Relativistic Dynamics in the Quantum Limit of Maximal Proper Acceleration

Quantum extension of the Newton’s second law of motion in the relativistic form (referred to as Einstein’s second law) is proposed in order to adjust it to the realm of elementary particles subjected to extreme accelerations. An underlying idea is the quantization of proper time, set in a close connection with the concept of maximal proper acceleration. The relativistic concept of proper time is identified with the quantum notion of evolution parameter. The postulated quantum of proper time is thought to depend on the particle mass: consequently relates to the Compton wavelength as and to the Caianiello’s maximal proper acceleration as Quantization of proper time makes the relativistic increase of particle mass/energy discrete, which impacts on the general shape of Newton’s second law, now including both velocity and acceleration limits. Introducing the acceleration-dependent term results in a gradual neutralization of the mass increase as determined by the mass-velocity relation, together with the increasing proper acceleration. The new formula satisfies the correspondence principle with respect to the classical (Newtonian and relativistic) cases, and to the relevant formula connecting Planck units of force, mass and acceleration. The obtained results are juxtaposed with the quantization of spacetime proposed by the Causal Sets approach to quantum gravity.