Boson-pair Casimir operator
= Boson-pair Casimir operator
{title2=$K_3^2-K_1^2-K_2^2=[(N_a-N_b)^2-1]/4$}
The identities $K_+K_-=N_aN_b$ and $K_-K_+=(N_a+1)(N_b+1)$ give the displayed quadratic <Casimir operator>. It commutes with all generators because the number difference does. A fixed number-difference sector therefore has a fixed Casimir value even though pair creation changes its total occupation.