Showing papers on "Symmetry (physics) published in 1996"

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Abstract: Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.

Nonholonomic Mechanical Systems with Symmetry

Abstract: This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

Universal Formula for Noncommutative Geometry Actions: Unification of Gravity and the Standard Model

Abstract: A universal formula for an action associated with a noncommutative geometry, defined by a spectral triple $(A,H,D)$, is proposed It is based on the spectrum of the Dirac operator and is a geometric invariant The new symmetry principle is the automorphism of the algebra $A$ which combines both diffeomorphisms and internal symmetries Applying this to the geometry defined by the spectrum of the standard model gives an action that unifies gravity with the standard model at a very high energy scale

Self-assembly of block copolymers

Abstract: Recent advances in the application of self-consistent field theory permit the calculation of the free energy of ordered phases with any symmetry. Hence it is now possible to study, within this theory, the complex phases of self-assembling polymer systems starting from standard model polymer Hamiltonians.

The role of symmetry in fundamental physics.

Abstract: Until the 20th century principles of symmetry played little conscious role in theoretical physics. The Greeks and others were fascinated by the symmetries of objects and believed that these would be mirrored in the structure of nature. Even Kepler attempted to impose his notions of symmetry on the motion of the planets. Newton’s laws of mechanics embodied symmetry principles, notably the principle of equivalence of inertial frames, or Galilean invariance. These symmetries implied conservation laws. Although these conservation laws, especially those of momentum and energy, were regarded to be of fundamental importance, these were regarded as consequences of the dynamical laws of nature rather than as consequences of the symmetries that underlay these laws. Maxwell’s equations, formulated in 1865, embodied both Lorentz invariance and gauge invariance. But these symmetries of electrodynamics were not fully appreciated for over 40 years or more.

Expanding Relativistic Shells and Gamma-Ray Burst Temporal Structure

Abstract: Many models of gamma-ray bursts (GRBs) involve a shell expanding at extreme relativistic speeds. The shell of material expands in a photon-quiet phase for a period t0 and then becomes gamma-ray active, perhaps due to inhomogeneities in the interstellar medium or the generation of shocks. Based on kinematics, we relate the envelope of the emission of the event to the characteristics of the photon-quiet and photon-active phases. We initially assume local spherical symmetry wherein, on average, the same conditions prevail over the shell's surface within angles the order of Γ–1, where Γ is the Lorentz factor for the bulk motion. The contribution of the curvature to the temporal structure is comparable to the contribution from the overall expansion. As a result, GRB time histories from a shell should have an envelope similar to "FRED" (fast rise, exponential decay) events in which the rise time is related to the duration of the photon-active phase and the fall time is related to the duration of the photon-quiet phase. This result depends only on local spherical symmetry and, since most GRBs do not have such envelopes, we introduce the "shell symmetry" problem: the observed time history envelopes of most GRBs do not agree with that expected for a relativistic expanding shell.Although FREDs have the signature of a relativistic shell, they may not be due to a single shell, as required by some cosmological models. Some FREDs have precursors in which the peaks are separated by more than the expansion time required to explain FRED shape. Such a burst is most likely explained by a central engine; that is, the separation of the multiple peaks occurs because the central site produced multiple releases of energy on timescales comparable to the duration of the event. Alternatively, there still could be local spherical symmetry of the bulk material, but with a low "filling factor"; that is, only a few percent of the viewable surface (which is already very small, 4πΓ–2) ever becomes gamma-ray active.Long complex bursts present a myriad of problems for the models. The duration of the event at the detector is ~t0/(2Γ2). The long duration cannot be due to large t0, since it requires too much energy to sweep up the interstellar medium. Nor can it be due to small Σ if the time variation is due to ambient objects, since the density of such objects is unreasonable (~1018Γ–4 pc–3 for typical parameters). Long events must explain why they almost always violate local spherical symmetry or why they have low filling factors.Both precursor and long complex events are likely to be "central engines" that produce multiple releases of energy over ~100 s. One promising alternative scenario is one in which the shell becomes thicker than the radius of the curvature within Γ–1. Then it acts as a parallel slab, eliminating the problems associated with local spherical symmetry.

