# Eikonal Blog

## 2010.02.16

### A su(1,1) realization

Filed under: mathematics — Tags: — sandokan65 @ 14:13

For assembly of bosonic particle creation $\hat{a}_{\underline{p}}^\dagger$ and annihilation $\hat{a}_{\underline{p}}$ operators, the following bilinear combinations

• $\hat{{\cal L}}_1 + i \hat{{\cal L}}_2 :\equiv \hat{a}_{\underline{p}}^\dagger \hat{a}_{-\underline{p}}^\dagger$,
• $\hat{{\cal L}}_1 - i \hat{{\cal L}}_2 :\equiv \hat{a}_{\underline{p}} \hat{a}_{-\underline{p}}$,
• $\hat{{\cal L}}_3 :\equiv \frac12 \left(\hat{a}_{\underline{p}}^\dagger \hat{a}_{\underline{p}} + \hat{a}_{-\underline{p}}^\dagger \hat{a}_{-\underline{p}} + 1 \right)$.

form a representation of the $su(1,1)$ algebra:

• $[\hat{{\cal L}}_3 , \hat{{\cal L}}_1] = + i \hat{{\cal L}}_2$,
• $[\hat{{\cal L}}_2 , \hat{{\cal L}}_3] = + i \hat{{\cal L}}_1$,
• $[\hat{{\cal L}}_1 , \hat{{\cal L}}_2] = - i \hat{{\cal L}}_3$.

Source: G.E. Volovik “Vacuum in Quantum Liquids and in General Relativity”; arXiv: gr-qc/0104046v1, 2001.04.15; http://arxiv.org/abs/gr-qc/0104046.