Permutations

There are $N!$ permutations of $N$ particles. We label the permutations by $N!$ indices $p$ and define a permutation operator $\hat{\Pi}_p$, for example

$$\hat{\Pi}_{p=(1,3)} \Psi(\xi_1,\xi_2,\xi_3) = \Psi(\xi_3,\xi_2,\xi_1)$$ (1.7)

$$\hat{\Pi}_{p=(1,2,3)} \Psi(\xi_1,\xi_2,\xi_3) = \hat{\Pi}_{p=(2,3)}\Psi(\xi_2,\xi_1,\xi_3)= \Psi(\xi_2,\xi_3,\xi_1)$$ (1.8)

We furthermore define the symmetrization operator $\hat{S}$ and the anti-symmetrization operator $\hat{A}$,

$$\hat{S} = \frac{1}{\sqrt{N!}}\sum_p \hat{\Pi}_p$$ (1.9)

$$\hat{A} = \frac{1}{\sqrt{N!}}\sum_p \hat{\Pi}_p {\rm sign}(p),$$ (1.10)

where ${\rm sign}(p)$ is the sign of the permutation which is either $-1$ or $+1$, ${\rm sign}(p)=({-1})^{n(p)}$ where $n(p)$ is the number of swaps required to achieve the permutation $p$.