Boson

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

A boson with spin one. A massive vector boson has three physical particle polarizations. A complex Proca field describes a charged particle and its antiparticle, each with three particle polarizations; the free classical field has three complex dynamical components after its constraints are enforced.

Bosonic exchange symmetry 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 this symmetry leads to Bose-Einstein statistics and the Bose-Einstein distribution. Exchange symmetry itself does not require thermal equilibrium.

Place one noninteracting spin-zero boson in each of two orthonormal orbitals $a,b$. Its normalized symmetric spatial wavefunction is $[a({\mathbf{r}}_1)b({\mathbf{r}}_2)+b({\mathbf{r}}_1)a({\mathbf{r}}_2)]/\sqrt{2}$. At coincident positions the probability density is $2|a(\mathbf{r}){|}^2|b(\mathbf{r}){|}^2$, twice the labelled distinguishable-product value. Coincidence at an exact point is understood as a density or the limit of small detector volumes. If both particles occupy the same orbital, the normalized product is already symmetric and this comparison gives factor one instead; the factor two is not a universal property of every two-boson state.

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.