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Kaku, Quantum Field Theory Page 47 (2.68/9)

  1. Feb 13, 2010 #1
    1. The problem statement, all variables and given/known data
    Here is equation (2.68)
    [tex](M^{ij})_{ab} = -i(\delta^i_a\delta^j_b - \delta^j_a\delta^i_b)[/tex]

    Here is equation (2.69) (abbreviated)
    [tex][M^{ij},M^{lm}]_{ab} = +i\delta^{jl}(M^{im})_{ab} +- ...[/tex]

    The problem is to show that (2.68) implies (2.69)

    2. Relevant equations



    3. The attempt at a solution
    [tex][M^{ij},M^{lm}]_{ab} = (M^{ij})_{ac}(M^{lm})_{cb} - (M^{lm})_{ac}(M^{ij})_{cb}[/tex]
    [tex]= -(\delta^i_a \delta^j_c - \delta^j_a \delta^i_c)(\delta^l_c \delta^m_b - \delta^m_c \delta^l_b) + (\delta^l_a \delta^m_c - \delta^m_a \delta^l_c)(\delta^i_c \delta^j_b - \delta^j_c \delta^i_b)[/tex]
    [tex]= -\delta^i_a\delta^{jl}\delta^m_b + \delta^m_a\delta^{jl}\delta^i_b +- ...[/tex]
    [tex]= -i\delta^{jl}(M^{im})_{ab} +- ...[/tex]

    This is (2.69) times a factor of -1. Am I wrong, or is Kaku. If Kaku, then what it correct. I have tried to find this equation in other books, but without success. I was able to confirm equation (2.88) on page 51 using equation (2.87). Note the leading minus sign on the right hand side of (2.68) which is not found on (2.87).
     
  2. jcsd
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