Showing papers on "Symmetry (physics) published in 1969"
TL;DR: In this article, the authors introduced the outgoing spherical (or circular cylinder) partial waves as a basis for the equation QT = − Re (Q) describing scattering for general incidence on a smooth object of arbitrary shape.
Abstract: Upon introducing the outgoing spherical (or circular cylinder) partial waves {ψn} as a basis, the equation QT = − Re (Q) is obtained for the transition matrix T describing scattering for general incidence on a smooth object of arbitrary shape. Elements of Q involve integrals over the object surface, e.g. Qmn = ±(i2)δmn+(k8π)∫dσ⋅∇[Re(ψm)ψn]. where the −, + apply for Dirichlet and Neumann conditions, respectively. For quadric (separable) surfaces, Q is symmetric. Symmetry and unitarity lead to a secular equation defining eigenfunctions for general bodies. Some apparently new closed‐form results are obtained in the low frequency limit, and the transition matrix is computed numerically for the infinite strip.
469 citations
TL;DR: In this article, it was shown that the strong interactions are almost invariant under the group of pseudoscalar octet, i.e., the octet is realized by eight Goldstone bosons.
Abstract: Starting with the modern developments of current algebra and the hypothesis of partially conserved axial-vector current, it has gradually become apparent that the strong interactions are almost invariant under the group $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$. In the limit that symmetry breaking is neglected, $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ does not appear as a symmetry of the particle states as $\mathrm{SU}(3)$ does, but rather as a symmetry realized by eight Goldstone bosons, i.e., the pseudoscalar octet. Most papers on $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ symmetry have been concerned with soft-meson theorems and their connection with effective Lagrangians. This paper is devoted to other aspects of the symmetry. Part of the paper is frankly pedagogical. The physics behind a symmetry realized by way of Goldstone bosons is brought out through a study of the $\ensuremath{\sigma}$ model. Then the general principles are stated abstractly and applied to the hadrons. One of the new results presented here is that there are two distinct ways in which $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ can be realized. In both cases there is an octet of massless pseudoscalar mesons. The two possibilities differ in the residual symmetry of the hadron spectrum: In one case, it is only $\mathrm{SU}(3)$; in the other, it is $\mathrm{SU}(3)$ times a discrete symmetry, which leads to parity doublets. It is conjectured that some of the observed parity doubling in nucleon resonances is a consequence of this new discrete symmetry. Symmetry breaking is discussed in detail and is found to be very complex. In particular, it is shown that, at least for the pseudoscalar-meson masses, octet enhancement can never occur for first-order perturbations around an $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$-symmetrical limit. Since octet enhancement is an empirical fact, one is forced to conclude that lowest-order perturbation theory is not a good approximation. In connection with octet enhancement, we show how one can use a principle of pole dominance in the angular momentum plane to replace scalar "tadpole" mesons with Regge trajectories.
437 citations
389 citations
TL;DR: In this article, it was shown that the Ricci tensor of an isometry group in an n-dimensional pseudo-Riemannian manifold is invertible, and that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes.
Abstract: Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n‐dimensional pseudo‐Riemannian manifold (a group is said to be orthogonally transitive if its surfaces of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n ‐ p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n ‐ 2)‐parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a symmetry axis is present), the invertibility of the Ricci tensor is sufficient to imply orthogonal transitivity; definitions are given of convection and of the flux vector of an isometry group, and it is shown that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes. The purpose of this work is to obtain results which have physical significance in ordinary space‐time (n = 4), the main application being to stationary axisymmetric systems; illustrative examples are given at each stage; in particular it is shown that, when the source‐free Maxwell‐Einsteinequations are satisfied, the Ricci tensor must be invertible, so that Theorem 2 always applies (giving a generalization of the theorem of Papapetrou which applies to the pure‐vaccuum case).
259 citations
TL;DR: In this paper, it was shown that assuming only a symmetry condition for the null hypothesis leads to effective bounds on the dispersion of the t-statistic, and the symmetry condition is weak enough to include all cases of independent but not necessarily identically distributed observations, each symmetric about the origin.
Abstract: The size and power of Student's t-test are discussed under weaker than normal conditions. It is shown that assuming only a symmetry condition for the null hypothesis leads to effective bounds on the dispersion of the t-statistic. (The symmetry condition is weak enough to include all cases of independent but not necessarily identically distributed observations, each symmetric about the origin.) The connection between Student's test and the usual non-parametric tests is examined, as well as power considerations involving Winsorization and permutation tests. Simultaneous use of different one-sample tests is also discussed.
237 citations
216 citations
TL;DR: In this paper, the collision operator is assumed to arise from a time independent transition probability, and includes the Boltzmann integral operator or Langevin differential operator as special cases, and is recovered only if the collision frequency is large, and not the general behavior for large t.
