Decomposition of Time-Ordered Products and Path-Ordered Exponentials

TL;DR

This paper introduces a decomposition formula for time-ordered products and exponentials, enabling explicit expressions and unifying previous formulas like Campbell-Baker-Hausdorff.

Contribution

It provides a new decomposition approach for time-ordered exponentials in terms of commutator integrals, generalizing previous results.

Findings

Derived an explicit exponential expression for time-ordered exponentials.

Unified Campbell-Baker-Hausdorff and nonabelian eikonal formulas as special cases.

Enabled summation over product orders to simplify complex operator exponentials.

Abstract

We present a decomposition formula for $U_n$, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities $C_m$, which are the integrals of time-ordered commutators of the same operators. The resulting factorization enables a summation over $n$ to be carried out to yield an explicit expression for the time-ordered exponential, an expression which turns out to be an exponential function of $C_m$. The Campbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtained previously are both special cases of this result.