General Relativity within the confines of a Hilbert Space

[Perturbative]

Introduction
If Quantum Mechanics is more fundamental than General Relativity as most Physicists believe, and Quantum Mechanics is described using Hilbert Spaces wouldn't finding a compatible version of General Relativity that operates within the confines of a Hilbert Space be of utmost importance to finding a unifying theory of Quantum Gravity or possibly a Grand Unified Theory/Theory of Everything?

Possible new Predictions?
Surely formulating General Relativity within the confines of a Hilbert Space would make adjustments to GR and new predictions as a result of those adjustments. Even if it does not lead to a consistent theory of Quantum Gravity, or a 'Theory of Everything' it would provide some useful information at the boundaries of QM and GR wouldn't it?

Is this an active area of research?
Has something like this already been tried? Is it a dead-end? Is it something that Mathematicians/Theoretical Physicists are actively pursuing?

Further Reading
If there are any links to further reading, introductory texts etc., especially on the Pure Mathematics behind Hilbert Spaces and into the techniques of Mathematical Physics being used to probe the boundaries of General Relativity, I would love to read about them.

 

[PeterDonis]

If Quantum Mechanics is more fundamental than General Relativity as most Physicists believe, and Quantum Mechanics is described using Hilbert Spaces wouldn't finding a compatible version of General Relativity that operates within the confines of a Hilbert Space be of utmost importance to finding a unifying theory of Quantum Gravity or possibly a Grand Unified Theory/Theory of Everything?

The current view is that classical GR is an effective field theory, like all other classical field theories such as Maxwell electrodynamics. Classical effective field theories don't need to be formulated in a Hilbert space; we do that with the underlying quantum field theories, for example quantum electrodynamics as the underlying quantum field theory of which Maxwell electrodynamics is the classical limit.

Has something like this already been tried?

Yes. From the late 1950's to the early 1970's, many researchers pursued a quantum field theory of a massless spin-2 field, basically the obvious QFT that would have classical GR as its classical limit the way QED has Maxwell electrodynamics as its classical limit. This theory can certainly be constructed and its properties are well understood. The problem from a fundamental physics point of view is that the theory is not renormalizable, which means that it is expected to only be an effective field theory (like classical GR but at the "next level down", so to speak), not a fundamental theory. Current research in quantum gravity is focused on finding the underlying fundamental theory for which the massless spin-2 field, and classical GR, are effective field theories in some low energy limit: the two current front runner candidates I'm aware of are string theory and loop quantum gravity.