An invisible extended Unruh-DeWitt detector

Abstract: We develop a localized particle detector model formulated as a massive quantum field on Minkowski spacetime with the spatial origin excised. To render the problem well-posed at the puncture, we impose boundary conditions at the excised point, which we take to be of Robin type. This setup yields a discrete sector, given by bound-state solutions of the radial equation with real, positive frequencies, which characterizes the detector. We construct the full two-point function and show its decomposition into: (i) the discrete radial bound-state sector, (ii) the boundary condition modified continuous sector, and (iii) the unmodified Dirichlet sector. We then compute the detector field's stress-energy tensor and prove its covariant conservation. For the specific localized modes in this setup, the discrete-sector contribution cancels in the complete stress-energy tensor, leaving only boundary-condition induced terms. Notably, the discrete modes crucial to localized field-based detectors emerge naturally from the boundary conditions, without ad hoc confining potentials, providing a fully relativistic framework that extends the traditional Unruh-DeWitt paradigm. This mechanism is not restricted to Minkowski spacetime: the same construction can be applied to massive fields on backgrounds with naked singularities, such as conical and global monopole spacetimes, offering a unified route to detector localization in a broad class of geometries.