# Particle statistics in fractal dimensions? [closed]

We know that fermions and bosons are the only two (indistinguishable) particle statistics for $$d\geq 3$$, and that anyons are for $$d=2.$$

What if the space were a fractal? Like the Sierpinski gasket, with $$d=1.585$$?

EDIT for clarity, to try and keep the questio open:

Here they have electrons in a Sierpinski gasket shaped potential.
The Sierpinski gasket has a fractal dimension.
If I want to permute the positions of indistinguishable particles, I have to do this in this fractal dimension. So this neither $$SO(2)$$ ($$d=2$$) not $$SO(1)$$.
The first homotopy (fundamental) group of such group, $$\pi_1(SO(d))$$ would give the number of distinct statistics: $$\mathbb{Z}$$ in $$d=2$$ (anyons) and $$\mathbb{Z_2}$$ in $$d\geq3$$ (fermions, bosons).

How would I go on about it here, in $$d = 1.58$$?

• I think the main confusion I have is that fractal dimensions are not dimensions, but a measure of the complexity of the shape/curve. – Kyle Kanos Dec 19 '18 at 11:00