Eikonal Blog


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.

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