In analogy with QFT, the partition function in canonical quantum gravity is defined as a functional integral over the metric tensor (which is now the quantum field),

$$ \int \mathcal{D} g \mathcal{D}\phi \exp{I_E(g,\phi)}$$ where $\phi$ are all matter fields and $I_E$ is the Euclidean Einstein-Hilbert action.

This path integral can be seen as a way to generate solutions to the Wheeler-deWitt equation, which is a canonically quantised version of the Hamiltonian constraint of GR; $$ \hat{H}\Psi=0$$

I am confused that there never seems to be a discussion on gauge-fixing, which is usually essential if you quantise something. In GR the gauge transformations are basically diffeomorphism, and I know superspace factors out all diffeomorphisms, but I don't see how this is implemented in either the path integral nor the WdW equation.

Passing over to the minisuperspace approximation, my worry becomes clearer. Let's only include homogeneous and isotropic metrics in the path integral. A general $0(4)$ metric in the 3+1 split is of the form $$ds^2 = N^2(\lambda)\text{d} \lambda ^2 + a^2(\lambda)\text{d} \Omega^2_3,$$ where $a$ is the scale factor and $N$ is the lapse. Importantly, the lapse $N$ incorporates all gauge freedom (it ensures that in making the 3+1 split we do not kill the reparametrisation invariance of GR). The path integral in this case is simply given by

$$ \int \mathcal{D} a \mathcal{D}N \exp{I_E(a,N)}$$

In other words, we integrate over the gauge freedom? What does this mean? How does this correspond to the usual gauge fixing paradigm in QFT?


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