How Does the Casimir Operator Function in SU(2) Algebra?

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SUMMARY

The Casimir operator for SU(2) algebra is defined as C2 = T11T11 + T12T21 + T21T12 + T22T22, where Tij = a+i aj. The correct value for C2 in SU(2) is \frac{N}{2} (\frac{N}{2} + 1). However, the direct calculation presented in the discussion yields a different result, specifically N(N + 1), indicating a potential error in the calculation process. The discussion also addresses the expression of the three generators of SU(2) in terms of the operators a_i and a^\dagger_i.

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This discussion is beneficial for theoretical physicists, mathematicians specializing in algebra, and students studying quantum mechanics, particularly those interested in the properties of SU(2) and its applications in particle physics.

hjlim
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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!
 
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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
 

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