An Exact Formulation of the Time-Ordered Exponential using Path-Sums

TL;DR

This paper introduces a novel path-sum formulation for the time-ordered exponential of time-dependent matrices, representing it as a branched continued fraction linked to self-avoiding walks, with bounds for sparse matrices.

Contribution

The paper presents a new exact formulation of the time-ordered exponential using path-sums, connecting it to graph walks and providing decay bounds for sparse matrices.

Findings

Path-sum formulation as a branched continued fraction

Elementary interpretation via self-avoiding walks

Super-exponential decay bounds for sparse matrices

Abstract

We present the path-sum formulation for $\mathsf{OE}[\mathsf{H}](t',t)=\mathcal{T}\,\text{exp}\big(\int_{t}^{t'}\!\mathsf{H}(\tau)\,d\tau\big)$, the time-ordered exponential of a time-dependent matrix $\mathsf{H}(t)$. The path-sum formulation gives $\mathsf{OE}[\mathsf{H}]$ as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on graphs and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.