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I am a bit confused about inverting operators, and calculating propagators on a curved spacetime. Consider the following example:

If I have a Lagrangian for a charged scalar field on a curved spacetime $$ \mathcal{L} = -\frac 14 F_{\mu\nu} F^{\mu\nu} + |D_\mu \phi|^2 $$ it would involve covariant derivatives of $A_\mu$, where $\nabla_\mu A_\nu = \partial_\mu A_\nu + \Gamma_{\mu\;\nu}^{\;\lambda} A_\lambda$, and derivatives of the scalar field of the form $D_\mu \phi = \partial_\mu \phi - iA_\mu \phi$. (Note that for a scalar field the covariant derivative is equal to the normal partial derivative $\nabla_\mu \phi = \partial_\mu \phi$)

Now, if I rewrite the scalar field as $\phi = \sqrt{m} e^{i\theta}$, $\phi^* = \sqrt{m} e^{-i\theta}$, I would find for it's kinetic term $$ |D_\mu \phi|^2 = \Big| (\partial_\mu - i A_\mu) \sqrt{m} e^{i\theta} \Big|^2 = m \Big( (\partial_\mu \theta)^2 - 2 (\partial_\mu \theta) A^\mu + A_\mu^2 \Big) $$ which involves (covariant) derivatives of the 'new' scalar field $\nabla_\mu \theta = \partial_\mu \theta$. If I explicitly write the partial derivatives in the partition function $Z = \int DA_\mu D\phi\; e^{iS}$, integrating out $\theta$ would now give me for the kinetic term in the Lagrangian (the above line) $$ A_\mu \; \left[m\left( g^{\mu\nu} - \frac{\partial^\mu \partial^\nu}{\Box} \right)\right] A_\nu $$ where the $\Box$ and $\partial_\mu$ operators are normal partial derivatives that are now working on $A_\mu$. Since the original expression only had simple partial derivatives it seems fair that after integrating out $\theta$, I should still only have partial derivatives.

However, this can't be right, since such a term contracted with the $A_\mu$ field would not be Lorentz invariant. Since we need to have a Lorentz invariant expression in the Lagrangian, and for the previous scalar field $\partial_\mu \theta = \nabla_\mu \theta$, it seems fair that I should be allowed to replace these partial derivatives by covariant derivatives. $$ A_\mu \; \left[m\left( g^{\mu\nu} - \frac{\nabla^\mu \nabla^\nu}{\nabla^\lambda \nabla_\lambda} \right)\right] A_\nu $$ Thus, intuitively I think the second result is correct. However, for the vector field $\nabla_\mu \neq \partial_\mu$, and I feel uncomfortable about the extra Christoffel connections that make the difference between the above expression and the previous one. Could anyone else give their thoughts on this and perhaps give another argument for why the above expression is (not) right?

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Integrating out the $\theta$ field is equivalent to substitution the EOM of the $\theta$ field in the action. Now, bear in mind while doing that you should be careful about $\sqrt{-g}$ when shifitng the derivatives during integration by parts, if you choose to use $\partial$. That is why retain the covariant derivatives even when you work with a scalar in curved spacetime.

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