Quantum Area Fluctuations from Gravitational Phase Space

We study the gravitational phase space associated to a stretched horizon within a finite-sized causal diamond in $(d+2)$-dimensional spacetimes. By imposing the Raychaudhuri equation, we obtain its constrained symplectic form using the covariant phase space formalism and derive the relevant quantum commutators by inverting the symplectic form and quantizing. Finally, we compute the area fluctuations of the causal diamond by taking a Carrollian limit of the stretched horizon in pure Minkowski spacetime, and derive the relationship $⟨(\mathrm{\Delta }A{)}^2⟩\ge \frac{2\pi G}{d}⟨A⟩$, showing that the variance of the area fluctuations is proportional to the area itself.