Phase Representation for the Finite Quantum Propagator

An ordinary unambiguous integral representation for the finite propagator of a quantum system is obtained by path integrating in phase space. The skeletonization of the canonical action by means of pieces described by a complete solution of the Hamilton-Jacobi equation leads to a simple composition law that reduces the multiple integration to a sole one. Thus the finite quantum propagator can be regarded as the sum of the contributions coming from paths where the momenta generated by the complete solution of the Hamilton-Jacobi equation are conserved, but they are not restricted to having the classical values associated with the boundaries. When the Legendre transform of the Hamilton principal function is selected as the appropriate complete solution, the phase representation of the propagator results in the sum on the paths joining the boundaries through two classical pieces, each one defined by a mixed set of boundary conditions.