Lens optics as an optical computer for group contractions

Abstract

It is shown that the one-lens system in para-axial optics can serve as an optical computer for contraction of Wigner's little groups and an analogue computer which transforms analytically computations on a spherical surface to those on a hyperbolic surface. It is shown possible to construct a set of Lorentz transformations which leads to a two-by-two matrix whose expression is the same as those in the para-axial lens optics. It is shown that the lens focal condition corresponds to the contraction of the O(3)-like little group for a massive particle to the E(2)-like little group for a massless particle, and also to the contraction of the O(2,1)-like little group for a space-like particle to the same E(2)-like little group. The lens-focusing transformations presented in this paper allow us to continue analytically the spherical O(3) world to the hyperbolic O(2,1) world, and vice versa.

14 pages, RevTeX, no figures

References in corpus (4)

• Stokes Parameters as a Minkowskian Four-vector
• Iwasawa Effects in Multi-layer Optics
• Interferometers and Decoherence Matrices
• Binary Representations of ABCD Matrices

Cited by in corpus (8)

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• Rotations associated with Lorentz boosts
• ABCD Matrices as Similarity Transformations of Wigner Matrices and Periodic Systems in Optics
• The language of Einstein spoken by optical instruments
• Lens optics and the continuity problems of the ABCD matrix
• Lorentz group theory and polarization of the light
• Factorizations for 3-rotations and polarization of the light in Mueller-Stokes an Jones formalisms
• Transitivity in the theory of the Lorentz group and the Stokes -- Mueller formalism in optics