# Can closed loops evade the spin-statistic theorem in 3 dimensions?

The famous spin-statistics result asserts that there are only bosons and fermions, and that they have integer and integer-and-a-half spin respectively. In two-dimensional condensed matter systems, anyons are also possible. They avoid this result due to a topological obstruction.

Suppose that "loops" were fundamental, rather than point-particles. In addition to normal relative motion, it would be possible for one loop to go through the hole of another, and a path of this sort would not be continuous with one that doesn't. Would this provide enough of a topological obstruction to be able to avoid the spin-statistics theorem?

The answer to your question is yes, your intuition is 100% correct. It all boils down to the topology of the configuration space $\mathcal C$, mainly the first homotopy group $\pi_1(\mathcal C)$ (which is non-zero in your example). See problem set 1, problem 3 from this course at Oxford. This exercise is precisely about loops in 3+1D! One has to argue that for any type of point-particle statistics in 2+1D (representations of the Braid group) there exist a corresponding loop statistics in 3+1D.

Note however that these loops will only lead to non-trivial statistics in 3+1D, in higher dimensions there will not be any topological obstruction. This is related to the fact that in higher dimensions, you can always untie knots.

More generally you can think about many different ways of getting non-trivial statistics. You can give your object more complicated internal structure than just point-particles (loops are just one example) or you can put your objects on topologically non-trivial manifolds. See for example this paper about so-called "projective ribbon permutation statistics", which is a way of having non-trivial statistics in higher dimensions but with "defect" that have some internal structure.

Well, yes and no. If you are interested in more general statistics for point-particles, you have to go to 2+1 dimensions where you have anyons. Under an exchange of two (abelian) anyons, the wave function changes by a phase $e^{i\pi\alpha}$. Here $\alpha = 1$ correspond to fermions, $\alpha=0$ correspond to bosons while for any phase $\alpha\in[0,1]$ you have any-ons. So in the sense of exchange statistics, anyons interpolate between fermions and bosons.
There is however another approach to take. In a famous paper, Haldane suggests the so-called exclusion statistics, which defines particle statistics in terms of a generalized Pauli exclusion principle (as you suggest). A natural question is then, does the interpolation of anyons between fermions and bosons lead to a interpolation of exclusion statistics? Murthy and Shankar seem to have tried to answer this question, and they find the corresponding exclusion parameter for the $\alpha$ anyon (equation (16)). However, I don't know enough about exclusion statistics and the state of the field to give many details. But you can learn a lot from reading some of the papers which cite Haldanes paper.