Subgroups Of SU Research Articles

Representations of anisotropic unitary groups

Let S U ( f ) SU(f) be the special unitary group of an anisotropic hermitian form f f over a field k k . Assume f f represents only one norm class in k k . The representations α : S U ( f ) → S L ( n , R ) \alpha :\,SU(f) \to SL(n,\,R) are characterized when R R is a commutative ring with 2 2 not a zero divisor and n = dim ⁡ f ⩾ 3 n = \dim f \geqslant 3 with n ≠ 4 , 6 n \ne 4,\,6 . In particular, a partial classification of the normal subgroups of S U ( f ) SU(f) is given when k k is the function field C ( X ) {\mathbf {C}}(X) .

Superinvariant chiral and vector fields

This work is concerned with the characterization of field supermultiplets on four-dimensional Minkowski space that are stable under the action of subgroups of the superconformal group SU(2,2/1). The most general scalar and vector superfields whose Lie derivative with respect to a fermionic tangent vector vanishes are determined. Invariance under subgroups of SU(2,2/1) with more than one odd generator is also discussed.

Nonlinear equations with superposition formulas and the exceptional group G2. II. Classification of the equations

Nonlinear equations with superposition formulas are obtained, corresponding to the action of the complex and real forms of the exceptional Lie group G2 on the homogeneous spaces G2/H. The isotropy group of the origin H is taken as one of the maximal parabolic subgroups of G2, or as one of the maximal reductive subgroups, leaving some vector space V∈C7 (or V∈R7) invariant. The parabolic subgroups, as well as the simple subgroups SL(3,C), SU(3), SL(3,R) or SU(2,1) lead to equations with quadratic or quartic nonlinearities. The semisimple subgroups SL(2,C)⊗SL(2,C), SU(2)⊗SU(2), and SU(1,1)⊗SU(1,1) lead to equations with quadratic nonlinearities and additional nonlinear constraints on the independent variables.

Classical and quantum mechanics on the unit ball in C n

In this paper we show that the complex unit ball B in C n is a strictly Hamiltonian SU(1, n) space and after reduction to the Heisenberg group (which is a subgroup of SU(1, n)) it consists of one parameter family of standard phase spaces (T ∗ R (n-1), − 4h μ dx k ∧ dp k) . We construct the quantum bundle over B and prove that the deformation of its holomorphic and metric structure gives the Hamiltonian on T ∗ R (n −1) . Moreover, we describe explicitly the coherent states and point out their connection with the path integrals.

Correspondence between the interacting boson model and the fermion dynamical symmetry model of nuclei.

The relation between the two kinds of dynamical symmetry models in nuclear structure physics, the well-known interacting boson model and the recently proposed fermion dynamical symmetry model, is studied from group theoretical point of view. A one-to-one correspondence is established between the generators, building blocks, irreducible basis vectors and the invariants of the subgroups SO(5), SO(6), and SU(3) contained in the dynamical group chains of the interacting boson model and those of the fermion dynamical symmetry model.

A Lie-theoretic setting for the classical interpolation theories

The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane.

Integrity bases for symmetry-conserving higher-order interaction terms in the interacting boson model

Using generating function techniques, the authors construct integrity bases for O(3) scalar operators in the enveloping algebra of G, where G is one of the subgroups of SU(6) in the three dynamical chains for the interacting boson model.

A grand unification preon model with E 6 metacolor

We have constructed a version of the chiral three-preon model E 6 × SO(10) that satisfies the condition of asymptotic freedom in the metacolor and composite color-flavor sectors. The construction is based on the global color-flavor symmetry group SU(18). By applying the 't Hooft anomaly matching condition to the subgroup SU(16) × SU(2) of SU(18), together with a few physical constraints, we obtain a unique solution that gives rise to precisely three generations of the spinorial representation 16 of SO(10) without exotics. Except for N = 18, no solution exists for the global color-flavor group satisfying metacolor asymptotic freedom ( N < 22) when SU( N) breaks to SU(16) × SU( N − 16).

