Inequivalent solutions for Dirac spin-(1/2) particles under conservation of parity I

Inequivalent invariance constraints upon solutions for Dirac spin-(1/2) particles under conservation of space inversion, time reversal, and charge conjugation have been established, respectively, from standard representation. For space inversion we explicitly show that the wave functions for zero mass neutrinos satisfy only one constraint. For free particles we show that the conventional plane wave solutions as well as Foldy-Wouthuysen representation conditionally comply with the inversion constraints. As a result, only two positive energy solutions and two negative energy solutions can be established. Instead, by law of conservation of parity one shall obtain four linearly independent plane wave solutions which hold not only for positive energies but also for negative energies. We explicitly point out why conventional approach fails to obtain such result. In contrast with free particles one expects intuitively that in a Coulomb field, one can establish twice as many as conventional solutions. Indeed, one shall find easily from inversion constraints that additional linearly independent degenerate bound states have to be established. We present the explicit result.