On possible quantization of the fundamental tensor in the relativistic regime

Abstract

AbstractInspired by the quantum gravity prescriptions, for instance, string theory, doubly‐special relativity, and quantum loop gravity, we apply the generalized relativistic noncommutative Heisenberg algebra on the eight‐dimensional Finsler manifold, the natural generalization of the four‐dimensional Riemann manifold. The Finsler manifold is conjectured to relax the quadratic restriction on the length measure, especially in the relativistic regime. The additional curvature coupled with the Finsler manifold is assumed to mimic the quantization on the Riemann manifold. The second ingredient we suggest is the existence of a minimum measurable length, whose value is determined from the relativistic generalized uncertainty principle, and then applying in to the coordinates of the Finsler manifold. The quantum‐induced corrections to the fundamental tensor are determined under extreme conditions of a relativistic energy. Whether this proposal is adequate for a full quantization, the various limitations of the proposed theory are elaborated and discussed, especially the limitations of the curved spacetime model assumed for the line element measure. We have conservatively formulated our conclusion that the quantization of the fundamental tensor might be conditioned to an adequate quantization of the line element measure, at least the one on the eight‐dimensional Finsler manifold.