Srednicki — Quantum Field Theory · Problem 78.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · background-field-gauge

Problem

Our one-loop corrections can be interpreted as functional determinants. Define

\mathbf{\square}_{\mathrm{R},(a,b)} \equiv \bar{D}^2 + gT^a_\mathrm{R}\bar{F}^a_{\mu\nu}S^{\mu\nu}_{(a,b)} \;, \tag{78.38}

where

\bar{D}_\mu = \partial_\mu - ig(T^a_\mathrm{R})\bar{A}^a_\mu

is the background-covariant derivative in the representation R, implicitly multiplied by the identity matrix for the

(a,b)

representation of the Lorentz group, and

S^{\mu\nu}_{(a,b)}

are the Lorentz generators for that representation; in particular,

S^{\mu\nu}_{(1,1)} = 0 \;, \tag{78.39}

S^{\mu\nu}_{(2,1)\oplus(1,2)} = \tfrac{i}{4}[\gamma^\mu,\gamma^\nu] \;, \tag{78.40}

(S^{\mu\nu}_{(2,2)})_{\alpha\beta} = -i(\delta^\mu_{\;\alpha}\delta^\nu_{\;\beta} - \delta^\nu_{\;\alpha}\delta^\mu_{\;\beta}) \;. \tag{78.41}

Show that the one-loop contribution to the terms in the quantum action that do not depend on the ghost fields is given by

\begin{aligned} \exp i\Gamma_{1-\mathrm{loop}}(\bar{A}, 0, 0; \bar{A}) &\propto (\det\,\mathbf{\square}_{\mathrm{A},(1,1)})^{+1} \notag\\ &\times (\det\,\mathbf{\square}_{\mathrm{A},(2,2)})^{-1/2} \notag\\ &\times (\det\,\mathbf{\square}_{\mathrm{R}_{\mathrm{DF}},(2,1)\oplus(1,2)})^{+1/2} \notag\\ &\times (\det\,\mathbf{\square}_{\mathrm{R}_{\mathrm{CS}},(1,1)})^{-1} \;. \tag{78.42} \end{aligned}

Verify that this expression agrees with the diagrammatic analysis in this section.