Regge Calculus of the Fundamental Particles

Feynman, as well as numerous other famous physicists, has postulated that space-time could be discrete rather than a continuum at the quantum level. However, this hypothesis has been only applied at the Planck level where Wheeler, Hawking and others argue for a space-time foam. We, in the following, will argue and demonstrate that such a discrete space-time is valid at the fundamental particle level as well. A discrete space-time, governed by Regge calculus, is used to model the fundamental particles and their interaction. In this calculus, curvature is distributed over the 2-simplex defect with energy-momentum concentrated on the legs of the 2-simplex. We show that a baryon, composed of 3 quarks, is equivalent to the triangular 2-simplex, where the quarks compose the legs. Mesons and leptons correspond to collapsed line and point 2-simplices, respectively, where curvature and Action remain well-behaved through the collapse. There are 6 invariants of the 2-simplex that we argue correspond to the 6 quark flavors. The 3 on-diagonal invariants have inner products of 2/3 while the 3 off-diagonal invariants have -1/3 inner products. The closure condition on these invariants yields confinement of the quarks. A fundamental tenet of the quantum theory is the Uncertainty Principle. We shall show that a discrete analogue of the Uncertainty Principle is a direct consequence of our approach