Action Of Symmetry Group Research Articles

Nonlinear realizations, the orbit method and Kohnʼs theorem

The orbit method is used to describe the centre-of-mass motion of the system of particles with fixed charge-to-mass ratio moving in homogeneous magnetic field and confined by harmonic potential. The nonlinear action of symmetry group on phase space is identified and compared with the one obtained with the help of the Eisenhart–Duval lift.

Enhanced inflation in the Dirac-Born-Infeld framework

We consider the Einstein equations within the Dirac-Born-Infield (DBI) scenario for a spatially flat Friedmann-Robertson-Walker (FRW) spacetime without a cosmological constant. We derive the inflationary scenario by applying the symmetry transformations which preserve the form of the Friedmann and conservation equations. These form invariance transformations generate a symmetry group parametrized by the Lorentz factor $\ensuremath{\gamma}$. We explicitly obtain an inflationary scenario by the cooperative effect of adding energy density into the Friedmann equation. For the case of a constant Lorentz factor, and under the slow roll assumption, we find the transformation rules for the scalar and tensor power spectra of perturbations as well as their ratio under the action of the form invariance symmetry group. Within this case and due to its relevance for the inflationary paradigm, we find the general solution of the dynamical equations for a DBI field driven by an exponential potential and show a broad set of inflationary solutions. The general solution can be split into three subsets and all these behave asymptotically as a power-law solution at early and at late times.

Group foliation of equations in geophysical fluid dynamics

The method of group foliation can be used to construct solutions to a system of partial differential equations that, as opposed to Lie's method of symmetry reduction, are not invariant under any symmetry of the equations. The classical approach is based on foliating the space of solutions into orbits of the given symmetry group action, resulting in rewriting the equations as a pair of systems, the so-called automorphic and resolvent systems, involving the differential invariants of the symmetry group, while a more modern approach utilizes a reduction process for an exterior differential system associated with the equations. In each method solutions to the reduced equations are then used to reconstruct solutions to the original equations. We present an application of the two techniques to the one-dimensional Korteweg-de Vries equation and the two-dimensional Flierl-Petviashvili (FP) equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest.

A first approximation for quantization of singular spaces

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group–that commutes with reduction–can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.

EXAMPLES OF FORCED SYMMETRY-BREAKING TO HOMOCLINIC CYCLES IN THREE-DIMENSIONAL EUCLIDEAN-INVARIANT SYSTEMS

We study perturbations of cubic planforms, proving there exists perturbations with homoclinic cycles between persistent steady states. Our results do not depend on the representation of the symmetry group of the lattice, and are thus quite general. The problem is studied using group theory rather than direct methods. We use the abstract action of the symmetry group of the perturbation on the group orbit to determine the existence of zero- and one-dimensional flow-invariant subspaces. The residual symmetry of the perturbation constrains the flows on these subspaces and, in certain cases, homoclinic cycles are guaranteed to exist. Cubic planforms are physically interesting due to their relevance to certain physical systems. Applications to reaction–diffusion systems, nonlinear optical systems and the polyacrylamide methylene blue oxygen reaction are discussed.

Quantum field theory and its symmetry reduction

The relation between symmetry reduction before and after quantization of a field theory is discussed using a toy model: the axisymmetric Klein–Gordon field. We consider three possible notions of symmetry at the quantum level: invariance under the group action, and two notions derived from imposing symmetry as a system of constraints in the manner of Dirac, reformulated as a first class system. One of the latter two turns out to be the most appropriate notion of symmetry in the sense that it satisfies a number of physical criteria, including the commutativity of quantization and symmetry reduction. Somewhat surprisingly, the requirement of invariance under the symmetry group action is not appropriate for this purpose. A generalization of the physically selected notion of symmetry to loop quantum gravity is presented and briefly discussed.

