A Numbers-Based Approach to a Free Particle’s Proper Spacetime

This paper contains a proposal for a free, nonzero-rest-mass particle’s proper spacetime, determined exclusively by the particle’s rest mass m and numbers. The approach defines proper time as de Broglie time, which is isomorphic to a sequence of natural numbers 1 , 2 , … , n that count de Broglie time units (h/c2)(m0-1) (see Ferber in Found Phys Lett 9:575, 1996). The approach is based on defining the spatial coordinate as proper following the constructive definition of positive and negative integers as all possible differences of ordered pairs of natural numbers multiplied by the Compton unit (h/c)(m0-1). The spatial and temporal coordinates that form the particle’s proper spacetime are constructed as Euclidean projections of the de Broglie time. The corresponding expression in the form of an energy-momentum relation reveals the existence, aside from the rest energy term mc², of an additional energy term of the same order of magnitude, which is related to large intervals of the m-particle’s proper space. The relation of the numbers-based approach to the foundations of the special theory of relativity and of quantum mechanics is discussed.