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.

Advertisements

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Create a free website or blog at WordPress.com.

%d bloggers like this: