Showing papers on "Symmetry (physics) published in 1986"

Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity

Abstract: It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.

Additional symmetries for integrable equations and conformal algebra representation

Abstract: We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified ‘dressing’ method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S1 algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators \(\lambda ^m \partial _\lambda \), with integer m, p. Some additional results about Kac-Moody algebra applications are presented.

Dilaton and chiral-symmetry breaking

Abstract: The spontaneous breaking of chiral symmetry in certain gauge models may also imply the spontaneous breaking of an approximate scale symmetry. This breaking will produce the dilaton as a pseudo-Goldstone boson of spontaneously broken scale invariance.

Statistics of energy levels without time-reversal symmetry: Aharonov-Bohm chaotic billiards

Abstract: A planar domain D contains a single line of magnetic flux Phi . Switching on Phi breaks time-reversal symmetry (T) for quantal particles with charge q moving in D, whilst preserving the geometry of classical (billiard) trajectories bouncing off the boundary delta D. If delta D is such that these classical trajectories are chaotic, the authors predict that T breaking will cause the local statistics of quantal energy levels to change their universality class, from that of the Gaussian orthogonal ensemble (GOE) of random-matrix theory to that of the Gaussian unitary ensemble (GUE). In the semiclassical limit this transition is abrupt; for statistics involving the first N levels, GUE behaviour requires that the quantum flux alpha identical to q Phi /h>>0.13N-14/. The special flux alpha =1/2 corresponds to 'false T breaking' and for this case GOE statistics are predicted. These predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry, for which delta D is a cubic conformal image of the unit disc.

Large-scale flow in turbulent convection: a mathematical model

Abstract: A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.

Running with symmetry

Abstract: Symmetry can simplify the control of dynamic legged systems In this paper, the symmetries studied describe motion of the body and legs in terms of even and odd functions of time A single set of equations describes symmetric running for systems with any number of legs and for a wide range of gaits Techniques based on symmetry have been used in laboratory experiments to control machines that run on one, two, and four legs In addition to simplifying the control of legged machines, symmetry may help us to understand legged locomotion in animals Data from a cat trotting and galloping on a treadmill and from a human running on a track conform reasonably well to the predicted symmetries

False time-reversal violation and energy level statistics: the role of anti-unitary symmetry

Abstract: The authors extend the classification of symmetries necessary to predict the universality class of spectral fluctuations of quantal systems whose classical motion is chaotic, by explaining that a system with neither time-reversal symmetry (T) nor geometric symmetry may display the spectral statistics of the Gaussian orthogonal ensemble (GOE), rather than those of the Gaussian unitary ensemble (GUE), provided it possesses instead some combination of symmetries which includes T. Such combinations constitute invariance under anti-unitary transformations (whose classical analogue are called anticanonical). For a particle in a magnetic field B plus scalar potential V, an example is TSx where Sx is a mirror reflection under which B and V are invariant. The authors illustrate this numerically for a single flux line in a hard-walled enclosure (Aharonov-Bohm quantum billiards), which also provides an example of an anti-unitary symmetry of non-geometrical origin; the spectral fluctuations are, as predicted, GOE rather than GUE.

Angular dependence of ultrasonic wave propagation in a stressed, orthorhombic continuum: Theory and application to the measurement of stress and texture

Abstract: A theory for ultrasonic wave propagation in a symmetry plane of a biaxially stressed, orthorhombic continuum is presented. Since many of the material parameters which appear in the analysis are unknown, in particular the third‐order elastic constants of polycrystalline metals, emphasis is placed on the angular dependence of the velocities. An expansion to first order in stress‐induced anisotropy and to second order in textural anisotropy reveals terms with twofold, fourfold, and sixfold symmetry. Scenarios are proposed for using various properties of this symmetry to deduce the difference in magnitude and directions of the principal stresses independent of textural anisotropy and the textural anisotropy independent of the stresses. Experimental results are presented for the cases of aluminum, 304 stainless steel, and copper.

