Gauge-Invariant Energy Functional in Relativistic Schroedinger Theory

Abstract

The non-invariant energy functional of the preceding paper is improved in order to obtain its gauge-invariant form by strictly taking into account the non-Abelian character of Relativistic Schrödinger Theory (RST). As an application of the results, the dichotomy of positronium with respect to singlet and triplet states is discussed (ortho-and para-positronium). The degeneracy of the ortho-and para-states occurs in RST if (i) the magnetic interactions are neglected (as in the conventional theory) and (ii) the anisotropy of the electric interaction potential is disregarded.In view of such a very crude approximation procedure, the non-relativistic positronium spectrum in RST agrees amazingly well with the conventional predictions.Indeed using here a very simple trial function with only two variational parameters yields deviations from the conventional predictions of some 10% (or less) up to large principal quantum numbers (n ≃ 100). This hints at the possibility that the exact RST calculations could yield predictions for the bound systems which come very close to (or even coincide with) those of the conventional quantum theory. Such a numerical correctness of the RST predictions supports a fluid-dynamic picture of quantum matter where any single particle appears as the carrier of a wave-like 1 structure (such a structure, however, is generally believed to be associated exclusively with statistical ensembles but not with single particles). Here, more detailed predictions of the positronium spectrum would be desirable but this will necessitate to first elaborate more exact solutions of the RST eigenvalue problem, especially with regard to the anisotropic and magnetic effects.