EXERCISE $2\mathrm{.}2$ Consequence of the indistinguishability of particles For this exercise we first need to show that any permutation $P$ of the permutation group $S_N$ is a unitary operator, i$.$e$.$

$P^{}=P^{-1}.$

Reminder: Permutations are defined as arrangements of the numbers $(1,2,$dots,$N)$ in a particular sequence and written as $P=(p_1,p_2,$dots,$p_N).$ The number of arrangements of $N$ numbers $(1,2,$dots,$N)$ is $a(N)=N!.$ The permutations of $N$ numbers form a group $S_N$ with $I=(1,2,$dots,$N)$ as identity element. A pair of numbers $p_i$ and $p_j$ in the permutation $P$ for which $p_i>p_j$ form an inversion. A permutation with an even number of inversions is called even, a permutation with an odd number of inversions is called odd. Each permutation is thus characterized by

$(P)=e^{F(P)}=+-1$

where $f(P)$ is the number of inversions and $(P)=+1$ for an even, $(P)=-1$ for an odd permutation.

Occupation number representation: Bosons and Fermions

$19$

The indistinguishability of quantum particles means that there can be no observable $O$ of the many$-$particle system that can be used to distinguish the particles.

Express this requirement as a relation between the observable $O$ and an arbitrary permutation $n{S}_N.$ You may want to start from the expressions for the expectation values of the observable $O$ in an arbitrary many$-$particle state $|$:$|$ and the permuted state $P|$:$|.$