Abstract: The calculation of 〈Pn[cosθ(t)]〉 is formulated for the symmetric top, and evaluated for several special models; Pn is the Legendre polynomial and θ(t) the angle made by the symmetry axis with its initial direction. The collision operator is presumed to arise from a time independent transition probability, and includes the Boltzmann integral operator or Langevin differential operator as special cases. Diffusional relaxation is recovered only if the collision frequency is large, and is not the general behavior for large t.
162 citations
TL;DR: In this article, the symmetry properties of the matrices describing the reflection and transmission of polarized radiation by a slab of randomly oriented particles are discussed and a complete set of such symmetry relations valid in the common case in which there is no birefringence or dichroism is given.
Abstract: In this paper we discuss symmetry properties of the matrices describing the reflection and transmission of polarized radiation by a slab of randomly oriented particles. A complete set of such symmetry relations valid in the common case in which there is no birefringence or dichroism is given. The derivation proceeds via 1) the symmetry properties of the phase matrix describing the scattering in a volume element and 2) the symmetry properties of the reflection and transmission matrices based on single scattering only. Birefringence and dichroism may occur if the particles do not have a plane of symmetry. The study of symmetry relations for this case is not carried beyond the stage of the phase matrix. Possible applications and some errors in the literature are pointed out.
147 citations
TL;DR: In this paper, a method based upon analogies with the Wigner boost technique is presented for setting up the nonlinear realizations of any continuous symmetry group, which are relevant to and useful in the treatment of cases of spontaneous symmetry breakdown.
Abstract: A method, based upon analogies with the Wigner boost technique, is presented for setting up the nonlinear realizations of any continuous symmetry group. It is argued that such realizations are relevant to and useful in the treatment of cases of spontaneous symmetry breakdown.
146 citations
TL;DR: In this article, the problem of averaging fourth-rank tensors with texture describing weight functions has been solved for orthotropic physical symmetry and for orthorhombic crystal symmetry.
Abstract: The problem of averaging fourth‐rank tensors with texture describing weight functions has been solved for orthotropic physical symmetry and for orthorhombic crystal symmetry. The results are presented in tabular form. The procedure for extending the tabular results to tetragonal, hexagonal, and cubic crystal symmetries is indicated. The solution requires the coefficients of the generalized spherical harmonic expansion of the weight function up to fourth order, and entails only those approximations required to obtain such coefficients from experimental data.
133 citations
TL;DR: The complete phonon dispersion relation for the symmetry directions in silver at room temperature has been measured by triple-axis neutron spectrometry as mentioned in this paper, but there is some disagreement with the time-of-flight results of Drexel et al.
TL;DR: In this article, the wave equation for elastic waves in an isotropic solid is solved generally in Cartesian and in circular cylindrical coordinates, and the solutions are applied in the study of a variety of guiding structures of circular and rectangular symmetry.
Abstract: The wave equation for elastic waves in an isotropic solid is solved generally in Cartesian and in circular cylindrical coordinates. The solutions are applied in the study of a variety of guiding structures of circular and rectangular symmetry. In general, the wave functions do not satisfy the boundary conditions, but in special cases they do. From a study of these special cases it is possible to arrive at some useful results and to general principles which give some insight into the behavior of waveguides in general. The results and observations obtained are compared and where appropriate, with corresponding results for electromagnetic waveguides.
TL;DR: In this article, the Raman active mode in CU2O whose symmetry properties are studied by means of group theory considerations is determined. But the results are restricted to the case of CU2Os.
TL;DR: In this paper, a general expression for the optical rotation of large helical polymers is developed and the assumption of periodic boundary conditions is shown to be valid, and it is shown that a major term in optical rotation is a direct result of the helical symmetry as reflected in the correct dipole selection rules.
Abstract: A general expression for the optical rotation of large helical polymers is developed. The assumption of periodic boundary conditions is shown to be valid. A major term in the optical rotation is found to be a direct result of the helical symmetry as reflected in the correct dipole selection rules.
TL;DR: In this paper, the configuration space approach to the three-particle problem is generalized to the case of four particles, and special coordinates are defined which have simple symmetry properties with respect to the exchange of identical particles.
Abstract: The configuration‐space approach to the three‐particle problem is generalized to the case of four particles. Special coordinates are defined which have simple symmetry properties with respect to the exchange of identical particles. The construction of a suitable orthogonal system is discussed. Some of these functions are given explicitly. It is pointed out that the use of this orthogonal system leads to a considerable simplification for a large number of four‐particle problems, namely, the approximate reduction of the Schrodinger equation to a finite system of coupled differential equations for functions that depend on one variable only.
TL;DR: In this paper, the temperature gradient effect on the stability of a plasma in the maximum-J configuration was studied and two possible instabilities due to temperature gradient were found in the axisymmetric system.
