Multiplicative partial integration and the Trotter Product Formula

Abstract For a long time it has been apparent that the Trotter Product Formula, a simple version of which reads exp {(A + B)t} = lim n→∞ exp {( At n ) · ( Bt n )} n , where A, B are (non-commuting) operators and t is real, is related to the formula for the multipicative integral of the sum of two functions. But the precise connection between the two has not been exposed in the literature. Our purpose is to fill this lacuna. The theory of the multiplicative integral is outlined in Section 1, and the formula for the integral of the sum of two functions and the Trotter Product Formula are discussed in Section 2. Possible extensions are alluded to in Section 3.