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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!