Abstract
It is argued that quantum-mechanical \ensuremath{\beta} functions can be derived consistently by using renormalization length scales implied by the quasiclassical minimization of Hamiltonian forms. Our renormalization-group results for the Dirac-Coulomb system can then be related analytically to ones obtained in ladder quantum electrodynamics in four space-time dimensions. Involving spinless relativistic two-body Hamiltonians, one can present exact energy- and \ensuremath{\beta}-function solutions for the Lorentz scalar linear potential. Inverse-square, other linear, and mixed linear plus Coulomb potentials can also be treated. The exact energy of this last potential has been discussed in conjunction with an alternative \ensuremath{\beta} function.
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