Symmetry, broken symmetry, and handedness in bimanual coordination dynamics

Abstract: The symmetrical dynamics of 1∶1 rhythmic bimanual coordination may be specified by an order parameter equation involving the relative phase between rhythmic components, and an interlimb coupling which determines the relative attractiveness of in-phase and anti-phase patterns. Symmetry breaking of these dynamics can occur via the difference in the natural frequencies, Δω, of the left and right rhythmic components, or by the intrinsic asymmetrical dynamics of the body. The latter is captured by additional terms that render the symmetrical coupling slightly anisotropic. A major prediction resulting from this step is that although Δω=0, as the frequency of coordination is increased, the asymmetrical coupling will increase and the symmetrical coupling will decrease. This results in a greater left-limb bias in left-handers and right-limb bias in right-handers. This “increased handedness” prediction was confirmed in an experiment in which 20 left-handed and 20 right-handed individuals performed 1∶ 1 coordination with hand-held rigid pendulums. Manipulations of left and right pendulum lengths controlled Δω, and the coupled frequency was determined by a metronome. Also confirmed was the prediction that the small shift in equilibria from in-phase and anti-phase due to the intrinsic asymmetry should be amplified in left-handers when Δω > 0 and in right-handers when Δω < 0. Further, the bias in left-handers was more consistent than the bias in right-handers, and a subgroup of right-handers was identified who performed similarly to left-handers. The coordination dynamics of functional asymmetry provides insights into the elementary synergy between the limbs, the dynamical mechanism that modulates it, and the nature of the asymmetry in left-handed and right-handed individuals.

On the stability of symplectic and energy-momentum

Abstract: This paper presents a detailed comparison of two implicit time integration schemes for a simple non-linear Hamiltonian system with symmetry: the motion of a particle in a central force field. The goal is to cstablisb analytical and numerical results pertaining to the stability properties of the implicit mid-point rule (the proto-typical implicit symplectic method) and a particular energy-momentum conserving scheme, and to compare the two schemes with respect to accuracy. While all results presented herein are withm the context of a simple model problem, the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non-linear solid mechanic.;: namely, the presence of large (and relatively slow) overall motions together with high-frequency internal motions. Dedicated to the memo O' of Juan Carlos Simo

Expanding relativistic shells and gamma-ray burst temporal structure

Abstract: Many models of gamma-ray bursts (GRBs) involve a shell expanding at extreme relativistic speeds The shell of material expands in a photon-quiet phase for a period t0 and then becomes gamma-ray active, perhaps due to inhomogeneities in the interstellar medium or the generation of shocks Based on kinematics, we relate the envelope of the emission of the event to the characteristics of the photon-quiet and photon-active phases We initially assume local spherical symmetry wherein, on average, the same conditions prevail over the shell's surface within angles the order of Γ–1, where Γ is the Lorentz factor for the bulk motion The contribution of the curvature to the temporal structure is comparable to the contribution from the overall expansion As a result, GRB time histories from a shell should have an envelope similar to "FRED" (fast rise, exponential decay) events in which the rise time is related to the duration of the photon-active phase and the fall time is related to the duration of the photon-quiet phase This result depends only on local spherical symmetry and, since most GRBs do not have such envelopes, we introduce the "shell symmetry" problem: the observed time history envelopes of most GRBs do not agree with that expected for a relativistic expanding shellAlthough FREDs have the signature of a relativistic shell, they may not be due to a single shell, as required by some cosmological models Some FREDs have precursors in which the peaks are separated by more than the expansion time required to explain FRED shape Such a burst is most likely explained by a central engine; that is, the separation of the multiple peaks occurs because the central site produced multiple releases of energy on timescales comparable to the duration of the event Alternatively, there still could be local spherical symmetry of the bulk material, but with a low "filling factor"; that is, only a few percent of the viewable surface (which is already very small, 4πΓ–2) ever becomes gamma-ray activeLong complex bursts present a myriad of problems for the models The duration of the event at the detector is ~t0/(2Γ2) The long duration cannot be due to large t0, since it requires too much energy to sweep up the interstellar medium Nor can it be due to small Σ if the time variation is due to ambient objects, since the density of such objects is unreasonable (~1018Γ–4 pc–3 for typical parameters) Long events must explain why they almost always violate local spherical symmetry or why they have low filling factorsBoth precursor and long complex events are likely to be "central engines" that produce multiple releases of energy over ~100 s One promising alternative scenario is one in which the shell becomes thicker than the radius of the curvature within Γ–1 Then it acts as a parallel slab, eliminating the problems associated with local spherical symmetry

Analytic and numerical study of preheating dynamics.