Symmetry rules for chemical reactions

Abstract: The symmetry properties of molecular orbitals and of reaction coordinates can be used to decide on the feasibility of selected chemical reaction mechanisms. Some reaction paths are shown to have a large energy barrier and are said to be “forbidden by orbital symmetry.” The reactions of molecules with no symmetry can also be analyzed by being compared to related symmetric molecules, where the molecular orbitals are topological identical.

Symmetries of differential equations

Abstract: The use of a symmetry to reduce the order of an nth-order differential equation is treated by considering the symmetry of an associated vector field. A particular choice of associated vector field leads to the usual extension of the Lie symmetry method. The possibility of other choices leads to a powerful generalisation. An algebraic classification of transformations arises naturally from the theory. It is shown to be equivalent to the geometric classification of transformations as contact and non-contact.

Homothetic and Conformal Symmetries of Solutions to Einstein's Equations

Abstract: We present several results about the nonexistence of solutions of Einstein's equations with homothetic or conformal symmetry. We show that the only spatially compact, globally hyperbolic spacetimes admitting a hypersurface of constant mean extrinsic curvature, and also admitting an infinitesimal proper homothetic symmetry, are everywhere locally flat; this assumes that the matter fields either obey certain energy conditions, or are the Yang-Mills or massless Klein-Gordon fields. We find that the only vacuum solutions admitting an infinitesimal proper conformal symmetry are everywhere locally flat spacetimes and certain plane wave solutions. We show that if the dominant energy condition is assumed, then Minkowski spacetime is the only asymptotically flat solution which has an infinitesimal conformal symmetry that is asymptotic to a dilation. In other words, with the exceptions cited, homothetic or conformal Killing fields are in fact Killing in spatially compact or asymptotically flat spactimes. In the conformal procedure for solving the initial value problem, we show that data with infinitesimal conformal symmetry evolves to a spacetime with full isometry.

Symmetry and stability in Taylor Couette flow

Abstract: We study the flow of a fluid between concentric rotating cylinders (the Taylor problem) by exploiting the symmetries of the system. The Navier–Stokes equations, linearized about Couette flow, possess two zero and four purely imaginary eigenvalues at a suitable value of the speed of rotation of the outer cylinder. There is thus a reduced bifurcation equation on a six-dimensional space which can be shown to commute with an action of the symmetry group $O(2) \times SO(2)$. We use the group structure to analyze this bifurcation equation in the simplest (nondegenerate) case and to compute the stabilities of solutions. In particular, when the outer cylinder is counterrotated we can obtain transitions which seem to agree with recent experiments of Andereck, Liu, and Swinney [1984]. It is also possible to obtain the “main sequence” in this model. This sequence is normally observed in experiments when the outer cylinder is held fixed.

Hopf bifurcation with dihedral group symmetry - Coupled nonlinear oscillators

Abstract: The theory of Hopf bifurcation with symmetry developed by Golubitsky and Stewart (1985) is applied to systems of ODEs having the symmetries of a regular polygon, that is, whose symmetry group is dihedral. The existence and stability of symmetry-breaking branches of periodic solutions are considered. In particular, these results are applied to a general system of n nonlinear oscillators coupled symmetrically in a ring, and the generic oscillation patterns are described. It is found that the symmetry can force some oscillators to have twice the frequency of others. The case of four oscillators has exceptional features.

Saturation level of the hopfield model for neural network

Abstract: The Hopfield model of a neural network is studied for p = αN, where p is the number of memorized patterns and N the number of neurons. The averaging over the quenched randomness is performed with the replica method, with replica symmetry broken once. It is shown that the critical value of α increases from 0.138 to 0.144, in excellent agreement with simulation results. For 0.138 < α ≤ 0.144 retrieval states exist only with replica symmetry breaking. Wherever the difference between the replica symmetric solution and the broken symmetry solution is numerically detectable, symmetry breaking improves the retrieval. At αc the number of errors decreases from 1.5% to 0.9%.

