How to Implement Current Conservation for SU(N) in the Adjoint Representation?

[Tian]

Homework Statement

In the "An introduction to Quantum Field Thoery" of Peskin and Schroeder, the equation(15.51) of the chapter 15.3 gives the classical equation of motion, so from this equation to derive the current conservation.

Relevant Equations

the classical equation of motion for SU(N), please see my picture

Here is my solution

 

[samalkhaiat]

$$D^{\mu}F_{\mu\nu} = - j_{\nu},$$ Differentiate this covariantly and anti-symmetrized to obtain $$\frac{1}{2}[D^{\mu}, D^{\nu}]F_{\mu\nu} = D^{\nu}j_{\nu}. \ \ \ \ (1)$$ Now, from the definition of the covariant derivative in the adjoint representation (acting on any matrix-valued field) $$D^{\mu}M \equiv \partial^{\mu}M + [A^{\mu} , M],$$ you can show that $$[D^{\mu} , D^{\nu}]M = [F^{\mu\nu} ,M]$$ Thus, for $M = F_{\mu\nu}$, eq(1) becomes $$D^{\nu}j_{\nu} = \frac{1}{2}[F^{\mu\nu} , F_{\mu\nu}] = 0.$$

 

[Tian]

Thank you veery much . It should be done in the adjoint repesentation.