Wigner’s new physics frontier: Physics of two-by-two matrices, including the Lorentz group and optical instruments

According to Eugene Wigner, quantum mechanics is a physics of Fourier transformations, and special relativity is a physics of Lorentz transformations. Since two-by-two matrices with unit determinant form the group SL(2, c) which acts as the universal covering group of the Lorentz group, the two-by-two matrices constitute the natural language for special relativity. The central language for optical instruments is the two-by-two matrix called the beam transfer matrix, or the so-called ABCD matrix. It is shown that the ABCD matrices also form the SL(2, C) group. Thus, it is possible to perform experiments in special relativity using optical instruments. Likewise, the optical instruments can be explained in terms of the symmetry of relativistic particles. Based on this review article, the last author(YSK) presented papers at a number of conferences, including the 8th International Wigner Symposium (New York, U.S.A., 2003), the 8th International Conference on Squeezed States and Uncertainty Relations (Puebla, Mexico, 2003), the Symmetry Festival (Budapest, Hungary, 2003), and the International Conference on Physics and Control (Saint Petersburg, Russia, 2003). electronic address:[email protected] electronic address: [email protected] electronic address: [email protected]