Define a fundamental length operator via Gram–Schmidt in JT gravity

Develop a detailed construction and analysis of a fundamental length operator on the boundary Hilbert space in JT gravity obtained by Gram–Schmidt orthonormalization of the states {V|ℓ⟩}, including a viable bulk path-integral representation and consistency with negative-energy effects.

Background

The operator \hat ℓ_fund used in prior work, defined by \int dℓ ℓ V|ℓ⟩⟨ℓ|V†, is natural from a path-integral perspective but problematic at the Hilbert-space level because the states V|ℓ⟩ are not orthonormal. The authors suggest that a better definition would orthonormalize these states via Gram–Schmidt.

They note that existing modified definitions do not account for negative-energy contributions highlighted in this paper and defer a detailed construction and analysis of such a Gram–Schmidt-based operator.