THE FUNCTIONAL INTEGRAL CONSTRUCTED DIRECTLY FROM THE HAMILTONIAN

Starting with any Hamiltonian and a countable basis we construct a functional integral for the matrix elements of the time evolution operator of the quantum system. Our functional integral consists of a real measure on continuous time paths along with a complex weighting function of the paths. The paths spend time at points corresponding to elements of the basis. We show directly, by cancelling paths, that the functional integral gives rise to unitary time evolution. We illustrate our general construction with a number of examples including the quantum mechanics of the Schrödinger and Dirac particles in one space dimension. © 1992.