Permutation operator

The permutation operator $P_{ij}$ is an operator that permutes the variables $q_{i}$ and $q_{j}$. It commutes with the Hamiltonian: $[P_{ij},H]=0$. For some wave function $\psi(q_{1},\ldots,q_{N})$ of $N$ generalized coordinates, the permutation operator acts on it as

The is a unitary operator since $P_{ij}^{2}=\hat{\mathbf{1}}$ and hence its eigenvalues are $\epsilon=\pm1$. $\epsilon=1$ represents a symmetry, $\epsilon=-1$ an antisymmetry.

The operator can be generalized to make multiple swaps in one application. In other words, it does a permutation of the coordinates. We define the operator as $P:q_{n}\to q_{P_{n}}$, where $P_{n}$ represents the new index after the permutation. The application is the same:

This operator is even if the number of swaps is even, and odd otherwise.

A state $\psi$ for which the operator $P_{ij}$ is even is said to be symmetric (under permutation) and a state for which it is odd is said to be antisymmetric (under permutation). In quantum physics, symmetric states are associated with bosons and antisymmetric ones with fermions (this fact is known as the symmetrization postulate). These constraints the explain quantum statistics (Bose-Einstein distribution and Fermi-Dirac distribution) and are a consequence of the indistinguishabilty of particles. (Anti)symmetry remains unchanged under superposition, which means that superimposing multiple (anti)symmetric states leads to an (anti)symmetric state.