The spin - statistics connection , the Gauss - Bonnet theorem and the Hausdorff dimension of the quantum paths

We obtain an explicit expression relating the writhing number, W [C], of the quantum path, C, with any value of spin, s, 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, h, associated to the fractal quantum curves of the particles: h 2 + 2s = W [C] = 1 4π ∮