Ray Representation Research Articles

Seismic recognition of local inhomogeneities

A simple method to contour local inhomogeneities using seismic data is proposed. It formalizes an approximate inversion method which is based on the interpretation of local inhomogeneities as making the differences between an actual seismic data set and a previous reference model. It uses the optimal statistical criteria of parameter estimation and recognition and the ray representation of the waves spreading. Any combination of direct, reflected and/or other types of waves may be used as the database. Inhomogeneities, having a size two times above the wavelength of the seismic waves, can be resolved. Laboratory experiments, using ultrasonic waves and analysis of data from field experiments, confirmed the theoretical results. The method can be used to search for ore bodies, kimberlite cubes, oiltraps, etc.

Ray representation for k-trees

k-trees have established themselves as useful data structures in pattern recognition. A fundamental operation regarding k-trees is the construction of a k-tree. We present a method to store an object as a set of rays and an algorithm to convert such a set into a k-tree. The algorithm is conceptually simple, works for any k, and builds a k-tree from the rays very fast. It produces a minimal k-tree and does not lead to intermediate storage swell.

Group structure of superluminal Lorentz transformations and their representation

Constructing the generalized superluminal Lorentz transformations for inertial frames with nonpreferred orientation, we have investigated their group theoretical and commutation properties. The extended homogeneous Lorentz group in terms of these transformations has been constructed and its generators have been derived; these have been shown to generate the ray representation of the resulting Poincaré group.

Wave beam shaping on diffraction of a whispering gallery wave at a convex cylindrical surface

LITERATURE CITED i. M.L. Ter-Mikaelyan, Izv. Akad. Nauk ArmSSR, Erevan (1969). 2. V.L. Gunzburg and V. N. Tsytovich, Transition Radiation and Transition Scattering [in Russian, Nauka, Moscow (1984). 3. V.G. Baryshevskii and I. D. Feranchuk, Zh. ~ksp. Teor. Fiz., 61, No. 3(9), 949 (1971). 4. V.G. Baryshevskii, Materials from the 2nd Symposium on Transition Radiation of High- Energy Particles [in Russian], Erevan (1983), p. 184. 5. L.D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media [in Russian], Nauka, Moscow (1982). WAVE BEAM SHAPING ON DIFFRACTION OF A WHISPERING GALLERY WAVE AT A CONVEX CYLINDRICAL SURFACE S. N. Vlasov, M. A. Shapiro, and E. V. Sheinina UDC 517.934+621.371 The problem of transforming a whispering gallery wave into a wave beam is re- solved numerically for the case of diffraction at a convex cylindrical surface. Conditions have been determined under which the conversion factor is close to unity. i. When the waves of a whispering gallery are propagated near a surface with a variable curvature the field separates from the surface in the vicinity of the point of flexure. In the case of linearly changing curvature [i], on radiation of the whispering gallery mode exhibiting only a single field variation along the transverse coordinate, a single-lobed radiation pattern is formed (a bell-shaped field structure), while on radiation of higher types of modes with two and more variations, the radiation pattern is multilobed [2]. On the other hand, as follows from the ray representation of the whispering gallery mode [3], the radiation pattern of the wave from the edge of a cylindrical concave surface is represented by the congruence of rays tangent on the segment of a simple caustic (Fig. i). Within the scope of such a representation, valid for higher types of modes, radiation from an edge exhibits a quasiuniform transverse structure in the far zone. However, the radiation pattern corresponds only to the main lobe of the radiation pattern; diffraction at the edge leads to the formation of side lobes~ From the standpoint of forming a bell-shaped field structure, there is interest in ana- lyzing the diffraction of whispering gallery waves from convex surfaces that do not disrupt the radiation pattern from an edge, but which block the formation of side lobes. Such a formulation of the problem, unlike [i] in which the coordinate system is connected to a dis- torted surface, corresponds to a greater degree to a system of radial coordinates [4]. It is difficult to analyze the lateral structure of the radiation in radial coordinates. A simplifying circumstance is the case in which the radiation is formed by a congruence of paraxial rays (Fig. i), and for the solution of the problem we can make use of Cartesian coordinates connected to a reference beam. 2. Our determination of the lateral structure of the radiation is based on the para- bolic equation method. Let the z coordinate be calculated in the direction of the reference beam and let the shape of the surface be described by the function x = x(z); the Cartesian coordinates (x, z) have been normalized to the radius of surface curvature a for z < 0 (Fig~ Institute of Applied Physics, Academy of Sciences of the USSR. Translated from Izvest- iya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 31, No. 12, pp. 1482-1487, December, 1988. Original article submitted January 27, 1987. 1070 0033-8443/88/3112-1070512.50 9 1989 Plenum Publishing Corporation

Galilean quantum kinematics

The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.

