By Bianca Dittrich, Christophe Goeller, Etera R. Livine, and Aldo Riello
Despite many years of research, quantum gravity remains a challenge. One of the reasons is that the many tools developed for perturbative quantum field theory are, in general, not applicable to quantum gravity. On the other hand, non-perturbative approaches have a difficult time in finding and extracting computable observables. The foremost problem here is a lack of diffeomorphism-invariant observables.
The situation can be improved very much by considering space-time regions with boundaries. This is also physically motivated, since one would like to be able to describe the physics of a given bounded region in a quasi-local way, that is without requiring a detailed description of the rest of the space-time outside. The key point is that the boundary can be used as an anchor, allowing to define observables in relation to this boundary. Then we can consider different boundary conditions, which translates at the quantum level into a rich zoo of boundary wave-functions. These boundary states can correspond to semi-classical boundary geometries or superpositions of those. The states can also describe asymptotic flat boundaries, thus allowing us to compare with perturbative approaches. In this context, holography in quantum gravity aims to determine how much of the bulk geometry can be reconstructed from the data encoded in the boundary state.
The boundary wave function Ψ are described by dual theories defined on the boundary of the solid torus. These 2D boundary theories, obtained by integrating over all the bulk degrees of freedom of the geometry, encode the full 3D quantum gravity partition function.
CQG+ 
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