Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths

TL;DR

This paper investigates how quantum measurements influence the fractal geometry of particle paths, showing that measurement dynamics can alter the Hausdorff dimension and linking measurement physics with quantum fractality.

Contribution

It provides a realistic model of measurement effects on quantum path fractality, extending Abbott et al.'s theoretical predictions by including measurement-induced perturbations and feedback control.

Findings

Measurement dynamics lower the Hausdorff dimension of quantum paths.

Feedback control stabilizes trajectories and allows tuning of fractal dimension.

Results reduce to Abbott et al.'s predictions when measurement effects vanish.

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.