Lorentz Group derivable from Polarization Optics

The Lorentz group is the fundamental language for space-time symmetries of relativistic particles. This group can these days be derived from the symmetries observed in other branches of physics. It is shown that this group can be derived from optical filters. The group O(2,1) is appropriate for attenuation filters, while the O(3) group describes phase-shift filters. The combined operation leads to a two-by-two representation of the six-parameter Lorentz group. It is shown also that the bilinear representation of this group is the natural language for the polarization optics. The Lorentz group serves useful purposes in many branches of physics. In this note, we would like to show that the bilinear representation of the six-parameter Lorentz group [1] is the natural language for polarization of light waves. In studying polarized light propagating along the z direction, the traditional approach is to consider the x and y components of the electric fields. Their amplitude ratio and the phase difference determine the degree of polarization. Thus, we can change the polarization either by adjusting the amplitudes, by changing the relative phase shift, or both. For convenience, we call the optical device which changes amplitudes an “attenuator,” and the device which changes the relative phase a “phase shifter.” Let us write the electric field vector as