Srednicki — Quantum Field Theory · Problem 70.6

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · bianchi-identity-yang-mills

Problem

The field strength in Yang-Mills theory is in the adjoint representation, and so its covariant derivative is

(D_\rho F_{\mu\nu})^a = \partial_\rho F^a_{\mu\nu} - igA^c_\rho(T^c_{\mathrm{A}})^{ab}F^b_{\mu\nu} . \tag{70.40}

Prove the Bianchi identity,

(D_\mu F_{\nu\rho})^a + (D_\nu F_{\rho\mu})^a + (D_\rho F_{\mu\nu})^a = 0 . \tag{70.41}