Refraction as an Example of a Photonic Bohm Aharonov Effect?

   In Newtonian mechanics a force changes momentum, for example it may change its magnitude only or its direction only or both. In the case of the Bohm Aharonov effect for a particle with rest mass there exists a potential which does not change momentum, but introduces a phase shift. Thus exp(ipx) for say an electron becomes exp(ip(x-xo)). Given that the form exp(-iEt+px) holds for both particles with rest mass and photons one might expect to find a Bohm Aharonov effect for photons. In (1), it is stated that a photonic Bohm Aharonov effect may be “induced by a dynamic modulation of a material’s permittivity”.

     In this  note we argue that the refraction of light seems to have the same form as a Bohm Aharonov effect in the sense that if one begins and ends in the same medium, the particle has the same energy E, momentum p, speed of light c and makes the same angle with the normal. In a Newtonian sense there has been no change to momentum, but the ray is shifted in the x direction from one that does not refract. It seems that this may be described by exp(i py*y) exp(i px(x-xo)) i.e. a phase shifted form suggesting that for the x direction one has a Bohm Aharonov effect.

   We try to link this phase shift in the x direction to more general ideas. We argue as in (2) that a lack of information related to speed (i.e. the index of refraction) in the x direction leads to a conservation law related to speed in the x direction. This speed, however, is a hypothetical one such that it acts as a hypotenuse with c/n(i) being a projection. This differs from the actual x component of velocity which is a projection of c/n(i). Thus this actual x speed is not conserved meaning that light moving in a new medium will struck the outer bound at a different x than it would have it had not moved in a new medium. Thus the beam coming out of the new medium has the same E and p vector, but a shifted x i.e. there is a phase shift of the Bohm Aharonov type we argue even though we do not formally deal with a potential.