Separating regions in JT gravity at finite G_N

Determine how to define and separate the bulk spacetime into right and left regions in Jackiw–Teitelboim gravity with matter at finite G_N, where the Hilbert space does not factorize and horizons are not sharply defined. Provide an operational, algebraic characterization of R and L subregions compatible with the type II_∞ structure of the algebras A_R and A_L.

Background

In JT gravity with matter, even though the bulk spacetime is locally AdS_2, at finite G_N the theory is fully quantum: horizons are not sharp and the Hilbert space does not factorize into left/right. The bulk operator algebra nevertheless decomposes as B(H)=A_R∨A_L with A_R′=A_L and both A_R and A_L are type II_∞.

An explicit, general separation into R and L bulk regions is not known in this quantum regime, and an algebraic definition would sharpen the notion of subregions when geometric horizons are fluctuating.