Integrable ergodic W* systems

A W* system (M, G, alpha ) consists of a W* algebra M (the 'algebra of observables'), a kinematical (locally compact, separable) group G and a representation alpha of G by automorphisms of M. Ergodic W* systems are used in algebraic quantum mechanics to describe elementary physical systems. An ergodic W* system is integrable in the sense of Connes and Takesaki (1977) if it admits 'observables', i.e. operators which transform 'suitably' under the action alpha . Elementary quantum systems, corresponding to irreducible ray representations of G on a Hilbert space, and elementary classical systems, corresponding to transitive representations of G on a phase space, fit into this scheme. This letter investigates the structure of integrable ergodic W* systems, in particular those of Abelian groups and those whose underlying W* algebra is of type I.

Pedestrian approach to two-cocycles for unitary ray representations of Lie groups

Necessary and sufficient conditions for unitary ray representations of connected Lie groups are reexamined. Thus a systematic constructive method is obtained for calculating the admissible exponent factors (two-cocycles). The gauge freedom of the unitary ray representation formalism is also considered. This introduces the distinction between trivial and genuine ray representations. A special gauge is then adopted, within which the two-cocycle is almost unique. The only prerequisite of the exponent factor calculus is the knowledge of the binary combination rules for the essential parameters of the group. The attained method affords a simple, general, and explicit (i.e., coordinate-dependent) two-cocycle calculus. The aim of this paper is merely instrumental.

A ray representation of surface diffraction by a multilayer cylinder

The exact expressions for the field produced by a point source radiating in the presence of an infinitely long cylindrical structure comprising layers of different materials is evaluated asymptotically for source and observation points that are widely removed from the cylinder surface and related so that the cylinder shadows the source from the observer. The diffracted field is interpreted in a ray-optic format. It is observed that surface-propagating waves along the layers can be slowed significantly as compared with those on an impenetrable cylinder of the same radius. This wave slowing requires that rays attaching to the cylinder refract into a layer-guided mode and refract a second time as the curved structure sheds energy into the scattered field. Consequently, rays attaching to and shedding from the cylinder are not tangent to it when the layer profile slows the surface-propagating waves. The analysis applies for both metallically backed and open-shell layers. Computed diffraction coefficient results are given for a coated metallic cylinder as a function of coating thickness.

Quantum kinematics of the harmonic oscillator

The formalism of non-Abelian quantum kinematics is applied to the Newtonian symmetry group of the harmonic oscillator. Within the regular ray representation of the group, the Schrödinger operator, as well as two other (new) invariant operators, are obtained as Casimir operators of the extended kinematic algebra. Superselection rules are then introduced, which permit the identification (and the explicit calculation) of the physical states of the system. Next, a complementary ray representation, attached to the space-time realization of the group, casts the Schrödinger operator into the familiar time-dependent space-time differential operator of the harmonic oscillator and thus, by means of the superselection rules, one obtains the time-dependent Schrödinger equation of the sytem. Finally, the evaluation of a Hurwitz invariant integral, over the group manifold, affords the well known Feynman space-time propagator 〈t′,x′‖t,x〉 of the simple harmonic oscillator. Everything comes out from the assumed symmetries of the system. The whole approach is group theoretic and ‘‘relativistic.’’ No classical analog is used in this ‘‘quantization’’ scheme.

Group-related coherent states

Coherent states defined with respect to an irreducible ray representation u: g→ug, g∈G, of an arbitrary locally compact separable group G are examined. It is shown that the following conditions (a)–(d) are equivalent: (a) u admits coherent states, (b) u is square integrable, (c) the W*-system implemented by u is integrable, and (d) u is a subrepresentation of the left regular c-representation, where c is the respective multiplier of u. Furthermore, the group theoretical background of what is called the ‘‘P-representation of observables’’ associated with coherent states is investigated: It is shown that the P-representation (which corresponds to a covariant semispectral measure) fulfills a certain maximality requirement. The P-representation can be used to represent the quantum system in question on the Hilbert space L2(G,dg) of square-integrable functions (with respect to Haar measure dg) on the kinematical group G.

Short-wave asymptotics of the solution of the problem of a point source in an inhomogeneous moving medium

The leading term is obtained for the short-wave asymptotics of the field of a point source in a small neighborhood of it and also an expression for the excitation coefficient of the wave in the region where the ray representation is valid for an infinite, inhomogeneous, vortex-free, moving medium with constant entropy and for a weakly vortical medium for which the speed of motion is small.