Symmetrised method for the calculation of the band structure of noncollinear magnets

Abstract: The one-electron Schrodinger equation for a noncollinear magnet (NCM) with arbitrary direction of atomic spin in the magnetic unit cell (MUC) is presented. By analogy with the usual KKR method the secular equation for an NCM is deduced. The symmetry of the problem is discussed on the basis of spin space groups. It is shown that the symmetry consideration often leads to the secular equation factorising anywhere in the Brillouin zone. The calculation of the electron spectrum of the spin spiral for BCC Fe has been carried out and the calculated energy of the spin wave is in good agreement with the experimental value.

A direct variational approach to Skyrme's model for meson fields

Abstract: The solutions of Skyrme's variational problem describe the structure of mesons in a field of weak energy. The problem consists in minimizing the corresponding energy among the functions from ℝ3 toS3 which have a fixed “degree” without making any symmetry assumptions. We prove the existence of minima and study their properties.

Quantum theory of the micromaser: Symmetry breaking via off-diagonal atomic injection.

Abstract: We propose and analyze an experiment in which atoms in a coherent superposition of atomic states are injected into a high-Q maser cavity. We show that the symmetry of the field in the cavity is ``broken'' in the same way as results from a classical signal injected into a laser cavity. This broken symmetry can be detected by monitoring the atomic excitation of a probe atom.

An introduction to anomalies

Abstract: These lectures are dedicated to the study of the recent progress and implications of anomalies in quantum field theory. In this introduction we would like to recapitulate some of the highlights in the history of the subject. In its original form,1 one considers a triangle diagram with two vector currents and an axial vector current. Requiring Bose symmetry and vector current conservation in the vector channels, one finds that the axial vector current is not conserved, leading to the breakdown of chiral symmetry in the presence of external gauge fields. The existence of this anomaly led to an understanding of π0 decay, and later on to the resolution of the U(1) problem2 in QCD. These anomalies correspond to the breakdown of global axial symmetries, and their existence does not jeopardize unitarity or renormalizability. More dangerous anomalies appear whenever chiral currents are coupled to gauge fields. For example, in four dimensions we can consider V-A currents coupled to gauge fields as in the standard Weinberg-Salam model, and compute the same triangle diagram with V-A currents on each vertex.3 Again one finds an anomaly, and unless the anomaly cancels after summing over all the fermion species, the theory will not be gauge invariant, implying a loss of renormalizability. If we recall the Feynman rules for non-Abelian gauge theories coupled to fermions in some representation Ta of the gauge group G, the anomaly for gauge currents is proportional to a purely group theoretic factor times a Feynman integral.

The anomaly structure of theories with external gravity

Abstract: The cohomology problem of the overall local symmetry group of theories with external gravity, including diffeomorphisms, local Lorentz, and gauge transformations, is studied, in order to determine all possible anomalies. To this end the nontrivial cohomology classes of the coupled system of two coboundary operators are classified in the abstract. Using this result and a technical assumption the nontrivial cohomology classes of the coboundary operator associated with diffeomorphisms are determined. These possible anomalies split in any dimension into two distinct families. Both are calculated (the second only in four dimensions). Using known results about gauge and local Lorentz anomalies, the possible anomalies of the overall local symmetry group are determined.

Symmetry-breaking for positive solutions of semilinear elliptic equations

Abstract: In a recent interesting paper, GIDAS, NI, and NIRENBERG [2] proved that positive solutions of the Dirichlet problem for second-order semi-linear elliptic equations on balls must themselves be spherically symmetric functions. Here we consider the bifurcation problem for such solutions. Specifically, we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution; when this happens, we say that the symmetry "breaks." To carry out this program, we rely on certain results in [6], where we studied the kernel of the associated linearized operator. This enables us to give some necessary conditions for symmetry to break. We also find a class of functionsf where these conditions are also sufficient; see equation (18) below. The problem is, of course, to show that, when zero comes into the spectrum of the linearized operator and our conditions are fulfilled, bifurcation into the non-radial direction actually occurs. This is done by showing first that the "bifurcation curve" is a smooth manifold near the bifurcation point, and then appealing to a theorem of VANDERBAUWHEDE [8], which gives sufficient conditions for bifurcation to occur in the presence of symmetries.

