Spacetime Path Formalism: Localized States

This note is an addendum to [9]. In that paper, I present a formalism for relativistic quantum mechanics in which the spacetime paths of particles are considered fundamental. These paths are parameterized by an invariant “fifth parameter” (in addition to the four coordinates of spacetime), which places the formalism within the class of “parameterized” or “worldline” approaches to quantum mechanics (see the references cited in [9]). One of my goals in [9] was to show how the standard results of the traditional formulation of relativistic quantum mechanics and quantum field theory can be reproduced in the new formalism. This included the emergence of on-shell particle states from the naturally off-shell foundations of the formalism, along with the usual inner product between wave functions of such states. Now, it is well known that there are issues with the ability to localize the position of particles in the usual formulation of relativistic quantum mechanics (see [5, 8]). This clearly should be addressed in any approach based on particle paths (see also, for example, [6, 7]). The intent of the present note is to show how it may be addressed in the context of formalism proposed in [9].