Probabilistic Microcausality in a Thermal Bath of Gravitons

We compute the (operator-valued) commutator of a massless scalar field $\phi$ coupled to gravity. We work in perturbations around Minkowski space, in transverse-traceless gauge at the lowest order in $G_N$. The commutator is composed of different operators, including terms with Dirac delta derivatives supported on the lightcone. These are responsible for ``bending" the Minkowski lightcone when evaluated on a classical/coherent state of gravitons, which allows to recover standard microcausality in the fixed-background limit. On more general gravitational states, metric fluctuations induce an uncertainty in the causal structure. We compute this effect on a thermal state of gravitons at temperature $T$ by evaluating the probability that $[\phi(t, \vec x), \phi(0)] \neq 0$. We find that the probability is Gaussian in $\vec x^{\, 2}$, centered on the lightcone and with time-growing variance $$ {\rm Var}( \vec x^{\, 2}) = \frac{16 \, G_N T t^3}{3}\, .$$ This result is obtained by subtracting a universal vacuum contribution, which is log-divergent in the UV and subleading in the large-time limit. As a source of finite size can effectively serve as a regulator in this case, the lightcone spread in the vacuum appears to be source-dependent.