Abstract: We analyze the phenomenon of preheating, ie, explosive particle production due to parametric amplification of quantum fluctuations in the unbroken symmetry case, or spinodal instabilities in the broken symmetry phase, using the Minkowski space O({ital N}) vector model in the large {ital N} limit to study the nonperturbative issues involved We give analytic results for weak couplings and times short compared to the time at which the fluctuations become of the same order as the tree level terms, as well as numerical results including the full back reaction In the case where the symmetry is unbroken, the analytical results agree spectacularly well with the numerical ones in their common domain of validity In the broken symmetry case, interesting situations, corresponding to slow roll initial conditions from the unstable minimum at the origin, give rise to a new and unexpected phenomenon: the dynamical relaxation of the vacuum energy That is, particles are abundantly produced at the expense of the quantum vacuum energy while the zero mode comes back to almost its initial value In both cases we obtain analytically and numerically the equation of state which in both cases can be written in terms of an effective polytropic index that interpolatesmore » between vacuum and radiationlike domination We find that simplified analyses based on the harmonic behavior of the zero mode, giving rise to a Mathieu equation for the nonzero modes, miss important physics Furthermore, such analyses that do not include the full back reaction and do not conserve energy result in unbound particle production Our results rule out the possibility of symmetry restoration by nonequilibrium fluctuations in the cases relevant for new inflationary scenarios Estimates of the reheating temperature and the inconsistency of a kinetic approach to thermalization when a nonperturbatively large number of particles are created are discussed {copyright} {ital 1996 The American Physical Society}« less

Eckardt frame theory of interacting electrons in quantum dots

Abstract: A transformation to a moving frame (the Eckardt frame) is used to study the quantum states of interacting electrons in parabolic quantum dots in the presence of a perpendicular magnetic field. The approach is motivated by examining ground-state pair-correlation functions obtained by exact diagonalization. The main results concern the physical nature of the electron states and the origin of magic numbers. Some of the states are found to be localized about a single minimum of the potential energy. They have well-defined symmetry and are physically analogous to molecules. They are treated approximately by antisymmetrizing Eckardt frame rotational-vibrational states. This approach leads to selection rules that predict all the magic angular momentum and spin combinations found in previous numerical work. In addition, it enables the ground-state energy and low-lying excitations of the molecular states to be calculated to high accuracy. Analytic results for three electrons agree very well with the results of exact diagonalization. States that are not localized about a single minimum are also studied. They do not have distinct spatial symmetry and occur only when selection rules and conservation laws allow tunneling between states localized on different minima. These states appear to be small system precursors of fractional quantum Hall liquids. \textcopyright{} 1996 The American Physical Society.

A study of conical intersection effects on scattering processes: The validity of adiabatic single‐surface approximations within a quasi‐Jahn–Teller model

Abstract: Conical intersections between Born–Oppenheimer potential energy surfaces create singularities which are known to have a direct effect on the symmetry of the nuclear wave functions. In this article is presented a quasi‐Jahn–Teller model to study the symmetry effects of these singularities on nonreactive and reactive scattering processes. Applying this model, we were able to determine in what way and to what extent the conical intersection affects the relevant S‐matrix elements. Having the results of this study available, conclusions concerning more realistic systems were derived.

Numerical Evidence for Spontaneously Broken Replica Symmetry in 3D Spin Glasses

Abstract: By numerical simulations of the $3d$ Ising spin glass we find evidence that spontaneous replica symmetry breaking theory and not the droplet model describes with good accuracy the equilibrium behavior of the system.

