Quantizing Null Hypersurfaces

In this talk, we first present a detailed analysis of the classical geometry of generic null hypersurfaces. We then reformulate the Einstein equations as conservation laws for the intrinsic geometric data on these hypersurfaces. Following this, we derive the symplectic structure and the corresponding Poisson bracket. Upon quantizing this phase space, we propose that the projected Einstein tensor obeys the operator product expansion of the stress tensor in a conformal field theory along null time. This hypothesis is supported by explicit computations in simplified scenarios, such as the absence of radiation and within the framework of perturbative gravity.

Notably, we discover a non-vanishing central charge, which counts the

local geometric degrees of freedom and diverges in the classical limit. We suggest that this central charge is a fundamental principle underlying the emergence of time in quantum gravity. If time permits, we will conclude by introducing a mesoscopic model of quantum gravity on null hypersurfaces, based on the concept of the "embadon," an operator that creates localized bits of area on cuts.