Character table for the 1080-element point-group-like subgroup of SU(3)

The character table for S(1080), the largest point-group-like subgroup of SU(3), is presented. So is the reduction of the first 28 SU(3) characters to S(1080) characters. These tables are needed in a recently proposed systematic description of SU(3)-invariant lattice gauge theories by effective S(1080)-invariant theories. Problems encountered in an alternative systematics, using only local S(1080)-invariant theories, are discussed.

The E 6 ⊗ SO(10) preon model based on global SU(18) color-flavor

We have constructed a version of the chiral three-preon model E 6 ⊗ SO(10) based on the global color-flavor symmetry group SU(18). By applying the 't Hooft anomaly matching condition to the subgroup SU(16)×SU(2) of SU(18) together with a few physical constraints, we obtain a unique solution that gives rise to three generations of the spinorial representation 16 of SO(10) without exotics. Except for N = 18, no solution at all exists for the global color-flavor group SU( N) (16 < N < 22) when SU( N) breaks to SU(16)×SU( N−16).

Some consequences of global horizontal symmetry

It is pointed out that the present SU(3) c ×SU(2) L ×U(1) gauge interactions with three families have a global horizontal symmetry (denoted hereby SU(3) H ) which is broken only by the weak charged hadron currentJ h. Also, with (u, c), (d, s), (v e, {ie437-1}) and (e −,μ −) as doublets of SU(2) H (subgroup of SU(3) H ),J h has simple transformation properties under this subgroup. Amplitude relations, using SU(2) H symmetry, for hadronic leptonic and semileptonic decays are given.

Symmetry chains and adaptation coefficients

Given a symmetry chain of physical significance it becomes necessary to obtain states which transform properly with respect to the symmetries of the chain. In this article we describe a method which permits us to calculate symmetry-adapted quantum states with relative ease. The coefficients for the symmetry-adapted linear combinations are obtained, in numerical form, in terms of the original states of the system and can thus be represented in the form of numerical tables. In addition, one also obtains automatically the matrix elements for the operators of the symmetry groups which are involved, and thus for any physical operator which can be expressed either as an element of the algebra or of the enveloping algebra. The method is well suited for computers once the physically relevant symmetry chain, or chains, have been defined. While the method to be described is generally applicable to any physical system for which semisimple Lie algebras play a role we choose here a familiar example in order to illustrate the method and to illuminate its simplicity. We choose the nuclear shell model for the case of two nucleons with orbital angular momentum l=1. While the states of the entire shell transform like the smallest spin representation of SO(25) we restrict our attention to its subgroup SU(6)×SU(2)T. We determine the symmetry chains which lead to total angular momentum SU(2)J and obtain the symmetry-adapted states for these chains.

On the doublet/triplet splitting and intermediate mass scales in locally supersymmetric SO(10)

In the light of the doublet/triplet splitting, the possibilities for an intermediate mass scale in locally supersymmetric SO(10) are analysed. It is found that the subgroup SU(4) c × SU(2) L × SU(2) R and more generally left-right symmetric models are unlikely to survive as intermediate symmetries since they imply too large values of the weak mixing angle. An alternative model using the subgroup SU(3) c × U(1) L × U(1) R is discussed. Requirements from global SUSY preservation impose an extra constraint and predictions for the grand unification and the intermediate masses are obtained at M X ≈ 6 × 10 15 GeV and M I ≈ 10 12 GeV.

SO(18) family unification in eight-dimensional space-time

We investigate an SO(18) gauge theory in an instanton-induced compactified manifold M 4 × S 4. The instanton is embedded in a diagonal subgroup SU(2) 2+3 of SO(18) ⊃ SO(10) × SU(2) 1 × SU(2) 2 × SU(2) 3 × SU(2) 4. The effective gauge theory in Minkowski space M 4 is described by SO(10) × SU(2) 1 × Su(2) 4 × SU(5), and a set of chiral fermions are obtained in the harmonic expansion of one complex SO(18) spinor representation of Weyl fermions in eight dimensions. These fermions contain four ordinary families and four mirror families. Some crucial roles of the isometry SO(5) are clarified concerning the family problems and the low-energy phenomenology.

