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.



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