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

[Jimmy Snyder]

Homework Statement

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)

Homework Equations

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).

 

[Jhero]

Hello,

Your attempt at a solution is correct. The factor of -1 in front of (2.69) is just a convention and does not change the validity of the equation. Different books may use different conventions, so it is always important to double check the signs in equations when using them in calculations. Keep up the good work!