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Analytic eigenenergies of the Dirac equation with finite degrees of freedom under a confining linear potential using basis functions localized in spacetime

Considering the propagation of fields in the spacetime continuum and the well-defined features of fields with finite degrees of freedom, the wave function is expanded in terms of a finite set of basis functions localized in spacetime. This paper presents the analytic eigenenergies derived for a confined fundamental fermion-antifermion pair under a linear potential obtained from the Wilson loop for the non-Abelian Yang-Mills field. The Hamiltonian matrix of the Dirac equation is analytically diagonalized using basis functions localized in spacetime. The squared lowest eigenenergy (as a function of the relativistic quantum number when the rotational energy is large compared to the composite particle masses) is proportional to the string tension and the absolute value of the Dirac's relativistic quantum number related to the total angular momentum, consistent with the expectation.