= Bosonic statistics from commuting creation operators
{title2=$[a_p^\dagger,a_q^\dagger]=0$}
Commuting <creation operators> make multiparticle states invariant under exchange of their labels. With $[a,a^\dagger]=1$, the normalized number states $(a^\dagger)^r|0\rangle/\sqrt{r!}$ exist for all $r\ge0$, giving unrestricted <bosonic occupation numbers>. The single-mode thermal sum is $Z=\sum_{r\ge0}e^{-\beta Er}=(1-e^{-\beta E})^{-1}$ 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.
Back to article page
