Decomposition of time-ordered products and path-ordered exponentials

We present a decomposition formula for Un, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities Cm, 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 Cm. The Campbell–Baker–Hausdorff formula and the non-Abelian eikonal formula obtained previously are both special cases of this result.