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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!

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# Casimir operator in su(2)

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