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American Journal of Physics

Dimension of a quantum-mechanical path

Larry F. Abbott, Mark B. Wise

Published Jan 1, 1981• 181 citations
Publisher

Abstract

We show that the observed path of a particle in quantum mechanics is a fractal curve with Hausdorff dimension two.
DOI
10.1119/1.12657

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Measurement-Induced Perturbations of Hausdorff Dimension in Quantum Paths

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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.

quant-pharXiv:2512.13046•The European Physical Journal Plus•Dec 16, 2025•0 citations
The notion of the dimension of a relativistic quantum mechanical path is illustrated. An elementary introduction to the concept of a fractal curve allows the discussion of the evolution of the quantum mechanical wave packet for free particles in terms of Hausdorff dimensions of quantum ‘‘trajectories.’’ The results of a previous work by Abbott and Wise can be generalized to relativistic quantum mechanics of spin-1/2 particles in the Foldy–Wouthuysen representation. A further extension to spin-zero particles is briefly outlined. By simple considerations, an understanding of the notion of a Hausdorff dimension and its values can be introduced.
We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$ < < < hhh $$
We obtain an explicit expression relating the writhing number, W[C]W[C]W[C], of the quantum path, CCC, with any value of spin, sss, of the particle which sweeps out that closed curve. We consider a fractal approach to the fractional spin particles and, in this way, we make clear a deeper connection between the Gauss-Bonnet theorem with the spin-statistics relation via the concept of Hausdorff dimension, hhh, associated to the fractal quantum curves of the particles: \frac{h}{2+2s}=W[C]=\frac{1}{4π}\oint_{C}d x_α\oint_{C}d x_β ε^{αβγ} \frac{(x-y)_γ}{|x-y|^3}.
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=2d=2d=2 to d=1d=1d=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.