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For abelain $U(1)$ gauge theory ($F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$) we can expand heat kernel operator $Q$ as \begin{align} Q &= -(\partial - iA)^2 + m^2 \\ & = - \partial^2 + m^2 + 2 i A \cdot \partial + i \partial \cdot A + A^2 \\ &\simeq - \partial^2 + m^2 + i F_{\mu\nu} x^\mu \partial^\nu + \frac{1}{4}F_{\sigma \mu} F^{\sigma\nu} dx^\mu dx_\nu \\ & = Q^0 + Q^1 \end{align} where \begin{align} &Q^0 =-\partial^2 + m^2 \\ &Q^1 =i F_{\mu\nu} x^\mu \partial^\nu + \frac{1}{4}F_{\sigma \mu} F^{\sigma\nu} dx^\mu dx_\nu \end{align} Here we impose :

( derivative of field strength can be ignored or is irrelevant so we set $A_\mu \rightarrow \frac{1}{2} F_{\alpha\mu} x^\alpha$. )


What i want to do its extension to non-abelian case ($F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu - i[A_\mu, A_{\nu}]$) \begin{align} Q &= -(\partial - iA)^2 + m^2 \\ & = - \partial^2 + m^2 + 2 i A \cdot \partial + i \partial \cdot A + A^2 \\ & = Q^0 + Q^1 \end{align} In this case how we can obtain $Q^0$, and $Q^1$ from using above method?

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