Reference [16] on which the present work is based develops the theory with a definite set of boundary conditions for the gravitational field at spatial infinity. Different boundary conditions, also compatible with the BMS symmetry but excluding solutions with “magnetic mass,” were considered earlier in [18].

This is a consequence of the duality between the states $|\uparrow ⟩$ and $|\downarrow ⟩$, and of the fact that (according to the rules of [26]), one must choose the wave functions $\mathrm{\Psi }[g_{ij}]$ multiplying $|\uparrow ⟩$ to have finite (noninfinite) overlap with the wave functions $\mathrm{\Xi }[g_{ij}]$ multiplying $|\downarrow ⟩$, which have themselves infinite norm since they fulfill the constraint equations ([26], Sec. 14.5.3), which yields infinity when integrating over the gauge orbits. Since the BRST condition implies no restriction on $\mathrm{\Psi }[g_{ij}]$, these functions can have (and must be chosen in order to have) finite integral along the gauge orbits so as to enforce duality. Of course, just as in standard quantum mechanics, the formal eigenstates $|g_{ij}(x)⟩$ considered in the text are not normalizable, but this has nothing to do with the constraint equations and can be handled by considering appropriate superpositions.

For this argument to work, it is necessary to have $C^k|\psi ⟩=0$ if $C^{\perp }|\psi ⟩=0$. Note also that as defined by (7), the BMS generators do not act on the variables of the nonminimal sector. One might consider a nonzero action by modifying (7) in a way that preserves the BRST-invariance condition, e.g., by adding the trivial term $\{\int d^{3}x({\mathcal{P}}^{\perp }{\pi }_{\perp }+{\mathcal{P}}^i{\pi }_i),\mathrm{\Omega }\}$. This does not change the key property that the subspace defined by (10) is BMS invariant.

The projected kernel $A_0^P[g_{ij}^{(2)},g_{ij}^{(1)}]$ is formally defined to be equal to ${\sum }_{\alpha ,\beta }{\psi }_{\alpha }[g_{ij}^{(2)}]({\psi }_{\alpha }|A_0|{\psi }_{\beta }){\psi }_{\beta }^{*}[g_{ij}^{(2)}]$ where the sum runs over a complete orthonormal set of solutions of the Wheeler-DeWitt equation in the physical (regularized) scalar product $(\psi |\chi )$ adapted to solutions of the Wheeler-DeWitt equation. It has been shown in [26] to be equal to (12). It follows that one can fold amplitudes either by summing over all intermediate states or over intermediate physical states only ([26], Sec. 16.5.3). The projected kernel $A_0^P[g_{ij}^{(2)},g_{ij}^{(1)}]$ fulfills the Wheeler-DeWitt equation in the argument $g_{ij}^{(2)}$ (and its complex conjugate, which is here the same, in the argument $g_{ij}^{(1)}$). Therefore, even for the identity $A_0=I$, the projected kernel is interesting and provides through (12) with $A=I$ a constructive procedure for getting solutions of the Wheeler-DeWitt equation.