A new length scale, and modified Einstein-Cartan-Dirac equations for a point mass

摘要

We have recently proposed a new action principle for combining Einsteinequations and the Dirac equation for a point mass. We used a length scale$L_{CS}$, dubbed the Compton-Schwarzschild length, to which the Comptonwavelength and Schwarzschild radius are small mass and large massapproximations, respectively. Here we write down the field equations whichfollow from this action. We argue that the large mass limit yields Einsteinequations, provided we assume wave function collapse and localisation for largemasses. The small mass limit yields the Dirac equation. We explain why theKerr-Newman black hole has the same gyromagnetic ratio as the Dirac electron,both being twice the classical value. The small mass limit also providescompelling reasons for introducing torsion, which is sourced by the spindensity of the Dirac field. There is thus a symmetry between torsion andgravity: torsion couples to quantum objects through Planck's constant $\hbar$(but not $G$) and is important in the microscopic limit. Whereas gravitycouples to classical matter, as usual, through Newton's gravitational constant$G$ (but not $\hbar$), and is important in the macroscopic limit. We constructthe Einstein-Cartan-Dirac equations which include the length $L_{CS}$. We finda potentially significant change in the coupling constant of the torsion drivencubic non-linear self-interaction term in the Dirac-Hehl-Datta equation. Wespeculate on the possibility that gravity is not a fundamental interaction, butemerges as a consequence of wave function collapse, and that the gravitationalconstant maybe expressible in terms of Planck's constant and the parameters ofdynamical collapse models.