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 analog computer that 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 2x2 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 spacelike 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. Since the traditional role of Wigner's little groups has been to dictate the internal space-time symmetries of massive, massless, and imaginary-mass particles, the one-lens system provides a unification of those symmetries.

Slide-rule-like property of Wigner’s little groups and cyclicSmatrices for multilayer optics

“…This condition does not depend on N. We have discussed a similar case in our previous paper [1]. In their recent paper [7], Baskal and Kim noted the same transition process for onelens optics. They noted that the camera focusing mechanism corresponds to contraction of Wigner little groups.…”

“…(45) through this process has been discussed in detail in Ref. [7] in connection with the contraction of Wigner's little groups. As we noted in Sec.…”

“…However, they produced many mathematical identities useful in other branches of physics, including classical layer optics which depends heavily on two-by-two matrices. The correspondence between the two-by-two and four-by-four representations of the Lorentz group has been repeatedly discussed in the literature [1,3,7]. In the two-by-two representation, we write the rotation matrix around the y axis as…”

“…We shall use only these three matrices in this paper. We use the following identity which Baskal and Kim introduced recently in their paper on lens optics and group contractions [7,11]. …”

“…However, in so doing, we are going through an important group theoretical aspect which became known to us only recently, namely on contractions of Wigner's little groups. This aspect was discussed in detail in a recent paper on lens optics [7]. Thus, we shall borrow some of the mathematical identities from that paper.…”