  This article is a review by the authors concerning the construction of a
Poincar${\rm \acute{e}}$ 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.
