Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

If I define T_{ij}= a^{+}_{i}a_{j}, then

C_{2}= T_{11}T_{11}+ T_{12}T_{21}+ T_{21}T_{12}+ T_{22}T_{22}is a second order casimir operator.

For SU(2), it's [tex]\frac{N}{2}[/tex] ([tex]\frac{N}{2}[/tex] + 1)

But as I calculate it directly,

C_{2}= a^{+}_{1}a_{1}a^{+}_{1}a_{1}+ a^{+}_{1}a_{2}a^{+}_{2}a_{1}+ a^{+}_{2}a_{1}a^{+}_{1}a_{1}+ a^{+}_{2}a_{2}a^{+}_{2}a_{2}=

a^{+}_{1}a_{1}a^{+}_{1}a_{1}+ a^{+}_{1}(a^{+}_{2}a_{2}+ 1)a_{1}+ a^{+}_{2}(a^{+}_{1}a_{1}+ 1)a_{2}+ a^{+}_{2}a_{2}a^{+}_{2}a_{2}=

N_{1}N_{1}+ N_{1}(N_{2}+ 1) + N_{2}(N_{1}+ 1) + N_{2}N_{2}= (N_{1}+ N_{2})^{2}+ N_{1}+ N_{2}= N(N + 1)

which is different from above. Can you let me know what is wrong with my argument? Thank you very much!

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Casimir operator in su(2)

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Casimir operator | Date |
---|---|

I Can we construct a Lie algebra from the squares of SU(1,1) | Feb 24, 2018 |

A On spectra | Feb 21, 2018 |

I Commutator between Casimirs and generators for Lorentz group | Apr 7, 2016 |

Why are the casimirs independent of the representation | May 5, 2009 |

Casimir operators | Oct 2, 2006 |

**Physics Forums - The Fusion of Science and Community**