Nontrivial local observables and impermeable and permeable boundary conditions for 1D kfgm particles

Abstract

Real solutions of the one-dimensional Klein-Fock-Gordon (KFG) equation make the usual two-vector current density vanish identically; as a result, the associated continuity equation becomes trivial, no nontrivial globally conserved quantity is obtained, and the usual current cannot distinguish between impermeable and permeable boundary conditions (BCs). This becomes an issue especially when the equation is used to describe a strictly neutral particle, namely a one-dimensional Klein-Fock-Gordon-Majorana (KFGM) particle. In this thesis, we address this limitation by introducing the simplest nontrivial local observables for the 1D KFGM system—an energy density and an energy current density. Working within the Feshbach-Villars (FV) framework on a finite interval, we derive these observables and establish their reality and conservation properties under the Majorana constraint. The acceptable BCs are obtained from the pseudo self-adjointness of the FV Hamiltonian together with two Majorana conditions. We show that the proposed energy density and energy current density, which form components ofan unusual energy-momentum tensor, satisfy a continuity equation and yield a conserved quantity for all acceptable BCs, whether the corresponding walls are impermeable or permeable. Additionally, the energy current density alone provides a complete local characterization of these BCs. By contrast, the standard energy current constructed from the usual energy-momentum tensor does not characterize all BCs correctly, and its associated energy density does not in general lead to a conserved global quantity. We further determine the conditions under which both sets of observables become equally satisfactory. This occurs only for four confining (impermeable) BCs and for a one-parameter family of nonconfining (permeable) BCs. These results highlight the fundamental role of BCs in first-quantized relativistic systems on finite intervals and provide a concrete way to select physically consistent boundary models for strictly neutral scalar particles.