Secondary bifurcations of Taylor vortices into wavy inflow or outflow boundaries

Abstract: Experiments of Andereck et al. (1986) with corotating cylinders, show that Taylor-vortex flow (TVF) can bifurcate into one of the following cellular flows: wavy vortices (WV), twisted vortices (TW), wavy inflow boundaries (WIB), wavy outflow boundaries (WOB). We describe here the structure of these different flows, showing how they result from simple symmetry breaking. Moreover we consider the codimension-two situation where WIB and WOB interact, since this is an observed physical situation.The method used in this paper is based on symmetry arguments. It differs notably from the Liapunov-Schmidt reduction used in particular by Golubitsky & Stewart (1986) on the same problem with counter-rotating cylinders. Here we take into account all the dynamics, instead of restricting the study to oscillating solutions. In addition to the standard oscillatory modes, we have a translational mode due to the indeterminacy of TVF under the shifts along the axis. We derive an amplitude-expansion procedure which allows the translational mode to depend on time. Our amplitude equations have nevertheless a simple structure because the oscillatory modes have a precise symmetry. They break, in general, the rotational invariance and they are either symmetric or antisymmetric with respect to the plane z = 0. Moreover, the most typical cases are when either of these modes has the same axial period as TVF or when their axial period is double this. This leads to four different cases which are shown to give WV, TW, WIB or WOB, all these flows being ‘rotating waves’, i.e. they are steady in a suitable rotating frame.Finally we consider the interaction between WIB and WOB that occurs when, at the onset of instability, the two critical modes arise simultaneously. In this case we show in particular that there may exist a stable quasi-periodic flow bifurcating from WIB or WOB. The two main frequencies are those of underlying WIB and WOB, while there may exist a third frequency corresponding to a slow superposed travelling wave in the axial direction.The method was used in the counter-rotating case for interacting non-axisymmetric modes (see Chossat et al. 1986). One of the original contributions here is not only to clarify the origin of all observed bifurcations from TVF, but also to handle the translational mode which may not stay small. This technique combined with centre-manifold and equivariance techniques may be helpful for many problems starting with orbits of solutions, such as the TVF considered here.

The bifurcation and secondary bifurcation of capillary-gravity waves in the presence of symmetry

Abstract: The bifurcation and secondary bifurcation of capillary-gravity waves is analysed when the surface tension is close to or equal to a value where the eigenspace of the critical phase speed has multiplicity two. The existence and multiplicity of solutions is seen, via the implicit function theorem, to be a special case of the secondary bifurcation phenomena, which occur when a double eigenvalue splits, under perturbation, into two simple eigenvalues in the presence of a symmetry in the problem.

Progress in Quantum Field Theory

Abstract: This article reviews many important theoretical ideas and methods in quantum field theory during about twenty years.Of which there are the importance of symmetry theory,gauge field theory,symmetry breaking and origin of mass,quantizative methods,renormalization and renormalizative effects.

Random-phase-approximation study of the response function describing optical second-harmonic generation from a metal selvedge.

Abstract: Optical second-harmonic generation in centrosymmetric metals stems from the breaking of inversion symmetry in the selvedge and from nonlocal effects. In the present work a random-phase-approximation description of the nonlocal nonlinear conductivity tensor of an inhomogeneous jellium is given. Main emphasis is devoted to a study of the electric dipole response from the selvedge, and to the magnetic dipole and electric quadrupole responses from the profile region and the bulk. A detailed analysis of the tensor symmetry schemes associated with the p\ensuremath{\cdot}A and A\ensuremath{\cdot}A interactions is presented. Dipole and quadrupole transitions are identified, and the role of direct and indirect quantum processes is discussed. On the basis of the so-called infinite-barrier model, a few numerical results are given.