Probabilistic causality from graviton fluctuations

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 $$ \text{Var}(δx) =\frac{4G_NTt}{3}\, , $$ where $δx$ 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.