non-Abelian Symmetry Group Research Articles

Principles of Symmetry, Dynamics and Spectroscopy

A Review of Matrix Algebra and Quantum Mechanics. Basic Theory and Applications of Symmetry Representations (Abelian Symmetry Groups). Basic Theory and Applications of Symmetry Representations (Non-Abelian Symmetry Groups). Theory and Applications of Higher Finite Symmetry and Induced Representations. Representations of Continuous Rotation Groups and Applications. Theory and Applications of Symmetry Representation Products (Finite Groups). Theory and Application of Symmetry Representation Products (Continuous Rotation Groups). Symmetry Analysis for Semiclassical and Quantum Mechanics: Dynamics with High Quanta. Appendices. Index.

Exploitation of the geometrical symmetry in the boundary element method with the group representation theory

We describe an implementation of the Boundary Element Method for the 2-D Laplace's problem where the domains under study present some geometrical symmetry. The boundary conditions do not share the symmetry so that intuitive reasoning cannot be used to take part of symmetry. So, we use the Group Representation Theory that presents a rationale in this context. It consists in reducing the original problem to a family of smaller ones, the global solution is obtained from superposition of the partial solutions. It leads to a substantial gain in memory volume and achieves large computational cost savings. We consider the non-abelian symmetry groups that represent the most general case. We present the case of the dihedral group D/sub 3/ as an example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Non-Abelian solitons in two-dimensional lattice field theories

For a class of self-interacting multicomponent scalar field theories with a global discrete non-Abelian symmetry group, mixed order–disorder correlation functions are defined in terms of Euclidean functional integrals. These correlation functions satisfy Osterwalder–Schrader positivity. From a representation of the correlation functions in terms of the transfer matrix, the dual algebra at fixed time is derived. This algebra implies parafermion operators showing non-Abelian braid group statistics. In a pure phase of spontaneous symmetry breaking for a related class of order–disorder correlation functions a convergent polymer representation is developed, emerging from a combined low- and high-temperature-type expansion. The infinite volume correlation functions of this class show exponential clustering in the disorder fields.

Phase structure of two-dimensional spin models and percolation

For a class of classical spin models in 2D satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially. The class contains discrete and continuous spin systems with Abelian and non-Abelian symmetry groups. For the discrete models our results imply that they show either long-range order or are in a soft phase characterized by powerlike decay of correlations; for the continuous models only the second possibility exists. The continuous models include a version of the plane rotator [O(2)] model; for this model we rederive, modulo two conjectures, the Frohlich-Spencer result on the existence of the Kosterlitz-Thouless phase in a very simple way. The proof is based on percolation-theoretic and topological arguments.

Lagrangian reduction and the double spherical pendulum

This paper studies the stability and bifurcations of the relative equilibrium of the double spherical pendulum, which has the circle as its symmetry group. The example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group.

Conformal field theories with nonabelian symmetry group

We discuss here exotic conformal field theories associated with the b− c systems on a class of algebraic curves with nonabelian symmetry group. In particular we derive explicitly the primary fields of these conformal field theories and their operator product expansions. The braiding relations between the primary fields show that they satisfy a nontrivial exotic statistics.

Symmetries on the moduli space of (2,2) superstring vacua

Symmetries of the space of (2,2) string vacua for c = 3, 6, 9 are discussed in the context of orbifoldized Landau-Ginzburg theories. A general method for finding the maximal symmetry groups on the moduli space of untwisted marginal operators is presented, by studying deformations of superpotentials. We give a complete discussion for the c = 3, 6 untwisted sectors. In particular we find non-abelian symmetry groups on the moduli space of c = 6 which are shown to correspond with automorphism groups of K3 surfaces. All these groups are subgroups of O(20, 4, Z). Some elements of these groups relate small volume target spaces to large ones.

Gauge and matter fields coupled to N = 2 supergravity

We consider the potential of a general matter system of N = 2 vector and scalar multiplets coupled to supergravity. For lagrangians that are initially quadratic in the matter fields we prove that the potential is either positive or unbounded from below. Our results have been obtained in the framework of a superconformal multiplet calculus, and we have verified that they can be derived from each of the three off-shell representations. As an example we consider SO(6) Yang-Mills theory coupled to scalar multiplets in the 10 + 10 representation, which, for suitably chosen parameters, leads to the potential of gauged N = 8 supergravity. We discuss possibilities for residual nonabelian symmetry groups after breaking of N = 8 super-symmetry to N = 1 or 2.

Fermi-Bose gas equivalence, non-abelian symmetry groups, phase transitions and all that

The equivalence triad of the two-dimensional massive Thirring model, sine-Gordon theory and the Coulomb gas is generalized to theories with non-abelian symmetry groups. The group and field theoretical properties of the resulting bosonic theories, connected with the characters of the corresponding groups, are investigated.

Asymptotic hadron mass spectrum with an internal non-abelian symmetry group

An expression for the asymptotic hadron mass spectrum with theSU(2) andSU(3) as internal symmetry groups is obtained in the context of the statistical bootstrap model. The result obtained withSU(2) symmetry is consistent with the one derived recently in the context of the bag model.

Twist as a Probe for Phase Structure

The use of twist is advocated as an efficient tool, in computer simulated lattice gauge theories, for a quick scan of phase diagrams and for determining phase structure.Physically, a twist is an excitation induced by a specific dislocation ("twisted boundary condition") in the system. We show that its effect can efficiently be measured and used to determine the thermodynamic phases of the system.An intriguing phenomenon occurs in case of nonabelian gauge theories: e.g. for SU(N), introducing a "simple" twist leaves the minimum action unchanged, (but changes the dimension of the zero action manifold), a double twist (two simple ones perpendicular to one another) (when N = 2 or 3) always raises it.Also reported are results obtained by applying this method to two-dimensional spin-spin systems, with the nonabelian symmetry groups O(3) and O(4). The results for O(4) are consistent with a disordered phase at all temperatures; those for O(3) are more difficult to interpret.

Appropriate linearizations of interacting theories with spontaneous symmetry breaking

Spontaneous breakdown of a symmetry group in a dynamical theory is connected with a rearrangement of group properties between quantized and classical degrees of freedom enforced by the classical asymmetry properties of the ground state. By disentangling the classical and quantized parts in the original basic field operators, designed for describing the short-distance behaviour, and by rearranging them in various ways, new dressed field operators with different symmetry properties, adapted to the description in the asymptotic region, can be constructed which, in general, admit more appropriate linearizations of the theory and hence offer better starting points for a perturbative treatment of the interaction. There are essentially two different dressing mechanisms of the basic field by which fields with particle interpretation are generated: an asymmetric dressing which relies strongly on the asymmetry of the ground state and a symmetric dressing which is only triggered by it. The formalism is exemplified for the nonlinear chiral model of Nambu and Jona-Lasinio. The main purpose of the paper, however, does not aim at improving the approximations in this model, but is thought of as a preparation for a similar treatment of more realistic theories with a spontaneous breakdown of a non-Abelian symmetry group and of the dilatation group in which these considerations may offer natural explanations for the occurrence of approximate higher symmetries and for the mysterious coexistence of two types of particles, the leptons and gauge particles on the one hand and the hadrons on the other hand.