Phonons as goldstone bosons

Abstract: The implications of the hidden, spontaneously broken symmetry for the properties of the sound waves of a solid are analyzed. Although the discussion does not go beyond standard wisdom, it presents some of the known results from a different perspective. In particular, I argue that, as a consequence of the hidden symmetry, the equations of motion for a sound wave necessarily contain nonlinear terms, describing phonon-phonon scattering and emphasize the analogy with the low energy theorems for pion-pion scattering.

Yukawa textures with an anomalous horizontal abelian symmetry

Abstract: The observed hierarchy of quark and lepton masses and mixings may be obtained by adding an abelian family symmetry to the Minimal Supersymmetric Model and coupling quarks and leptons to an electroweak singlet scalar field. In a large class of such models, this symmetry suffers from anomalies which must be compensated by the Green-Schwarz mechanism; this in turn fixes the electroweak mixing angle to be $\sin^2 \theta_W = 3/8$ at the string scale, without any assumed GUT structure. The analysis is extended to two distinct generalisations of the Standard Model: neutrino masses and mixings and R-parity violating interactions.

Geometric phase effects in H+O2 scattering. I. Surface function solutions in the presence of a conical intersection

Abstract: The general vector potential (gauge theory) approach for including geometric phase effects in accurate 3D quantum scattering calculations in hyperspherical coordinates is presented. A hybrid numerical technique utilizing both the DVR (discrete variable representation) and the FBR (finite basis representation) is developed. This method overcomes the singular behavior of the vector potential terms giving accurate surface function solutions to the complex Hermitian nuclear Schrodinger equation. The hybrid DVR/FBR technique is applied explicitly to HO2 for zero total angular momentum. The resulting complex surface functions include the geometric phase effects due to the C2v conical intersection. The O2 permutation symmetry is implemented to give real double‐valued surface functions which exhibit both even and odd symmetry. The surface function eigenvalues are compared to calculations without the geometric phase. The results indicate that geometric phase effects should be significant for H+O2 scattering even a...

Principle of nongravitating vacuum energy and some of its consequences.

Abstract: For Einstein's General Relativity (GR) or the alternatives suggested up to date the vacuum energy gravitates. We present a model where a new measure is introduced for integration of the total action in the D-dimensional space-time. This measure is built from D scalar fields $\varphi_{a}$. As a consequence of such a choice of the measure, the matter lagrangian $L_{m}$ can be changed by adding a constant while no gravitational effects, like a cosmological term, are induced. Such Non-Gravitating Vacuum Energy (NGVE) theory has an infinite dimensional symmetry group which contains volume-preserving diffeomorphisms in the internal space of scalar fields $\varphi_{a}$. Other symmetries contained in this symmetry group, suggest a deep connection of this theory with theories of extended objects. In general {\em the theory is different from GR} although for certain choices of $L_{m}$, which are related to the existence of an additional symmetry, solutions of GR are solutions of the model. This is achieved in four dimensions if $L_{m}$ is due to fundamental bosonic and fermionic strings. Other types of matter where this feature of the theory is realized, are for example: scalars without potential or subjected to nonlinear constraints, massless fermions and point particles. The point particle plays a special role, since it is a good phenomenological description of matter at large distances. de Sitter space is realized in an unconventional way, where the de Sitter metric holds, but such de Sitter space is supported by the existence of a variable scalar field which in practice destroys the maximal symmetry. The only space - time where maximal symmetry is not broken, in a dynamical sense, is Minkowski space. The theory has non trivial dynamics in 1+1 dimensions, unlike GR.

Noether symmetries in Bianchi universes

Abstract: We use our N\"other Symmetry Approach to study the Einstein equations minimally coupled with a scalar field, in the case of Bianchi universes of class A and B. Possible cases, when such symmetries exist, are found and two examples of exact integration of the equations of motion are given in the cases of Bianchi AI and BV.

Symmetries of discrete dynamical systems

Abstract: Differential–difference equations of the form un=Fn(t,un−1,un,un+1) are classified according to their continuous Lie point symmetry groups. It is shown that for nonlinear equations, the symmetry group can be at most seven‐dimensional. The integrable Toda lattice is a member of this class and has a four‐dimensional symmetry group.