Discrete scalar field and general relativity

Abstract

What is the nature - continuous or discrete - of matter and of its fundamental interactions? The physical meaning, the properties and the consequences of a discrete scalar field are discussed; limits for the validity of a mathematical description of fundamental physics in terms of continuum fields are a natural outcome of discrete fields with discrete interactions. Two demarcating points (a near and a far) define a domain where no difference between the discrete and the standard continuum field formalisms can be experimentally detected. Discrepancies, however, can be observed as a continuous-interaction is always stronger below the near point and weaker above the far point than a discrete one. The connections between the discrete scalar field and gravity from general relativity are discussed. Whereas vacuum solutions of general relativity can be retrieved from discrete scalar field solutions, this cannot be extended to solutions in presence of massive sources as they require a true tensor metric field. Contact is made, on passing, with the problem of dark matter and the rotation curve of galaxies, with inflation and the accelerated expansion, the apparent anomaly in the Pioneer spacecraft acceleration, the quantum Hall effect, high-$T_{c}$ superconductivity, quark confinement, and with Tsallis generalized one-parameter statistics as some possible manifestation of discrete interaction and of an essentially discrete world.