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 $T$ by computing the probability that $[\ddot{I}•(t,\vec{x}),\ddot{I}•(0)]\ne 0$. We find that the probability distribution for ${\vec{x}}^2$ is Gaussian, centered on the classical light cone, with a time-growing variance $$\mathrm{Var}({\vec{x}}^2)=\frac{16G_{N}T{t}^3}{3}.$$ This result is obtained after subtracting a universal vacuum contribution, which is logarithmically UV divergent and subleading at late times.

11 pages + appendices; v2: more concise (letter-like) version. References added. Results unchanged