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

TL;DR

This work addresses the difficulty of obtaining a conserved charge for real solutions of the 1D Klein-Fock-Gordon equation by introducing nontrivial local observables—three energy densities and two energy currents—that live in an unusual energy-momentum tensor. By analyzing the pseudo self-adjoint Feshbach-Villars Hamiltonian and enforcing Majorana conditions, the authors derive a comprehensive BC framework that distinguishes impermeable from permeable boundaries and yields a continuity equation under appropriate conditions. The results show that certain BCs (four impermeable and a one-parameter permeable set) render both the energy-current densities and the standard energy-momentum components equally satisfactory for describing confinement, while in general, the new observables provide a more complete BC characterization for 1D KFG Majorana particles. These insights illuminate the crucial role of boundary conditions in low-dimensional relativistic quantum systems and have potential implications for Casimir-type analyses and related boundary problems.

Abstract

Real solutions of the 1D Klein-Fock-Gordon (KFG) equation automatically cancel out the usual two-vector current density; consequently, the respective continuity equation is trivially satisfied, and a globally conserved quantity cannot be obtained. Additionally, distinguishing between impermeable and permeable boundary conditions (BCs) at a given point is not possible. We address these first-quantized conflicts by using the simplest nontrivial local observables, i.e., an energy density and an energy current density, which allows us to characterize a strictly neutral 1D KFG particle, i.e., a 1D KFG-Majorana (KFGM) particle, when it is confined to an interval and when it is restricted, e.g., in an interval with transparent walls. All the BCs for this system are extracted from the pseudo self-adjointness of the Feshbach-Villars (FV) Hamiltonian plus two Majorana conditions. We show that these energy densities are components of an unusual energy-momentum tensor and can satisfy a continuity equation, leading to a conserved quantity for all available BCs. Moreover, the energy current density can characterize all the BCs as either impermeable or permeable. In contrast, the commonly used energy current density -- a component of the usual energy-momentum tensor -- cannot characterize all the BCs. Additionally, this quantity and its respective energy density -- another component of the usual energy-momentum tensor -- may not lead to a conserved quantity. We also obtain the BCs for which the abovementioned densities and those commonly used are equally satisfactory. In fact, this occurs only for four impermeable BCs and a one-parameter set of permeable BCs. Our results highlight the important role played by the BCs when they are imposed on a system in which particles occupy a finite region.