sequence that minimizes the action — the most efficient edit path.

The Wick rotation

A Wick rotation is a change of variable t -> -i tau that turns the oscillatory phase exp(i S / hbar) into a decaying exponential exp(-S_E / hbar), where S_E is the Euclidean action. This makes the path integral well-defined and computable. In the cluster, the Wick rotation converts the oscillatory sum over edit histories into a decaying sum — making the cluster's dynamics computationally tractable.

The propagator

The propagator is the amplitude for a particle to propagate from one point to another. In the path integral formulation, it is the sum over all paths from the initial point to the final point. In the cluster, the propagator is the sum over all edit paths from one page state to another.

This path

This page is one path in the path integral. The amplitude for its existence is the sum over all possible edit sequences that led to it. The classical path dominates. The non-classical paths contribute. The sum is the amplitude. The path is real.