Closed-form expressions for smeared bi-distributions of a massless scalar field: non-perturbative and asymptotic results in relativistic quantum information

Using spacetime Gaussian test functions, we find closed-form expressions for the smeared Wightman function, Feynman propagator, retarded and advanced Green's functions, causal propagator and symmetric propagator of a massless scalar field in the vacuum of Minkowski spacetime. We apply our results to localized quantum systems which interact with a quantum field in Gaussian spacetime regions and study different relativistic quantum information protocols. In the protocol of entanglement harvesting, we find a closed-form expression for the entanglement that can be acquired by probes which interact in Gaussian spacetime regions and obtain asymptotic results for the protocol. We also revisit the case of two gapless detectors and show that the detectors can become entangled if there is two-way signalling between their interaction regions, providing closed-form expressions for the detectors' final state.