Toward construction of a consistent field theory with Poincare covariance in terms of step-function-type basis functions showing confinement/deconfinement, mass-gap and Regge trajectory for non-pure/pure non-Abelian gauge fields

This article is a review by the authors concerning the construction of a Poincaré covariant (owing to spacetime continuum) field-theoretic formalism in terms of step-function-type basis functions without ultraviolet divergences. This formalism analytically derives confinement/deconfinement, mass-gap and Regge trajectory for non-Abelian gauge fields, and gives solutions for self-interacting scalar fields. Fields propagate in spacetime continuum and fields with finite degrees of freedom toward continuum limit have no ultraviolet divergence. Basis functions defined in a parameter spacetime are mapped to real spacetime. The authors derive a new solution comprised of classical fields as a vacuum and quantum fluctuations, leading to the linear potential between the particle and antiparticle from the Wilson loop. The Polyakov line gives finite binding energies and reveals the deconfining property at high temperatures. The quantum action yields positive mass from the classical fields and quantum fluctuations produces the Coulomb potential. Pure Yang-Mills fields show the same mass-gap owing to the particle-antiparticle pair creation. The Dirac equation under linear potential is analytically solved in this formalism, reproducing the principal properties of Regge trajectories at a quantum level. Further outlook mentions a possibility of the difference between conventional continuum and present wave functions responsible for the cosmological constant.