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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.

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# I General Relativity within the confines of a Hilbert Space

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