The physical interpretation of point interactions in one-dimensional relativistic quantum mechanics

Abstract

We investigate point interactions in one-dimensional relativistic quantum mechanics using a distributional approach based on Schwartz's theory of distributions. From the properties of the most general covariant distribution describing relativistic point interactions we obtain the physical parameters associated with the point potentials that behave as a scalar, a pseudo-scalar and a vector under Lorentz transformations. Then, we establish a one-to-one relationship between these physical parameters and the well-known set of four parameters giving the boundary conditions at the singular point(s), which define a self-adjoint Hamiltonian. By considering the non-relativistic limit, we obtain the most general point interaction in the Schr"odinger equation in terms of these four physical point potentials. Finally, we study the symmetries of the relativistic point interactions under space inversion, time reversal and charge conjugation, and investigate how requirements of invariance under these symmetry transformations can be used to restrict the set of physical parameters.