I am currently doing some conformal field theory (in four dimensions) and want to show the invariance of Maxwell's equations under conformal transformations, in particular \begin{align} \partial_\mu\big(J^\mu_{~~~\rho}(x)J^\nu_{~~~\sigma}(x)F^{\rho\sigma}\circ g(x)\big) \end{align} where $g$ is a conformal transformation and $J_{\mu\nu}=\partial_\mu g_\nu$ its Jacobian.

Using the product rule, the derivative consists of three terms, one of which gives \begin{align} J^\mu_{~~~\rho}(x)J^\nu_{~~~\sigma}(x)\partial_\mu\big(F^{\rho\sigma}\circ g(x)\big)=\det(J(x))^\frac{1}{2}J^\mu_{~~~\rho}(x)\big(\partial_\xi F^{\xi\rho}\big)\circ g(x). \end{align} This is exactly the desired form as it gives the coupling to the conformally transformed current.

However, I do not see why the two terms, where the derivative acts on the Jacobians, vanish.

I feel like there is some differential equation following from the very definition of conformal transformations \begin{align} J^\mu_{~~~\rho}(x)J^\nu_{~~~\sigma}(x)\eta^{\rho\sigma}=\omega^2(x)\eta^{\mu\nu}, \end{align} where $\omega^2=\det(J(x))^\frac{1}{2}=J^\mu_{~~~\nu}J_\mu^{~~~\nu}$ but I do not see it.

I am happy for every hint.


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