Duality-invariant Einstein-Planck relation and the speed of light at very short wavelengths

We propose a generalized Einstein-Planck relation for photons which is invariant under the change l/alP to alP/l, l being the photon wavelength, lP Planck’s length, and a a numerical constant. This yields a wavelength-dependent speed of light v(l)= c/(1+a^2(lP/l)^2), with c the usual speed of light in vacuo, indicating that the speed of light should decrease for sufficiently short wavelengths. We discuss the conceptual differences with the previous proposals related to a possible decrease of the speed of light for very short wavelengths based on quantum fluctuations of the space-time, as well as its consequences on Heisenberg’s uncertainty principle up to second order in lP.