Casimir operator in su(2)

1. Mar 1, 2010

hjlim

Hi all,

If I define Tij = a+i aj, then

C2 = T11T11 + T12T21 + T21T12 + T22T22 is a second order casimir operator.

For SU(2), it's $$\frac{N}{2}$$ ($$\frac{N}{2}$$ + 1)

But as I calculate it directly,
C2 = a+1 a1a+1 a1 + a+1 a2a+2 a1 + a+2 a1a+1 a1 + a+2 a2a+2 a2 =
a+1 a1a+1 a1 + a+1 (a+2a2 + 1)a1 + a+2 (a+1a1 + 1)a2 + a+2 a2a+2 a2 =
N1N1 + N1(N2 + 1) + N2(N1 + 1) + N2N2 = (N1 + N2)2 + N1 + N2 = N(N + 1)

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

2. Mar 3, 2010

torquil

How are the three generators of SU(2) expressed in terms of the $$a_i$$ and $$a^\dagger_i$$?

Btw, you have a typo in the next to last term on the first line of the big calculation. The subscript 1 should be a 2. It doesn't influence your calculation.

Torquil