Search

Paraxial lens optics is discussed to study the continuity properties of thebeam transfer matrix. The two-by-two matrix for the one-lens camera-like system can be converted to an equi-diagonal form by a scale transformation, leaving the off-diagonal elements invariant. It is shown that the matrix remains continuous during the focusing process, but this transition is not analytic. However, its first derivative is still continuous, which leads to the concept of ‘tangential continuity’. It is then shown that this tangential continuity is applicable tomatrices pertinent to periodic optical systems, where the equi-diagonalization is achieved by a similarity transformation using rotations. It is also noted that both the scale transformations and the rotations can be unified within the framework of Hermitian transformations. [ABSTRACT FROM PUBLISHER]

It is not always possible to diagonalize the optical ABCD matrix, but it can be brought into one of the four Wigner matrices by a similarity transformation. It is shown that the four Wigner matrices can be combined into one matrix with four branches. This result is illustrated in terms of optical activities, laser cavities, and multilayer optics. [ABSTRACT FROM AUTHOR]

The mathematics of Lorentz transformations, called the Lorentz group, continues to play an 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 are 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 that 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. © 2005 Pleiades Publishing, Inc. [ABSTRACT FROM AUTHOR]

The presence of Killing-Yano tensors implies the existence of nongeneric supercharges in spinning point particle theories on curved backgrounds. Dual metrics are defined through their associated nondegenerate Killing tensors of valence two. Siklos spacetimes, which are the only nontrivial Einstein spaces conformal to nonflat pp-waves are investigated with regard to the existence of their corresponding Killing and Killing-Yano tensors. It is found that a class of Siklos space-times admit dual metrics and nongeneric supercharges. [ABSTRACT FROM AUTHOR]

Second-rank nondegenerate Killing tensors for some subclasses of space-times admitting parallel null one-planes are investigated. Lichéerowicz radiation conditions are imposed to provide a physical meaning to space-times whose metrics are described through their associated second-rank Killing tensors. Conditions under which the dual space-times retain the same physical properties are presented. [ABSTRACT FROM AUTHOR]

We study the Lax tensors of the separable coordinates in (2+1) dimensions. The Lax tensors of the dual manifolds are investigated. [ABSTRACT FROM AUTHOR]

The tensorial form of the Lax pair equations are given in a compact and geometrically transparent form in the presence of Cartan's torsion tensor. Three-dimensional space–times admitting Lax tensors are analyzed in detail. Solutions to Lax tensor equations include interesting examples as separable coordinates and the Toda lattice. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]