We propose a generalized Einstein-Planck relation for photons which is invariant under the change $\lambda /a{l}_P$ to $a{l}_P/\lambda$, $\lambda$ being the photon wavelength, $l_P$ Planck’s length, and $a$ a numerical constant. This yields a wavelength-dependent speed of light $v(\lambda )=c/(1+a^2(l_P/\lambda {)}^2)$, 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 $l_P$.

• Received 11 July 2011

DOI:https://doi.org/10.1103/PhysRevD.84.107303