BORN'S RECIPROCAL GENERAL RELATIVITY THEORY AND COMPLEX NON-ABELIAN GRAVITY AS GAUGE THEORY OF THE QUAPLECTIC GROUP: A NOVEL PATH TO QUANTUM GRAVITY
Abstract
Born's reciprocal relativity in flat space–times is based on the principle of a maximal speed limit (speed of light) and a maximal proper force (which is also compatible with a maximal and minimal length duality) and where coordinates and momenta are unified on a single footing. We extend Born's theory to the case of curved space–times and construct a reciprocal general relativity theory (in curved space–times) as a local gauge theory of the quaplectic group and given by the semidirect product 
, where the non-Abelian Weyl–Heisenberg group is H(1, 3). The gauge theory has the same structure as that of complex non-Abelian gravity. Actions are presented and it is argued why such actions based on Born's reciprocal relativity principle, involving a maximal speed limit and a maximal proper force, is a very promising avenue to quantize gravity that does not rely in breaking the Lorentz symmetry at the Planck scale, in contrast to other approaches based on deformations of the Poincaré algebra, quantum groups. It is discussed how one could embed the quaplectic gauge theory into one based on the U(1, 4), U(2, 3) groups where the observed cosmological constant emerges in a natural way. We conclude with a brief discussion of complex coordinates and Finsler spaces with symmetric and nonsymmetric metrics studied by Eisenhart as relevant closed-string target space backgrounds where Born's principle may be operating.
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