Srednicki — Quantum Field Theory · Problem 70.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · covariant-derivative-product-rule

Problem

Consider a field φi\varphi_i in the representation R1\mathrm{R}_1 and a field χI\chi_I in the representation R2\mathrm{R}_2. Their product φiχI\varphi_i\chi_I is then in the direct product representation R1⊗R2\mathrm{R}_1\otimes\mathrm{R}_2, with generator matrices given by eq. (70.13). a) Prove the distribution rule for the covariant derivative, [Dμ(φχ)]iI=(Dμφ)iχI+φi(Dμχ)I.(70.38)[D_\mu(\varphi\chi)]_{iI} = (D_\mu\varphi)_i\chi_I + \varphi_i(D_\mu\chi)_I . \tag{70.38} b) Consider a field φi\varphi_i in the complex representation R. Show that ∂μ(φ†iφi)=(Dμφ†)iφi+φ†i(Dμφ)i.(70.39)\partial_\mu(\varphi^{\dagger i}\varphi_i) = (D_\mu\varphi^\dagger)^i\varphi_i + \varphi^{\dagger i}(D_\mu\varphi)_i . \tag{70.39} Explain why this is a special case of eq. (70.38).

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