Emergence of Spacetime from Fluctuations

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Abstract
We use a result of Hawking and Gilkey to define a euclidean path integral of gravity and matter fields in the eigenbasis of the wave operators. On one hand, working in the eigenbasis of the wave operators means working exclusively with geometric invariants and this avoids the need to mod out the diffeomorphism group and makes it possible to carry out the path integral explicitly. On the other hand, working in the eigenbasis of the wave operators does not enforce the existence of a coordinate basis. As a consequence, this path integral also describes physical setups that are pre-geometric in the sense that they do not admit a mathematical representation in terms of fields on a spacetime. We focus on the regime in which a representation in terms of a spacetime and matter fields emerges. We find that the geometric properties of the emergent spacetime, such as its {volume and} number of dimensions, {depend on the energy scale considered and} on the balance of bosonic and fermionic species.