The Language of Einstein Spoken by Optical Instruments

Abstract

Einstein had to learn the mathematics of Lorentz transformations in order to complete his covariant formulation of Maxwell's equations. The mathematics of Lorentz transformations, called the Lorentz group, continues playing its important role in optical sciences. It is the basic mathematical language for coherent and squeezed states. It is noted that the six-parameter Lorentz group can be represented by two-by-two matrices. Since the beam transfer matrices in ray optics is largely based on two-by-two matrices or ABCD matrices, the Lorentz group is bound to be the basic language for ray optics, including polarization optics, interferometers, lens optics, multilayer optics, and the Poincaré sphere. Because the group of Lorentz transformations and ray optics are based on the same two-by-two matrix formalism, ray optics can perform mathematical operations which correspond to transformations in special relativity. It is shown, in particular, that one-lens optics provides a mathematical basis for unifying the internal space-time symmetries of massive and massless particles in the Lorentz-covariant world.

Lorentz Transformation, Poincaré Vectors and Poincaré Sphere in Various Branches of Physics

“…The same results are valid in PT, for the action of an orthogonal dichroic device of strength p d on a P-sphere Σ Moreover, the same results are valid in all of the fields and problems whose underpinning algebra is that of Lorentz transformation, e.g., multilayer optics [18][19][20][21][22][23][24], geometrical optics [25,26], laser cavities [27], and quantum optics [28]. After identifying the corresponding Poincaré vectors, one applies Equation 7, which leads to the same conclusions, in physical terms corresponding to the investigated fields.…”

“…Similarly, the Lorentzian underlying mathematics structure of various problems was recognized in other fields of physics and these problems were treated in terms of Lorentz group or of various subgroups of Lorentz group: interferometry, geometrical optics, laser cavity optics, quantum optics, etc. ( [25][26][27][28] and included references).…”

Vectorial Pauli algebraic approach in polarization optics. II. Interaction of light with the canonical polarization devices

“…Its roots stand in the isomorphism between the group of transformations SL(2, C), used all along this paper (in a pure operatorial representation) and the Lorentz group O(3, 1) which describe the transformations in the special relativity. Well-known, this isomorphism was largely exploited in the last decade in the quasirelativistic formulation of the theory of polarization and, more generally, of the ''two-state'' (''two-beam'') systems [27,[32][33][34][35][36].…”

Pauli algebraic forms of normal and nonnormal operators

“…47 Bearing in mind that SL͑2,c͒ is locally isomorphic to the six-parameter Lorentz group SO͑3,1͒, a physical system that can be analyzed in terms of SL͑2,c͒ language can be equally explained in the language of the Lorentz group. 48 From the viewpoint of the group theory, the nonHermitian polarizer [Eq. (37)] is an example of Wigner rotation: The two Hermitian polarizers of a sandwich such as Eq.…”