Relativistic Heisenberg principle for vortices of light from Planck to Hubble scales

We discuss, in the relativistic limits, Heisenberg's uncertainty principle for the electromagnetic orbital angular momentum (OAM) of monochromatic fields. The Landau and Peierls relativistic approach applied to the Heisenberg indetermination between the azimuthal angle and the OAM state , when a limit speed is present, exposes the conceptual limits of a possible formulation of the uncertainty principle for OAM states as it leads to the Ehrenfest's paradox of the rotation of a rigid disk in special relativity (SR). To face this problem, in a way similar to Einstein's discussion of a rotating rigid disk, one has to adopt a general relativistic approach and include the limits from smallest and largest scales present in nature, such as the Planck scale and the Hubble horizon.