Surfaces having no regular hypermaps

The central question of this paper is the Genus Question: For which N is it possible to draw a regular map or hypermap on the non-orientable surface of characteristic −N? We answer this question for all N from −1 to 50, and we display a body of theorems and techniques which can be used to settle the question for more complicated surfaces. These include: two ways to diagram an action of symmetry group, an equivalence relation on vertices (rotation centers in general), several applications of Sylow theory, and some non-Sylow observations on the size of the symmetry group.

Symmetry and gauge freedom

The classical field theories that underlie the quantum treatments of the electromagnetic, weak, and strong forces share a peculiar feature: specifying the initial state of the field determines the evolution of some degrees of freedom of the theory while leaving the evolution of some others wholly arbitrary. This strongly suggests that some of the variables of the standard state space lack physical content—intuitively, the space of states of such a theory is of higher dimension than the corresponding space of genuine physical possibilities. The structure of such theories can helpfully be characterized in terms of the action of symmetry groups on their space of states; and the conceptual problems surrounding their strange behavior can be sharpened in light of the observation that it is usually possible to eliminate the redundant variables associated with these symmetries—which turn out to be precisely those variables whose evolution is unconstrained by the dynamical laws of the theory. This paper discusses this approach, uses it to frame questions about the interpretation of classical gauge theories, and to reflect (pessimistically) on our prospects of reaching satisfactory answers to these questions.

Enhanced Gauge Symmetry and Braid Group Actions

Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi--Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible fashion. Extensions of this result are given for nontrivial $B$-fields on K3 surfaces, explaining physical restrictions on the $B$-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local $A_{2n}$-configuration in a threefold having global $\Z/2$ monodromy.

The principle of symmetric criticality in general relativity

We consider a version of Palais' principle of symmetric criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. Given a group action on a space of fields, PSC asserts that for any group-invariant Lagrangian, the equations obtained by restriction of Euler–Lagrange equations to group-invariant fields are equivalent to the Euler–Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric and show that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in the context of transverse symmetry group actions, provides a generalization of the well-known unimodularity condition that arises in spatially homogeneous cosmological models. The other condition seems to be new. These results are illustrated with a variety of examples from general relativity.

Homogeneous Bent Functions, Invariants, and Designs

We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are the six variable degree three functions constructed in l16r. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be a matrix group of order 10752. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.

Rotational-vibrational relative equilibria and the structure of quantum energy spectrum of the tetrahedral molecule P4

We find relative equilibria (RE) of the rotating and vibrating tetrahedral molecule P4 and study the correspondence of these RE's to the extremal quantum states in the vibration-rotation multiplet and to the extrema of the semi-quantum rotational energy surfaces obtained for a number of excited vibrational states. To compute the energy of RE's we normalize the full rotation-vibration Hamiltonian H of P4 in the approximation of nonresonant modes ν E 2 and ν F_2 3 and find the stationary points of the resulting normal form (known as reduced effective Hamiltonian H eff ) which is defined on the reduced phase space CP 2 × CP 1 × S 2 . Most of these points are fixed points of the symmetry group action on CP 2 × CP 1 × S 2 . To explain our results in more detail we introduce numerical values of the parameters of H, such as the cubic force constants, using an atom-atom harmonic potential with one adjustable parameter. This simple model gives correct qualitative description of the rotational structure of the lowest excited vibrational states ν 2 , ν 3 and ν 2 + ν 3 of P4.

Spacetime as Manifold of Internal Symmetry Orbits in External Symmetries

Interactions and particles in the standard model are characterized by the action of internal and external symmetry groups. The four symmetry regimes involved are related to each other in the context of induced group representations. In addition to Wigner's induced representations of external Poincare group operations, parametrized by energy-momenta, and the induced internal hyperisospin representations, parametrized by the standard model Higgs field, the external operations, including the Lorentz group, can also be considered to be induced by the internal operations of the hypercharge–isospin group. In such an interpretation nonlinear spacetime is parametrized by the orbits of the internal action group in the external action group.

