In a non-abelian gauge theory there is a "fundamental" gauge field $A_\mu^a$ with gauge index $a$ often called connection. Although $ A_\mu^a$ is not gauge invariant, gauge invariant quantities can be obtained from it. In particular the partition function in the path integral formulation is obtained schematically as: \begin{equation} \mathcal{Z}[J] = \int [\text{d} A_\mu] e^{i \mathcal{S}[A, \partial A] + i \int J \cdot A} \end{equation} Moreover, the associated field strength is given by \begin{equation} \left[\mathcal D_\mu,\mathcal D_\nu \right] \psi = -i \mathcal F_{\mu \nu} \psi \end{equation}

On the other hand, for gravity the usual logic is that from the "fundamental" metric tensor $g_{\mu \nu}$ one can construct connections (Christoffel symbols) $\Gamma^\alpha_{\mu \nu}$. Subsequently curvature quantities are obtained as commutators \begin{equation} \left[\nabla_\mu,\nabla_\nu \right] u^\alpha = R^\alpha_{\; \lambda \mu \nu}u^\lambda \end{equation} It is then clear that gravity is nothing but a gauge theory with gauge group $\mathcal{Diff}$ with the identifications \begin{equation} A_\mu^a \longleftrightarrow \Gamma_\mu^a \, , \hspace{0.2cm} a = \{ \alpha, \beta\} \end{equation} meaning that the indexes of the gauge group for gravity are a pair of spacetime indexes. It would be logical for me at first sight to define the path integral of (quantum) gravity in analogy with gauge theories as \begin{equation}\label{eq1} \mathcal{Z}[J] = \int [\text{d} \Gamma_\mu] e^{i \mathcal{S}[\Gamma, \partial \Gamma] + i \int J \cdot \Gamma} \end{equation} However what is usually done is the following (again schematically) \begin{equation} \mathcal{Z}[J] = \int [\text{d} g_{\mu \nu}] e^{i \mathcal{S}[g, \partial g] + i \int J \cdot g} \end{equation}

The two definitions differ by a Jacobian $[\text d \Gamma] = [\text d g] \frac{\delta \Gamma}{\delta g}$ which is non trivial since $\Gamma$ is non-linear in the metric. Is there a reason to choose the usual definition instead of mine? I get that GR gives a well defined Cauchy problem for $g_{\mu \nu}$ but in principle I don't see what is wrong with my definition.



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