A simple argument that small hydrogen may exist

Abstract

This paper examines whether a compact electron-proton configuration (small hydrogen) with a characteristic radius of a few femtometers is excluded by basic relativistic kinematics and simple stationarity constraints. Motivated by earlier discussions of formally deep relativistic energy scales in Dirac-based treatments, a phenomenological, virial-inspired energy-balance framework that incorporates relativistic kinetic energy, finite-size regularization of the central field, and order-of-magnitude spin-magnetic and spin-orbit contributions is developed in this paper. Within this framework, self-consistent characteristic scales associated is obtained with a hypothetical compact configuration without invoking Dirac or quantum-electrodynamics (QED) bound-state eigenvalues. The resulting scales-namely, a central energy scale of about 260 keV and a characteristic spin-dependent scale of order of 100 keV-define concrete experimental and observational energy ranges of interest. The present study does not establish the existence, formation probability, lifetime, or dynamical stability of such states. Rather, it shows that relativistic kinematics, finite-size effects, and virial-inspired stationarity constraints do not, by themselves, rule out compact stationary electron-proton configurations within the assumptions of the model. If such states were realized in nature and possessed radiative or interaction channels, those states may have implications for astrophysics, fusion concepts, and dark-matter phenomenology.