D.L. Bulathsinghala, K A I L Wijewardena Gamalath, Implementation of a Quantized Line Element in Klein-Gordon and Dirac Fields, ILCPA Volume 48, International Letters of Chemistry, Physics and Astronomy (Volume 48)
https://www.scipress.com/ILCPA.48.68
Abstract:
    In this paper an ansatz that the anti-commutation rules hold only as integrated average over time intervals and not at every instant giving rise to a time-discrete form of Klein-Gordon equation is examined. This coarse-grained validation of the anti-commutation rules enables us to show that the relativistic energy-momentum relation holds only over discrete time intervals, fitting well with the time-energy uncertainty relation. When this time-discrete scheme is applied to four vector notations in relativity, the line-element can be quantized and thereby how the physical attributes associated with time, space and matter can be quantized is sketched. This potentially enables us to discuss the Zeno’s arrow paradox within the classical limit. As the solutions of the Dirac equation can be used to construct solutions to the Klein-Gordon equation, this temporal quantization rule is applied to the Dirac equation and the solutions associated with the Dirac equation under such conditions are interpreted. Finally, the general relativistic effects are introduced to a line-element associated with a particle in relativistic motion and a time quantized line-element associated with gravity is obtained.
Keywords:
    Chronon, Continuity Equation, Dirac Equation, Energy-Time Uncertainty Relation, Klein-Gordon Equation, Probability Densities, Quantum Gravity, Time Quantization, Tortoise Paradox, Zeno’s Achilles Paradox, Zeno’s Arrow Paradox