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We determine the Hausdorff dimension of a particle path, $D_{\rm H}$, in the
recently proposed `smeared space’ model of quantum geometry. The model
introduces additional degrees of freedom to describe the quantum state of the
background and gives rise to both the generalised uncertainty principle (GUP)
and extended uncertainty principle (EUP) without introducing modified
commutation relations. We compare our results to previous studies of the
Hausdorff dimension in GUP models based on modified commutators and show that
the minimum length enters the relevant formulae in a different way. We then
determine the Hausdorff dimension of the particle path in smeared momentum
space, $\tilde{D}_{\rm H}$, and show that the minimum momentum is dual to the
minimum length. For sufficiently coarse grained paths, $D_{\rm H} = \tilde{D}_{\rm H} = 2$, as in canonical quantum mechanics. However, as the
resolutions approach the minimum scales, the dimensions of the paths in each
representation differ, in contrast to their counterparts in the canonical
theory. The GUP-induced corrections increase $D_{\rm H}$ whereas the
EUP-induced corrections decrease $\tilde{D}_{\rm H}$, relative to their
canonical values, and the extremal case corresponds to $D_{\rm H} = 3$,
$\tilde{D}_{\rm H} = 1$. These results show that the GUP and the EUP affect the
fractal properties of the particle path in fundamentally different, yet
complimentary, ways.

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