Lens optics as an optical computer for group contractions

TL;DR

This paper demonstrates that a simple lens system can act as an optical computer to perform group contractions related to Lorentz transformations, linking spherical and hyperbolic geometries in relativistic physics.

Contribution

It introduces a novel analogy between lens optics and Lorentz group contractions, enabling optical computation of transformations between different symmetry groups.

Findings

Lens focal condition corresponds to group contraction from O(3) to E(2).

Optical transformations can analytically connect spherical and hyperbolic geometries.

The approach models Lorentz transformations using simple optical systems.

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.