FEYNMAN TYPE FORMULAE FOR QUANTUM EVOLUTION AND DIFFUSION ON MANIFOLDS AND GRAPHS

摘要

The Feynman formula is a representation of either a Schrodinger semi-group e~(tH) or a Schrodinger group e~(it H) by limits of superpositions of some multiple integrals, over Cartesian powers of some space E, and elementary functions; here H is a classical Hamilton function and H is a corresponding quantum mechanical Hamiltonian, i.e. a pseudo-differential operator whose Weyl symbol is H. In the former case the multiple integrals in the Feynman formulae approximate some integrals with respect to some diffusion type measure on a set of functions which take values in E and are defined on a real interval (such functions are also called paths, or trajectories, in E); in the latter case the multiple integrals approximate integrals with respect to a pseudo-measure on the same set. The pseudo-measure is called the Feynman pseudo-measure and the corresponding integrals are called the Feynman path integrals~a.