Born reciprocity and discretized Finsler structure: An approach to quantize GR curvature tensors on three-sphere

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At relativistic energies and finite magnetic fields, the noncommutative relation of distance and momentum, the Heisenberg uncertainty principle, the fundamental theory of quantum mechanics, is conjectured to get modifications. Results from various rigorous approaches to quantum gravity, such as string theory, loop quantum gravity and doubly special relativity support the generalization of the noncommutative relation of the distance and momentum operators and the emergence of a minimal measurable length. With the relativistic four-dimensional generalized uncertainty principle (RGUP) in curved spacetime and Born reciprocity principle, the distance–momentum duality symmetry, we suggest to generalize Riemann to Finsler geometry. The Finsler structure allows the direct implementation of RGUP with its quantum-mechanical nature on a free particle with mass m, so that the Finsler structure [Formula: see text] can be expressed as [Formula: see text], from which the quantized fundamental tensor can be deduced. We present a systematic analytic and numerical evaluation of the additional geometric structures and connections which exclusively emerged from the proposed quantization approach on three-sphere. When limiting the discussion on the Einstein tensor, we find that the emerged curvatures, i.e. additional sources of gravitation, are dominant almost everywhere on the three-sphere. The nature of those curvatures is radically distinct from the ones of the classical Einstein tensor. For instance, the additional curvatures are no longer smooth or continuous.