Back-action from inertial and non-inertial Unruh-DeWitt detectors revisited in covariant perturbation theory
We investigate the back-action from a spatially pointlike particle detector on a quantum scalar field, as characterised by the expectation value of the field's stress-energy tensor, without conditioning on a measurement of the detector. First, assuming the field to be initially in a zero-mean Gaussian Hadamard state in a globally hyperbolic spacetime, we evaluate the field's two-point function in second-order perturbation theory by techniques of covariant curved spacetime quantum field theory, which allow a full control of the time and space localisation of the interaction, and do not rely on field mode decompositions or non-local particle countings. The detector's two-point function splits into a deterministic and a fluctuating part, and we show that this split is maintained in the back-action. We then specialise to a two-level Unruh-DeWitt detector, prepared in an energy eigenstate, for which the back-action is fully fluctuating. We compute the renormalised stress-energy tensor for a massless scalar field in $(3+1)$-dimensional Minkowski spacetime for a general detector trajectory, using the manifestly causal two-point function. We present explicit analytic and numerical results for an inertial detector and a uniformly linearly accelerated detector, switched on in the asymptotic past. The energy flux into and out of the accelerated detector accounts exactly for the energy gained and lost by the detector in its transitions due to the Unruh effect. The same holds for the outward flux associated with de-excitations of the inertial detector, which has a vanishing excitation rate and no inward flux. A novelty with the accelerated detector is two regions of negative energy density when the detector is initially prepared in its ground state, one near the Rindler horizon that bounds the causal future of the trajectory, the other in the far future of the trajectory.