Abstract: The temperature gradient effect on the stability of a plasma in the “maximum‐J configuration” is studied and two possible instabilities due to temperature gradient are found in the axisymmetric system.
TL;DR: A group theoretical analysis on the symmetry properties of the normal modes of vibration of crystals with calcite and α-corundum structures was carried out in this paper, where the authors showed that both of these structures have the spa...
Abstract: A group theoretical analysis is carried out on the symmetry properties of the normal modes of vibration of crystals with the calcite and α-corundum structures. Both of these structures have the spa...
TL;DR: In this article, the vibrational spectrum of C 4 H 6 Fe(CO) 3 has been studied and an assignment of the frequencies on the basis of the local symmetry of the Fe( CO) 3 fragment (C 3v ) explains most of the observations, but the FeC and O stretching modes of E symmetry are both split.
TL;DR: In this article, the most general effective Hamiltonians including spin-orbit coupling, crystal fields of lower symmetry and spin-spin coupling in various orders of perturbation theory are given for three symmetry groups.
TL;DR: In this article, it was shown that for face-centred cubic and body-centered cubic structures the inclusion of these extra anti-unitary symmetry operations leads to no extra degeneracy, but that in the hexagonal close-packed structure, for certain magnetization directions and for certain points and lines of symmetry on the top face of the Brillouin zone, these extra symmetry operations lead to additional degeneracies which were not noted by Falicov and Ruvalds.
Abstract: The essential degeneracies that may arise in the electronic band structure of a magnetic metal have been investigated by Falicov and Ruvalds. Because of the existence of a spontaneous magnetic moment the operation of time reversal θ is no longer a symmetry operation of the crystal. However, the crystal still possesses some compound anti-unitary symmetry operations, ignored by Falicov and Ruvalds, which consist of products of θ with various space-group operations. It is shown that for face-centred cubic and body-centred cubic structures the inclusion of these extra anti-unitary operations leads to no extra degeneracy, but that in the hexagonal close-packed structure, for certain magnetization directions and for certain points and lines of symmetry on the top face of the Brillouin zone, these extra anti-unitary symmetry operations lead to additional degeneracies which were not noted by Falicov and Ruvalds.
TL;DR: In this article, complete correlation is established between upper state wave functions, the corresponding transitions to these states, and the symmetry of the states under the full point group C 3 v for any arbitrary choice of phases for the rotation and vibration basis functions.
TL;DR: In this article, the ground Γ7 state of Pu3+ in cubic symmetry sites in CaF2, SrF2 and BaF2 were measured at T = 1.2°K by EPR techniques.
Abstract: The g values of the ground Γ7 state of Pu3+ in cubic symmetry sites in CaF2, SrF2, and BaF2 are 1.297 ± 0.001, 1.250 ± 0.002, and 1.187 ± 0 004, respectively, as measured at T = 1.2°K by EPR techniques. By fitting the g values, estimates of the crystalline‐field parameters are derived. It is shown that crystalline‐field effects for Pu3+ in the alkaline‐earth fluorides are the same order of magnitude as in the lanthanide series.
TL;DR: In this article, the Kramers-Kronig relations, sum rules, and symmetry properties for the electrodynamic transport tensors of a linear plasma medium were reviewed and extended.
Abstract: The purpose of this paper is to review and to extend, wherever possible, the Kramers-Kronig relations, sum rules, and symmetry properties for the electrodynamic transport tensors of a linear plasma medium. For complete generality, we consider both nonrelativistic and relativistic plasmas with and without external magnetic fields. Our study is carried out first within the framework of classical electrodynamics. We then exploit the statistical-mechanical fluctuation-dissipation theorem to further obtain the Onsager symmetry relations and Kubo sum-rule frequency moments. Of special significance is the emergence of a variety of new Kramers-Kronig formulae andf-sum rules for the inverse dispersion tensor.
TL;DR: In this paper, the authors considered the application of thermogalvanomagnetic coefficients of basic equations determining space-time symmetry restrictions, and showed that these coefficients can be expanded in powers of the components of the magnetic field, and the symmetry restrictions imposed on the expansion coefficients are discussed.
Abstract: Further development is given of the application to thermogalvanomagnetic coefficients of basic equations determining space-time symmetry restrictions. Material omitted from tables published by the author is supplied. Also, the thermogalvanomagnetic coefficients are expanded in powers of the components of the magnetic field, and the symmetry restrictions imposed on the expansion coefficients are discussed. The method is apparently also applicable to other transport coefficients and for other fields.
TL;DR: In this paper, the general projection technique for getting translationally-invariant configuration-space four-particle states of arbitrary symmetry and total orbital angular momentum, out of harmonic-oscillator states in the single relative coordinates was determined.