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Casimir operator in su(2)

  1. Mar 1, 2010 #1
    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 [tex]\frac{N}{2}[/tex] ([tex]\frac{N}{2}[/tex] + 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. jcsd
  3. Mar 3, 2010 #2
    How are the three generators of SU(2) expressed in terms of the [tex]a_i[/tex] and [tex]a^\dagger_i[/tex]?

    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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