Particle exchange operator ($P_{12}$)

For two particles, the exchange operator sends $|\alpha ⟩\otimes |\beta ⟩$ to $|\beta ⟩\otimes |\alpha ⟩$. It has eigenvalues $+1$ on symmetric states and $-1$ on antisymmetric states.

Physical states of identical particles have definite symmetry under every particle exchange operator: symmetric for bosons and antisymmetric for fermions.

Bosons are identical particles whose total multiparticle state is symmetric under exchange. In three-dimensional relativistic quantum theory they have integer spin.

Bose statistics is the symmetric exchange statistics of identical bosons. Commuting creation operators build the symmetric bosonic Fock space; a normalized one-particle mode has occupation states $(a^{†}{)}^n|0⟩/\sqrt{n!}$ for every nonnegative integer $n$. At thermal equilibrium these statistics lead to the Bose-Einstein distribution. Exchange symmetry itself does not require thermal equilibrium.

Commuting creation operators make multiparticle states invariant under exchange of their labels. With $[a,a^{†}]=1$, the normalized number states $(a^{†}{)}^r|0⟩/\sqrt{r!}$ exist for all $r\ge 0$, giving unrestricted bosonic occupation numbers. The single-mode thermal sum is $Z={\sum }_{r\ge 0}{e}^{-\beta Er}=(1-e^{-\beta E}{)}^{-1}$ and the mean occupation is $(e^{\beta E}-1{)}^{-1}$. Thus the exchange symmetry and the Bose-Einstein distribution are two expressions of the same bosonic algebra.

Fermions are identical particles whose total multiparticle state is antisymmetric under exchange. In three-dimensional relativistic quantum theory they have half-integer spin.

For two identical spin-one-half fermions with no additional antisymmetric internal factor, the antisymmetric spin-one-half singlet state combines with an even-$\ell$ symmetric orbital state, while the symmetric spin-one-half triplet state combines with an odd-$\ell$ antisymmetric orbital state.

No two identical fermions may occupy the same one-particle quantum state. At zero temperature, noninteracting fermions fill the available one-particle states in increasing order of energy.

The spin-statistics theorem associates integer-spin particles with bosonic exchange symmetry and half-integer-spin particles with fermionic exchange symmetry in relativistic quantum field theory under the standard locality and positivity assumptions.

For two spin-one particles, the spin tensor product has a six-dimensional symmetric subspace and a three-dimensional antisymmetric subspace. A symmetric spatial state combines with symmetric spin, while an antisymmetric spatial state combines with antisymmetric spin, so that the total bosonic state remains symmetric.

If $E_n=E_0+n\Delta$, the two-particle level $2E_0+N\Delta$ has degeneracyEach unequal orbital pair contributes six symmetric-spin states and three antisymmetric-spin states; an equal orbital pair exists only for even $N$ and contributes six states.