Showing papers on "Symmetry (geometry) published in 1970"

Search for a Universal Symmetry Group in Two Dimensions

Abstract: Recently several authors have proposed a universal symmetry group and demonstrated the classical validity of the concept. Supposedly, an SU(n) symmetry group could be constructed for whatsoever system of n degrees of freedom. The claim is assuredly valid for all classically degenerate systems, but is in contradiction with most of the well‐known and widely accepted solutions of Schrodinger's equation. We examine the reasons for this discrepancy on the quantum‐mechanical level. The construction of the universal symmetry group requires ladder operators, which in most cases are the ladder operators of Infeld and Hill. Complications which owe their origin entirely to numerical relationships imposed by quantization prevent these operators from forming a von Neumann algebra and, in turn, an SU(n) group of constants of the motion. Two important effects are those imposed by anisotropy, wherein not all quanta have the same size, and by non‐Cartesian coordinates, wherein quanta in some dimensions are restricted in size by those in other dimensions. These effects can be seen quite clearly in two‐dimensional systems: anisotropy in the Cartesian anisotropic harmonic oscillator, and conflict between dimensions in the polar form of the isotropic oscillator. Further complications arise when the two effects are combined, as in the harmonic oscillator or hydrogen atom with ``excess'' angular momentum. Enough residue of the universal symmetry concept remains that many similarities between the hydrogen atom and harmonic oscillator may be understood, including the fact that some levels of the hydrogen atom, which ordinarily transform according to an orthogonal group, may form irreducible representations of the unitary group.

Einstein Spaces with Symmetry Groups

Abstract: In this paper, we construct all possible groups of motion (symmetry groups) for empty Einstein spaces admitting a diverging, geodesic, and shear‐free ray congruence. (Minkowski space is excluded throughout the discussion.) It is proved that any such Einstein space cannot admit a symmetry group with dimension greater than four. Although the field equations are not solved completely for spaces with groups of dimension one or two, a generalization of the Kerr spinning‐mass solution is obtained from the 2‐dimensional class. It is shown that all such spaces with 4‐dimensional symmetry groups are well known: Schwarzschild, NUT (Newman, Unti, and Tamborino), and a particular hypersurface orthogonal Kerr‐Schild metric. The only member of these spaces admitting a 3‐dimensional symmetry group is a Petrov Type III hypersurface orthogonal metric.

Symmetries of the Racah Coefficients

Abstract: A new symmetry of the Racah coefficients is derived using a property of a generalized hypergeometric function of unit argument. The symmetry is similar in appearance, though not derivation, to that given by Regge.

Permutation symmetry control in concerted reactions

Abstract: The Woodward–Hoffmann selection rules are obtained from permutation symmetry by means of the Valence Bond method.

Educational toy device

Abstract: An educational toy device comprising multiple groups of cubes wherein each group is interconnected by means of a double hinge and each cube in each group is also connected by means of a double hinge thereby permitting any number of cube combinations for forming various configurations. Each of the cubes in each group is provided with a selected numeral on each of its six faces in a manner whereby the numerical sum on the faces of at least any pair of cubes adjacent another pair along a common line of symmetry is the same regardless of the various cube combinations.

Interpretation of the Symmetry of the Clebsch‐Gordan Coefficients Discovered by Regge

Abstract: An interpretation is proposed for the intriguing symmetry of the Clebsch‐Gordan coefficients discovered in 1958 by Regge. The interpretation is based on the observation that, in the reduction of the Kronecker product of two irreducible representations of an SU(2) group, there appears in a natural way another SU(2) group, which is almost independent of the original one. The Regge symmetry is interpreted as the symmetry under the interchange of these two SU(2) groups. In more picturesque language, the Regge symmetry is the symmetry under the interchange of the ``two‐ness'' of the two angular momenta being added with the ``two‐ness'' of SU(2). It follows from this interpretation that a symmetry of the same nature is present in the generalized Clebsch‐Gordan coefficient that appears in the reduction of the Kronecker product of n (and not two, except when n = 2) irreducible representations of SU(n).

A Set of Plane Measure Zero Containing all Finite Polygonal Arcs

Abstract: We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type. In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.

