Quantifying the failure of Schrödinger dynamics in the free expansion of relativistic particles
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It is known that the Schrödinger equation fails in describing the dynamics of highly energetic particles. We propose to quantify this lack of Lorentz covariance by evaluating the probability of a particle to be measured outside the set of light cones which are compatible with its initial wave function. We consider a simple case of a particle released from a box, which, in turn, is inside a larger container. It is shown that besides the increasing error at relativistic energies, there may be a complete breakdown, with Schrödinger dynamics allowing for deterministic, superluminal signaling for Lorentz factors above 129. In addition, we give an exact asymptotic expression for the violation in local causality by employing the stationary exponent method, from which the Compton wave length of the particle naturally arises as the relevant scale for the stationary points.
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