Lorentz Group in Ray and Polarization Optics

While the Lorentz group serves as the basic language for Einstein's special theory of relativity, it is turning out to be the basic mathematical instrument in optical sciences, particularly in ray optics and polarization optics. The beam transfer matrix, commonly called the $ABCD$ matrix, is shown to be a two-by-two representation of the Lorentz group applicable to the three-dimensional space-time consisting of two space and one time dimensions. The Jones matrix applicable to polarization states turns out to be the two-by-two representations of the Lorentz group applicable to the four-dimensional space-time consisting of three space and one time dimensions. The four-by-four Mueller matrix applicable to the Stokes parameters as well as the Poincar\'e sphere are both shown to be the representations of the Lorentz group.