Relativistic model of spontaneous wave-function localization induced by nonHermitian colored noise

Abstract

We develop a quantum field theory based on random nonHermitian actions, which upon quantization lead to stochastic nonlinear Schr\"{o}dinger dynamics for the state vector. In this framework, Lorentz and spacetime translation symmetries are preserved only in a statistical sense: the probability distribution of the action remains invariant under these transformations. As a result, the theory describes ensembles of quantum-state trajectories whose probability distributions remain invariant under changes of reference frame. As a concrete example, we augment the Dirac action with a purely imaginary term coupling the fermion density operator to a universal colored noise. This noise is constructed by solving the d'Alembert equation with white noise as its source, using a generalized stochastic calculus in 1+3 dimensions. We demonstrate that the colored noise drives stochastic localization of wave packets and derive the localization length analytically. Remarkably, the localization length decreases as the size of the observable universe increases. Our model thus provides a potential framework for relativistic spontaneous wave-function collapse. While establishing consistency with Born's law remains an open challenge, the present work constitutes a step toward embedding collapse models into a Lorentz-invariant quantum field theory.