  Sum-over-histories quantization of particle-like theory in curved space is
discussed. It is reviewed that the propagator satisfies the Schrodinger
equation respective wave equation with a Laplace-like operator. The exact
dependence of the operator on the choice of measure is shown.
  Next, modifications needed for a manifold with a boundary are introduced, and
the exact form of the equation for the propagator is derived. It is shown that
the Laplace-like operator contains some distributional terms localized on the
boundary. These terms induce proper boundary conditions for the propagator.
This choice of boundary conditions is explained as a consequence of a
measurement of particles on the boundary.
  The interaction with sources inside of the domain and sources on the boundary
is also discussed.
