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 covariant 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.