GEOMETRICAL OPTICS AS AN ABELIAN U(1) GAUGE THEORY IN A VACUUM SPACE-TIME

Abstract

As a consequence that geometrical optics (the eikonal equation) can be derived from Maxwell’s equations and Maxwell’s theory is nothing but an Abelian U(1) gauge theory, we propose that geometrical optics could also be treated as an Abelian U(1) gauge theory. We formulate geometrical optics as an Abelian U(1) gauge theory in a (3+1)-dimensional vacuum space-time as an approximation of the weak field. We show the explicit form of the phase, the gauge potential, and the field strength tensor related to the refractive index. We calculate numerically the refractive index and the magnetic field using the suitable parameters that we choose to mimic the real condition of nature. We obtain (without unit) the values of the refractive index  =1.0001 to represent a vacuum space-time and the amplitude ρ = 0.55853 related to magnitude of the magnetic field  = 0.10452 to represent the weak field.  The view of geometrical optics as gauge theory could be generalized or related to topological field theory where geometrical optics could have a topological structure in the case of the weak field.