Covariance of the Schr\"odinger equation under low velocity boosts

Abstract

It is known that the Schroedinger equation is not covariant under Galilei boosts, unless the phase of its solutions are shifted simultaneously. It is argued that the phase shift is not a coordinate transformation, because it depends on the mass of the Schroedinger particle. The phase shift also cannot be derived from low speed Lorentz boost. It is proposed to extend the Galilei boost with two terms of order v^2/c^2 to avoid these issues and to guarantee covariance of the Schroedinger kinetic energy and momentum. The extensions imply that proper time and relativity of simultaneity are essential features of Schroedinger quantum mechanics.