Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths

TL;DR

The paper addresses whether real measurement backaction alters the fractal geometry of quantum paths beyond Abbott et al.'s idealized result. It develops a framework where sequential position measurements are modeled with Gaussian meters, leading to a Lindblad-type master equation and a distinction between nonselective and selective evolution, the latter incorporating feedback to stabilize trajectories. Key findings show that nonselective measurements reduce the emergent Hausdorff dimension from $d=2$ toward smaller values as measurement strength increases, while selective evolution requires feedback to restore $d=2$ regardless of initial conditions, with the Abbott limit recovered for weak measurement ($D\to\infty$). The work provides a realistic link between quantum fractality and detector physics, quantifying how detectors reshape spacetime statistics at quantum scales and suggesting avenues to connect these ideas with quantum gravity and relativistic detector theories.

Abstract

In a seminal paper, Abbott et al. analyzed the relationship between a particle's trajectory and the resolution of position measurements performed by an observer at fixed time intervals. They predicted that quantum paths exhibit a universal Hausdorff dimension that transitions from $d=2$ to $d=1$ as the momentum of the particle increases. However, although measurements were assumed to occur at intervals of time, the calculations only involved evaluating the expectation value of operators for the free evolution of wave function within a single interval, with no actual physical measurements performed. In this work we investigate how quantum measurements alter the fractal geometry of quantum particle paths. By modelling sequential measurements using Gaussian wave packets for both the particle and the apparatus, we reveal that the dynamics of the measurement change the roughness of the path and shift the emergent Hausdorff dimension towards a lower value in nonselective evolution. For selective evolution, feedback control forces must be introduced to counteract stochastic wave function collapse, stabilising trajectories and enabling dimensionality to be tuned. When the contribution of the measurement approaches zero, our result reduces to that of Abbott et al. Our work can thus be regarded as a more realistic formulation of their approach, and it connects theoretical quantum fractality with measurement physics, quantifying how detectors reshape spacetime statistics at quantum scales.