Compact quantum systems: Internal geometry of relativistic systems

A generalization is presented of the kinematical algebra so(5), shown previously to be relevant for the description of the internal dynamics (Zitterbewegung) of Dirac’s electron. The algebra so(n+2) is proposed for the case of a compact quantum system with n degrees of freedom. Associated wave equations follow from boosting these compact quantum systems. There exists a contraction to the kinematical algebra of a system with n degrees of freedom of the usual type, by which the commutation relations between n coordinate operators Qi and corresponding momentum operators Pi, occurring within the so(n+2) algebra, go over into the usual canonical commutation relations. The so(n+2) algebra is contrasted with the sl(l,n) superalgebra introduced recently by Palev in a similar context: because so(n+2) has spinor representations, its use allows the possibility of interpreting the half-integral spin in terms of the angular momentum of internal finite quantum systems. Connection is made with the ideas of Weyl on the possible use in quantum mechanics of ray representation of finite Abelian groups, and so also with other recent works on finite quantum systems. Possible directions of future research are indicated.

Hamiltonian interpretation of anomalies

A family of quantum systems parametrized by the points of a compact space can realize its classical symmetries via a new kind of nontrivial ray representation. We show that this phenomenon in fact occurs for the quantum mechanics of fermions in the presence of background gauge fields, and is responsible for both the nonabelian anomaly and Witten's SU(2) anomaly. This provides a hamiltonian interpretation of anomalies: in the affected theories Gauss' law cannot be implemented. The analysis clearly shows why there are no further obstructions corresponding to higher spheres in configuration space, in agreement with a recent result of Atiyah and Singer.

A 3-cocycle in quantum mechanics

We find the existence of a non-vanishing 3-cocycle in the unitary, gauge-invariant, ray representations of translations for the quantum mechanics of a charged particle in a background magnetic monopole field. The 3-cocycle can be removed by passing to the central extension of the Lie algebra of translations, or the algebra can be made associative by imposing the Dirac quantization condition.

Cohomology of gauge groups: Cocycles and Schwinger terms

The relation between anomalies in space-time, anomalous Schwinger terms in equal-time commutators, and the cohomology of the relevant gauge groups is discussed. The consistency condition for the Schwinger terms is derived. Integral formulas are given for the phases occurring in the ray representations of the space gauge group at fixed time as well as for the higher cocycles.

On the Connection between Evanescent Field Tracking and Generalized Ray Representation in Optics

Correspondence is established between Choudhary and Felsen's method of evanescent field tracking and Lee's method of obtaining exact-ray equivalent equations from the scalar wave equation. However, the former results in real trajectories which deviate from the direction of the average Poynting vector; the exact-ray paths, on the other hand, are both codirectional with the Poynting vector, as well as being consistent with energy conservation.

Simply subducible groups. Ray groups and projectors for double groups and space groups

It is shown how the theory of ray representations of a group may be recast in terms of ordinary representations of a new entity, a ray group. When these ray groups are simply subducible, a recently developed method of symmetry analysis based directly on the character tables may be applied. This leads to a complete reduction of all space groups and double groups. Tables for this purpose are included in an appendix.

Effect of the branch-cut on the transformation between modes and rays

In this paper the effect of the branch-cut on the transformation between the modes and rays is discussed. Characteristics of the bottom reflection coefficient on the ζ plane were analyzed in detail and a certain branch-cut was taken. Once the integral path of the generalized ray is deformed to the real axis of the ζ plane, the ray representation consists of ‘‘canonical’’ generalized ray and a ‘‘lateral term.’’ The mode representation and the ray representation are given as follows: Mode representation=∑∞nφn(r,z,z0) +∮b. Ray representation =∑∞lBl(r,z,z0) +𝒥∞l∮bl. It was found that the ‘‘canonical’’ generalized ray Bl and the normal mode φn were still connected by the Poisson summation relation strictly, as proved by our previous paper [Gao and Shang, J. Sound Vib. 80, 105–115 (1982)]. The sum of the ‘‘lateral term’’ of the ‘‘canonical’’ generalized ray was exactly equal to the part of the ‘‘lateral’’ wave in the mode representation:∮b=∑∞l∮bl.

Improved ray representation for planar optical waveguides

Scalar wave optics of planar optical waveguides is converted into an exact ray description by nonlinear mapping relations which map ray functions into mode functions and vice versa. An exact duality invariant is derived and shown to be closely related to the Lewis invariant. The theory is also shown to reduce to geometric optics in the limit of vanishing wavelength.

Generalized Ray Representation for Optical Fibre Helical Rays

An exact ray representation for optical fibre helical rays is presented. The ray representation is related to the usual mode representation by a set of non-linear mapping relations which map ray functions into mode functions and viceversa. This duality of optical representations is shown to satisfy an invariant constraint closely related to the Lewis invariant.