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In Khare's book of fractional statistics and quantum theory, when discussing why we need fractional statistics he arrives at the configuration space enter image description here

for a system of two identical particles in $d$ spatial dimensions.
Q1: I do not see how the second equality is justified. ($P_{d-1}$ is the $d-1$ dimensional real projective space, and $\langle 0,\infty \rangle$=$(0,\infty)-\{0\}$).

Later on, he writes this: enter image description here

Q2: He seems to imply that $(\mathbb R^2 - \{0\})/\mathbb Z_2 = \mathbb R P_{1}$. But actually $\mathbb R P_{1} = (\mathbb R^2 - \{0\})/\sim $ with $x\sim y$ iff they lie on the same line. Maybe there is a physical reason to make this identification $(\mathbb R^2 - \{0\})/\mathbb Z_2 = \mathbb R P_{1}$?

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Denote

$$ M_d := (\mathbb{R}^d -\{0\})/\{x \sim - x\} \ . $$

To answer Q2: for the equation (2.14) to hold, it is not necessary that $M_d = \mathbb{R}P_{d-1}$. It's enough if these two spaces are homotopy equivalent, which they are. Indeed, consider an equivalence class $[x] \in M_d$. $[x]$ may be written as $e^\lambda [\hat{x}]$, where $\lambda \in \mathbb{R}$ and $\hat{x}$ is a unit vector in $\mathbb{R}^d$. Then consider the following map:

$$ F:[0,1] \times M_d \mapsto M_d \ , \quad (t,e^\lambda [\hat{x}]) \mapsto e^{(1-t)\lambda} [\hat{x}] \ .$$

Then $F(0,\cdot) = \text{id}_{M_d}(\cdot)$, but $F(1,\cdot)$ maps $M_d$ on the projective space.

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