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We present a decomposition formula for $U_n$, an integral of time-orderedproducts of operators, in terms of sums of products of the more primitivequantities $C_m$, which are the integrals of time-ordered commutators of thesame operators. The resulting factorization enables a summation over $n$ to becarried out to yield an explicit expression for the time-ordered exponential,an expression which turns out to be an exponential function of $C_m$. TheCampbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtainedpreviously are both special cases of this result.Comment: 31 pages, Revtex with two postscript figure

Decomposition of time-ordered products and path-ordered exponentials