Identical particles

Remarks

The orthogonal projections onto the two subspaces are $P_\pm = \frac{1}{n!} \sum_{\sigma \in S_n} (\pm 1)^{\sigma} U_\sigma$, where $(+1)^\sigma = 1$ and $(-1)^\sigma = \sgn(\sigma)$.

Example: let $\mathfrak{h} = \C^2$ with orthonormal basis $e_1, e_2$ and $n = 2$. The vector $\psi = (e_1 \otimes e_2 - e_2 \otimes e_1)/\sqrt{2}$ satisfies $U_{(12)} \psi = -\psi$, and $(12)$ is the only nontrivial permutation, so $\psi \in \mathcal{H}_-^{(2)}$. In fact $\mathcal{H}_-^{(2)}$ is the line spanned by $\psi$, while $\mathcal{H}_+^{(2)}$ has dimension $3$.

Non-example: the vector $e_1 \otimes e_2$ lies in neither subspace, since $U_{(12)}(e_1 \otimes e_2) = e_2 \otimes e_1$ is not $\pm e_1 \otimes e_2$. It describes two distinguishable particles, one in state $e_1$ and one in state $e_2$.

In wave function language, with $\mathfrak{h} = L^2(\R^3)$, bosonic wave functions satisfy $\psi(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = \psi(x_1, \dots, x_n)$ and fermionic ones change by $\sgn(\sigma)$. Spin is included by taking $\mathfrak{h} = L^2(\R^3) \otimes \C^{2s+1}$. The spin–statistics theorem of relativistic quantum field theory says that particles of integer spin are bosons and particles of half-integer spin are fermions.