Abstract
This paper is a review by the authors concerning the construction of a Poincaré covariant (owing to space–time 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 space–time continuum and fields with finite degrees of freedom toward continuum limit have no ultraviolet divergence. Basis functions defined in a parameter space–time are mapped to real space–time. The authors derive a new solution comprised of classical fields as 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 produce 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.
Highlights
This paper is a review concerning our published[1,2,3] and reported[4] works about a quantum field-theoretic approach for non-Abelian Yang–Mills gauge fields[5,6,7] using the localized basis functions with finite degrees of freedom in the space–time continuum toward the continuum limit
We have reviewed the authors’ research concerning the construction of a consistent Poincare covariant field-theoretic formalism in terms of step-functiontype basis functions without ultraviolet divergences for nonpure/pure non-Abelian Yang–Mills gauge fields
In the present formalism, fields are expressed in terms of the stepfunction-type basis functions with finite degrees of freedom toward the continuum limit in a parameter space–time continuum mapped to real space–time continuum
Summary
This paper is a review concerning our published[1,2,3] and reported[4] works about a quantum field-theoretic approach for non-Abelian Yang–Mills gauge fields[5,6,7] using the localized basis functions with finite degrees of freedom in the space–time continuum toward the continuum limit. The confined particle and antiparticle of the gauge source require the energy (mass) for the deconfinement.[82] The present formalism given in this paper is Poincare invariant with a cutoff to avoid ultraviolet divergences as mentioned in Sec. 2 and the gauge invariance as described which states that this formalism demonstrates the existence of the non-Abelian Yang–Mills fields. The order of magnitude of this energy difference would be the same as the order of magnitude of the matter (atoms). In Sec. 8, we summarize the conclusions
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