  Clear, persuasive arguments are brought forward to motivate the need for
highly precise measurements of the electron/muon orbital g, i.e. gL. First, we
briefly review results obtained using an extended Dirac equation, which
conclusively showed that, as a consequence of quantum relativistic corrections
arising from the time-dependence of the rest-energy, the electron gyromagnetic
factors are corrected. It is next demonstrated, using the data of Kusch & Foley
on the measurement of deltaS minus 2 deltaL together with the modern precise
measurements of the electron deltaS where deltaS identically equal to gS minus
2, that deltaL may be a small, non-zero quantity, where we have assumed
Russel-Saunders LS coupling and proposed, along with Kusch and Foley, that gS =
2 plus deltaS and gS = 1 plus deltaL. Therefore, there is probable evidence
from experimental data that gS is not exactly equal to 1; the expectation that
quantum effects will significantly modify the classical value of the orbital g
is therefore reasonable. Finally, we show that if, as suggested by the results
obtained from the modified Dirac theory, deltaS and deltaL depend linearly on a
dimensionless parameter DELTA such that the gyromagnetic factors are considered
corrected as follows; gS = 2 plus 2 DELTA and gL = 1 minus DELTA, then the
Kusch-Foley data implies that the correction DELTA approximately equals 1.0
times 10-3 . Modern, high precision measurements of the electron and muon
orbital gL are therefore required, in order to properly determine by
experiments the true value of gL minus 1, perhaps to about one part in a
trillion as was recently done for gS minus 2.
