  The standard derivation of Schroedinger's equation from a Lorentz-invariant
Feynman path integral consists in taking first the limit of infinite speed of
light and then the limit of short time slice. In this order of limits the light
cone of the path integral disappears, giving rise to an instantaneous spread of
the wave function to the entire space. We ascribe the failure of Schroedinger's
equation to retain the light cone of the path integral to the very nature of
the limiting process: it is a regular expansion of a singular approximation
problem, because the boundary conditions of the path integral on the light cone
are lost in this limit. We propose a distinguished limit, which produces an
intermediate model between non-relativistic and relativistic quantum mechanics:
it produces Schroedinger's equation and preserves the zero boundary conditions
on and outside the original light cone of the path integral. These boundary
conditions relieve the Schroedinger equation of several annoying, seemingly
unrelated unphysical artifacts, including non-analytic wave functions,
spontaneous appearance of discontinuities, non-existence of moments when the
initial wave function has a jump discontinuity (e.g., a collapsed wave function
after a measurement), the EPR paradox, and so on. The practical implications of
the present formulation are yet to be seen.
