Kinematics at infinite momentum

J. Opt. B: Quantum Semiclass. Opt.

Georgieva

Kim

et al. 2004

It has been almost 100 years since Einstein formulated his special theory of relativity in 1905. He showed that the basic space-time symmetry is dictated by the Lorentz group. It is shown that this group of Lorentz transformations is not only applicable to special relativity, but also constitutes the scientific language for optical sciences. It is noted that coherent and squeezed states of light are representations of the Lorentz group. The Lorentz group is also the basic underlying language for classical ray optics, including polarization optics, interferometers, the Poincaré sphere, one-lens optics, multi-lens optics, laser cavities, as well multilayer optics.

Section: Wigner's Little Groupsmentioning

confidence: 99%

Lorentz group in classical ray optics

J. Opt. B: Quantum Semiclass. Opt.

Georgieva

Kim

et al. 2004

“…(11) is negative while the remaining three are positive, and it can be written in the form of Eq. (14). If it is greater than 2 cosh ρ, all the elements are positive, and the core matrix should be written as Eq.…”

Section: Contractions Of the Little Groupsmentioning

confidence: 99%

“…Unlike the traditional procedures, the contraction presented this paper is an analytic transformation, which provides a reversible process from Eq. (14) to Eq. (16) through Eq.…”

Section: Lens Optics and Group Contractionsmentioning

confidence: 99%

“…The question is then whether the contraction of O(3) to E(2) necessarily mean the contraction of the O(3)-like little group to E(2)-like little group. This conceptual question also has been discussed extensively in the literature [10,13,14]. In this paper, we only use the results which can be represented in the two-by-two matrix representation of the Lorentz group.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Lens optics as an optical computer for group contractions

Kim

2003

Phys. Rev. E

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.

“…11 The algebra of the O(3, 1) Lorentz group can be contracted to the algebra of E(2)-like groups by means of a similar limiting procedure. The generators of its rotation subgroup are boosted to represent the little group for a massive particle in relativistic motion.…”

Section: Introductionmentioning

confidence: 99%

Dirac Gamma Matrices as Representations of the Algebra of Little Groups

2000

Mod. Phys. Lett. A

The algebra of O(3, 3) is contracted to that of E(2)-like groups in the infinite-momentum limit. It is shown through this process that there are 18 different possibilities for the representations of E(2)-like algebra if they are to be expressed in terms of Dirac γmatrices. It is also pointed out that the generators of this algebra are independent of the particular representation of the γ-matrices. Possible implications of this analysis is discussed.