Image Formation of a Reflecting Cone for an Off-Axis Source

Abstract: For a point source placed in front of a reflecting cone, expressions have been found for the shape of the virtual image and for the amplitude of the real-image field. Both the virtual and the real images have the same axis of symmetry, which passes through the vertex of the cone and lies in the plane containing the point source and the cone axis. The angle between the axis of symmetry and the axis of the cone is equal in magnitude but opposite in sign to the angle between the latter and the line joining the point source and the cone vertex. The projection of the virtual image onto a plane perpendicular to the axis of symmetry is an ellipse. Both the real and virtual images have two planes of symmetry that are the planes containing, respectively, the major and minor axes of the virtual image. The irradiance distribution along the axis of symmetry in the real-image field varies as the square of a Bessel function of zero order and of the first kind. The analytical expression found for the real-image field gives the expected behavior of the field when the cone becomes a plane. It also obeys the energy-conservation law.

Some comments on the theory of space-group representations. II. Basic domain and representation domain

Abstract: For pt. I see ibid., vol. 3, no.3, 610 (1970). Tables of space-group representations now available are incomplete for those space groups which do not have the full symmetry of the Bravais lattice on which they are based. It is shown that this defect can be overcome without recourse to further tabulation for all space groups except Pa3(Th6). This particular space group is considered separately, and is shown to be exceptional in that alone out of all the 230 space groups it possesses two k-vectors, which are related by an operation of the holosymmetric point group, but whose groups of k possess small representations of different dimensionalities. Nor does the addition of time reversal remove this anomaly.

IX - Symmetry, point groups, and character tables. Part II: Classification of molecules into point groups

Abstract: Examines translational symmetry, introduces point groups, and presents a convenient procedure for assigning the point group of a molecule.

Symmetry Properties of the Clebsch-Gordan Coefficients of Space Groups

Abstract: The symmetry properties of the Clebsch-Gordan coefficients of space groups are described. The connection with time-reversal symmetry is studied in an appendix. Es werden die Symmetrieeigenschaften der Clebsch-Gordan-Koeffizienten der Raumgruppen beschrieben. Der Zusammenhang mit der Zeitumkehrsymmetrie wird in einem Anhang untersucht.

IX - Symmetry, point groups, and character tables. Part III: Character tables and their significance

Abstract: Examines character tables and the symmetry properties of translational and rotational motion.

Algorithm for determining the symmetry and stacking properties of the planes (hkl) in a Bravais lattice

Abstract: An algorithm is presented which enables one to determine in detail the symmetry and stacking properties of the planes (hkl) in an arbitrary Bravais lattice characterized by the quantities a, b, c, cos α, cos β, cos γ.

Gravity interpretation using the fourier integral; discussion and reply

Abstract: The authors do not state explicitly that equations (1) and (12) express gravity anomalies of the two-dimensional cylinder and fault respectively. These structures have an infinite extension along the strike, and all the gravity profiles perpendicular to the strike are identical. For this reason it is admissible to apply the one-dimensional Fourier transform to obtain the one-dimensional amplitude spectra shown in Figures (1) and (5). The situation, however, is different for the sphere, which does not have a one-dimensional gravity profile in the above sense, but rather a two-dimensional field with cylindrical symmetry. Instead of using the one-dimensional Fourier transform as given by (6) along a straight line, we must apply the two-dimensional Fourier transform over the whole area. Because of the circular symmetry, one radial variable will suffice in place of the two Cartesian coordinates x and y, and the two-dimensional Fourier transform can be expressed as Hankel transform, so that equation (6) becomes F s ( ω )=2 π α D ∫ ∞ 0 J 0 ( ω ξ ) ( D 2 + ξ 2 ) 3 / 2 − ξ d ξ =2 π α e − D ω .

On the geometry of chiral dynamics. II

Abstract: Previous work on the geometry of group space appropriate to the problem of chiral symmetry is extended to cover the 'other fields' with the object of elucidating the relation between those fields which transform linearly and those which do not. The different fields just correspond to different choices of the local coordinates which must be set up to define 'spinors'. The results for specific quantities such as transformation matrices and covariant derivatives agree with those of other papers.