SU(3) ⊗ SU(2) ⊗ U(1) from D = 11 supergravity

In this paper we show that D = 11 supergravity admits an infinite discrete class of solutions having the phenomenological group SU(3) ⊗ SU(2) ⊗ U(1) as a symmetry of the internal space M 7. These solutions lead, in dimensional reduction, to SU(3) ⊗ SU(2) ⊗ U(1) gauge fields. In general all these spaces produce a complete breaking of supersymmetry except in one case where N = 2 supersymmetry survives. The parameter which classifies the solutions is a rational number q/ p which describes the embedding of the stability subgroup SU(2) ⊗ U(1) ⊗ U(1) of M 7 in SU(3) ⊗ SU(2) ⊗ U(1). For all q/ p ≠ 1 the holonomy group is SO(7) and all supersymmetries are broken. For q/ p = 1 the holonomy group is SU(3) and two supersymmetries survive. In this last case we can also find a solution with internal photon curl F αβγδ ≠ 0 . It breaks all sypersymmetries.

Three-photon coherence and SU (4) symmetries

The problem of three-photon coherence in four-level optical systems is solved using the unitary symmetries of SU (4). The nonlinear nutation appears as a rotation in one of the SU (2) subgroups of SU (4). Four constants of motion are identified.

Manifestly Broken Local Symmetries at LargeN

It is shown how, in the large-N limit of SU(N) gauge theory, a local symmetry can be truly broken. This is illustrated by use of the exact solution of the two-dimensional theory in the presence of a vanishingly small symmetry-breaking source term. The model is also solved at finite N in order to study the approach to N = infinity. It is found that any finite-dimensional subgroup of SU(infinity) can be broken.

Magnetic monopoles in SU(3) and SU(2) gauge theories without Higgs' scalars

We show that certain known singular solutions for pure SU(3) Yang-Mills theory carry magnetic charges with respect to both U(1) subgroups of SU(3). A topological characterisation in terms of monopoles is given to the SU(2) singular solutions of Wu and Yang.

Structure of electroweak interactions in case of many generations

The structure of any arbitrary SU(N) model of grand unification is analyzed with respect to the electroweak subgroup SU(2)/sub W/ x U(1). The renormalized value of the Weinberg angle appears to be the same as in the case of the SU(5) model. The possibility of introducing Cabibbo-type mixing parameters into the SU(N) model is shown as a result of diagonalization of the gauge fields of the theory. Some natural requirements allow the choice of a specific set of models. The content of the two SU(8) models 8+56+56+8 and 28+70+28 is presented. A possible Higgs sector for these models is considered.

Matter-coupled Yang-Mills system in Minkowski space. I. Invariant solutions in the presence of scalar fields

We obtain classical solutions to the Minkowskian field equations of a Yang-Mills system in the presence of an isotriplet of massless scalar fields with a quartic self-interaction. The gauge group is SU(2). The conformal properties of the equations allow their mapping on compactified Minkowski space [diffeomorphic to SU(2)\ifmmode\times\else\texttimes\fi{}U(1)] upon which SU(2,2) acts globally. The solutions are obtained by using as Ans\"atze fields invariant under certain subgroups of SU(2,2). The characterization of invariant fields in the context of gauge theories is reviewed and the theory is applied to explicitly construct the invariant Ans\"atze. The following (or combinations of the following) group actions on SU(2)\ifmmode\times\else\texttimes\fi{}U(1) are used: (i) Left $\mathrm{SU}{(2)}_{L}$ translations, (ii) right $\mathrm{SU}{(2)}_{R}$ translations, (iii) left action of the product $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{R}$, (iv) U(1) translations. A class of $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$-invariant solutions is found. In addition, all $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}{(1)}_{R}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$-invariant solutions (both real complex) are determined. These new solutions represent interaction modes for some of the previously found solutions to the pure Yang-Mills equations. They lead to everywhere regular configurations with finite energy on ordinary Minkowski space.