Indefinite special unitary group ($SU(p,q)$)

The indefinite special unitary group preserves a Hermitian form of signature $(p,q)$ and has determinant one: $M^{†}JM=J$, where $J=\mathrm{diag}(I_p,-I_q)$. With both $p,q$ positive it is noncompact, unlike the group preserving a positive-definite form. Its Lie algebra satisfies $X^{†}J+JX=0$ and $\mathrm{Tr}X=0$.

Matrices $\left(\begin{array}{cc}\alpha & \beta \\ {\beta }^{\ast }& {\alpha }^{\ast }\end{array}\right)$ with the displayed constraint form a real three-dimensional noncompact Lie group. A real Lie algebra basis is $m_1={\sigma }_x/2$, $m_2={\sigma }_y/2$, $m_3=i{\sigma }_z/2$, with $[m_1,m_2]=m_3$, $[m_2,m_3]=-m_1$, and $[m_3,m_1]=-m_2$.

Pair creation and annihilation in two independent oscillator modes obey $[K_{-},K_{+}]=2K_3$ and $[K_3,K_{\pm }]=\pm K_{\pm }$. The Hermitian components $K_1=(K_{+}+K_{-})/2$ and $K_2=(K_{+}-K_{-})/(2i)$ satisfy $[K_1,K_2]=-i{K}_3$. Multiplying all three Hermitian components by $i$ gives the real Lie bracket signs of SU(1,1). Both occupations change equally, so their difference is conserved. This realizes the algebra used in two-mode squeezed states.

The identities $K_{+}{K}_{-}=N_a{N}_b$ 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.