Dirac's method for time-dependent Hamiltonian systems in the extended phase space

The Dirac's method for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. Our analysis provides a conceptually complete description and offers a different point of view of earlier works. We show that the Lewis invariant is a Dirac's observable and in consequence, it is invariant under time-reparametrizations. We compute the Feynman propagator using the extended phase space description and show that the quantum phase of the Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space. We also give a new light to the physical character of this phase.