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

TL;DR

This work develops a quantum field theory built on a random nonHermitian action that preserves statistical Lorentz symmetry, achieved by coupling fermions to a universal colored noise $h(x)$ constructed from white noise via a 1+3D stochastic calculus. Quantization yields a nonlinear stochastic evolution for the state vector with an energy-production rate that remains finite due to the absence of second-order noise terms, and the framework predicts wave-packet localization with a localization length $r_c$ that scales as $r_c \propto 1/(\Lambda\,\tilde{\eta}^2)$, decreasing as the observable universe radius $\Lambda$ grows. In the nonrelativistic limit the mechanism is explicit: the physical state is multiplied by a local, randomly peaked factor $e^{m\eta\Theta(t,\mathbf{x})}$, with $\Theta$ built from the cumulative noise along past light cones. The paper also extends the construction to curved spacetime (FLRW), showing the infrared divergence is removed by cosmic expansion, and reports numerical simulations that corroborate the localization behavior. However, the Born-rule compatibility remains an open challenge, inviting connections to decoherence frameworks such as quantum Darwinism for potential resolution and further extensions to particle-antiparticle dynamics.

Abstract

We develop a quantum field theory based on random nonHermitian actions, which upon quantization lead to stochastic nonlinear Schrödinger 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.

Figures (3)

• Figure 1: Supporting regions of $\Theta(t, \textbf{x})$ at different spatial positions ($\textbf{x}_1, \textbf{x}_2, \textbf{x}_3$) are illustrated using different colors for clarity.
• Figure 2: ($a$) Squared wave function at different final times with $m\eta = 2$. ($b$) Averaged $\text{IPR}$ as a function of $\eta$ at $t_f = 1$. We choose $l_z = 30$ and $\Lambda = 10$.
• Figure 3: Averaged inverse participation ratio (IPR) as a function of: ($a$) the number of simulation runs, with $\Delta t = 0.1$, $\Delta \rho = 0.1$, and $\Delta z = 0.02$; ($b$) the time partitioning step $\Delta t$, with $\Delta \rho = 0.1$ and $\Delta z = 0.02$; ($c$) the radial partitioning step $\Delta \rho$, with $\Delta t = 0.1$ and $\Delta z = 0.02$; and ($d$) the axial partitioning step $\Delta z$, with $\Delta t = 0.1$ and $\Delta \rho = 0.1$. The parameters used in these simulations are $\Lambda = 10$, $l_z = 10$, $t = 10$, and $\eta = 5$.