Probabilistic Causality from Graviton Fluctuations

Abstract

We compute the commutator of a scalar field minimally coupled to gravity at leading order in $G_N$. The commutator is operator-valued, with terms involving derivatives of Dirac deltas supported on the Minkowski light cone. When evaluated on classical/coherent graviton states, these terms ``bend" the support of the commutator in precisely the way required to recover standard causality on a classical curved spacetime. However, these terms are also associated with a variance and are thus a source of uncertainty in the causal relations between events. We quantify this effect for a thermal state of gravitons at temperature $T$. In the large-time limit, the uncertainty of the lightcone position is given by $$\text{Var}(\delta x)=\frac{4G_{N}Tt}{3},$$ where $\delta x$ is the displacement from the classical lightcone at $|\vec{x}|=t$. We obtain this result by subtracting a universal vacuum contribution that is logarithmically UV divergent and subleading at late times.

10 pages + appendices; v3: discussions added. Discussion on black holes expanded and corrected