Renormalizing rectangles and other topics in random matrix theory

We consider random Hermitian matrices made of complex or real $M\times N$ rectangular blocks, where the blocks are drawn from various ensembles. These matrices have $N$ pairs of opposite real nonvanishing eigenvalues, as well as $M-N$ zero eigenvalues (for $M>N$.) These zero eigenvalues are ``kinematical" in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large $N,M$ limit, in which the ``rectangularity" $r={M\over N}$ is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large $N$ renormalization techniques. In addition to the kinematical $\delta$-function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if $|r-1|$ is held fixed as $N\rightarrow\infty$, the $N$ non-zero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the non-zero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. As $r\rightarrow 1$ the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal that $r\rightarrow 1$ drives a cross over to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.

Inversion-subversion of short DNA sequences by a nuclear vesicle on a kinked chromatid

The paper discusses a mechanism whereby symmetry group actions synergised by a nuclear vesicle and a kink yield either single or double inversions and/or other misorientations of relatively short DNA sequences and genes on small chromatid sections.

Action of Symmetry Groups

This paper studies maps which are invariant under the action of the symmetry group S,. The problem originates in social choice theory: there arc k individuals each with a space of preferences X, and a social choice map phi:Xk - X which is anonymous i.e. invariant under the action of a group of symmetries. Theorem 1 proves that a full range map Psi: Xk -> X exists which is invariant under the action of S, only if, for all i>1, the elements of the homotopy group IL(X) have orders relatively prime with k. Theorem 2 derives a similar results for actions of subgroups of the group Sk. Theorem 3 proves necessary and sufficient condition for a parafinite CW complex X to admit full range invariant maps for any prime number k:X must be contractible.

Self-dual Yang-Mills fields ind=4 and integrable systems in 1?d?3

The Ward correspondence between self-dual Yang-Mills fields and holomorphic vector bundles is used to develop a method for reducing the Lax pair for the self-duality equations of the Yang-Mills model ind=4 with respect to the action of continuous symmetry groups. It is well known that reductions of the self-duality equations lead to systems of nonlinear differential equations in dimension 1≤d≤3. For the integration of the reduced equations, it is necessary to find a Lax pair whose compatibility conditions is these equations. The method makes it possible to obtain systematically a Lax representation for the reduced self-duality equations. This is illustrated by a large number of examples.

Breaking of permutation symmetry and diagonal group action: Nielsen model and the standard model as low-energy limit

We point out that, on the basis of a theorem by L. Michel, every model based on the symmetryG ⊗ ... ⊗G has a generic low-energy limit the breaking to diagonal action, i.e., toG. This applies in particular to the proposal by Nielsen to consider such a model withG the standard model group.

Galerkin projections and the proper orthogonal decomposition for equivariant equations

In this paper we are interested in Galerkin projections of certain partial differential evolution equations, that retain the symmetries of the original nontruncated equations. In particular, we extend several results concerning the Galerkin projections, based on the proper orthogonal decomposition (which is also known as the Karhunen-Loève expansion) for equations which are equivariant under the action of compact Abelian symmetry groups. We provide sufficient conditions for numerical schemes, based on the linear and the nonlinear Galerkin method, to retain the symmetries of the original system. We perform a rational analysis of methods which use symmetry to enlarge the ensemble size of a data set. Our analysis has the potential for computational savings.

A geometrical approach to the Hamilton-Jacobi form of dynamics and its generalizations

The paper under review is devoted to the task of expressing the principles of classical dynamics in a coordinate-free manner and aims to show that the Hamilton-Jacobi formalism is as complete as the Hamiltonian or Lagrangian treatment of dynamical systems. This concerns mainly the incorporation of symmetry group actions in Hamilton-Jacobi theory. Stated in geometrical form, the unknown in this theory becomes a Lagrangian submanifold in a phase space, instead of a function over the configuration manifold in the ordinary formulation. The time-dependent Hamilton-Jacobi theory is also cast in this form after promoting time and its conjugate variable to the level of dynamical variables. Once this is done the analysis copies that of the time-independent case. The paper ends with interesting comments on the relationship between the Hamilton-Jacobi theory and Schrodinger's quantum mechanics. Plenty of other comments, examples and remarks are spread throughout the text. The exposition is quite informative and lucid.