A subluminous Schroedinger equation

TL;DR

This paper introduces a modified approach to deriving Schroedinger's equation that preserves the light cone structure, addressing unphysical artifacts and providing a more physically consistent intermediate model between non-relativistic and relativistic quantum mechanics.

Contribution

It proposes a distinguished limit in the path integral formulation that retains light cone boundary conditions, improving physical consistency over traditional derivations.

Findings

Preserves light cone boundary conditions in Schroedinger's equation

Removes unphysical artifacts like discontinuities and paradoxes

Provides an intermediate model between non-relativistic and relativistic quantum mechanics

Abstract

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.