Detection time of Dirac particles in one space dimension
We consider particles emanating from a source point inside an interval in one-dimensional space and passing through detectors situated at the endpoints of the interval that register their arrival time. Unambiguous measurements of arrival or detection time are problematic in the orthodox narratives of quantum mechanics, since time is not a self-adjoint operator. By contrast, the arrival time at the boundary of a particle whose motion is being guided by a wave function through the deBroglie-Bohm guiding law is well-defined and unambiguous, and can be computationally feasible provided the presence of detectors can be modeled in an effective way that does not depend on the details of their makeup. We use an absorbing boundary condition for Dirac's equation (ABCD) proposed by Tumulka, which is meant to simulate the interaction of a particle initially inside a domain with the detectors situated on the boundary of the domain. By finding an explicit solution, we prove that the initial-boundary value problem for Dirac's equation satisfied by the wave function is globally well-posed, the solution inherits the regularity of the initial data, and depends continuously on it. We then consider the case of a pair of particles emanating from the source inside the interval, and derive explicit formulas for the distribution of first arrival times at each detector, which we hope can be used to study issues related to non-locality in this setup.