# Exchange statistics from topology of configuration space

I'm trying to understand Leinaas and Myrheim's famous 1976 argument for exchange symmetry of the wavefunction.

If we consider the configuration space $$\mathcal{M}_2$$ for two identical particles moving in $$d$$ spatial dimensions we have

$$\mathcal{M}_2 = \mathbb{R}^d \times r(d,2)$$

where the first factor parametrizes the center of mass coordinate and the second factor $$r(d,2) =(\mathbb{R}^d / \mathbb{Z}_2)$$ corresponds to relative motion.

Leinaas and Myrheim's argument can be summarized as follows. If one artificially removes the origin from the orbifold space $$r(d,2)$$ then one of the resulting factors is the real projective space of dimension $$d-1$$,

$$r(d, 2) - \{0\} = (0,\infty) \times \mathbb{RP}^{d-1}$$

It is known that the first homotopy group of real projective space is $$\pi_1(\mathbb{RP}^{n}) = \mathbb{Z}_2$$ for $$n > 1$$ while $$\pi_1(\mathbb{RP}^{n}) = \mathbb{Z}$$ for $$n=1$$.

The physical dimension $$d = 3$$ (corresponding to $$n>1)$$ thus allows for the possibility of double-valued wavefunctions on the configuration space $$\mathcal{M}_2$$ which lift to single-valued wave functions on $$\mathbb{R}^d \times \mathbb{R}^d$$ satisfying exchange antisymmetry.

My concern is that the above argument depended on artificially removing the origin from the relative space $$r(d,2)$$ which physically corresponds to disallowing coincident particle positions. This seems to be contradicted by bosonic statistics which allow for coincident particle positions.

• I think this argument is done without assuming anything about bosons and fermions, deriving from classical configuration space. As a result of this argument then you get symmetry/anti-symmetry of wave functions which decides whether they can be in same state. Also, bosons can be in same quantum state which doesn't imply same position. Commented May 21 at 18:55