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On the Compton clock and the undulatory nature of particle mass in graphene systems

In undulatory mechanics the rest mass of a particle is associated to a rest periodicity known as Compton periodicity. In carbon nanotubes the Compton periodicity is determined geometrically, through dimensional reduction, by the circumference of the curled-up dimension, or by similar spatial constraints to the charge carrier wave function in other condensed matter systems. In this way the Compton periodicity is effectively reduced by several order of magnitudes with respect to that of the electron, allowing for the possibility to experimentally test foundational aspects of quantum mechanics. We present a novel powerful formalism to derive the electronic properties of carbon nanotubes, in agreement with the results known in the literature, from simple geometric and relativistic considerations about the Compton periodicity as well as a dictionary of analogies between particle and graphene physics.