The de Broglie Relations: Lorentz Invariance and Photons
Keywords: quantum optics, anisotropic media, special theory of relativity, covariance
A simple covariant derivation of the de Broglie relations breaks down when one considers propagation of light in an anisotropic medium. This breakdown results in two distasteful conclusions. The first is that the de Broglie relations are not completely universal and their covariance is restricted. The second is that wave-particle duality is not universal. In this paper a resolution of these problems is suggested, a resolution that is an extrapolation of ideas of de Broglie and Andrade e Silva. The relations are generalized for propagation in anisotropic media by defining a new phase wave with wave vector 
in the direction of energy transport and distinct from the conventional electromagnetic wave. The introduction of follows the method by which de Broglie associated a wave with an electron. The magnitude of is 
u/c2 where is the angular frequency and u the velocity of energy transport. The physical meaning of and the covariance of physical laws in the presence of matter are discussed. An experiment is described that would test the existence of the phase wave associated with the new wave vector . This experiment is essentially a Hanbury Brown-Twiss correlation experiment carried out within an anisotropic crystal.
u/c2 where is the angular frequency and u the velocity of energy transport. The physical meaning of 