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

1. Feb 13, 2010

### Jimmy Snyder

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

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

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

2. Relevant equations

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

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).