Newton mechanics, Galilean relativity, and special relativity in $\alpha$-deformed binary operation setting

Abstract

We define new velocity and acceleration having dimension of (Length)^{\alpha}/(Time)(Length)α/(Time) and (Length)^{\alpha}/(Time)^2,(Length)α/(Time)2, respectively, based on the fractional addition rule. We discuss the formulation of fractional Newton mechanics, Galilean relativity and special relativity in the same setting. We show the conservation of the fractional energy, characterize the Lorentz transformation and group, and derive the expressions of the energy and momentum. The two body decay is discussed as a concrete illustration.