Relations between the electromagnetic form factors of the 0−- and 1−-mesons from the collinear groupsSU(3)⊗SU(3)⊗U(1) andSU(6) W

The relations between the electromagnetic matrix elements of the 18 low lying pseudoscalar and vector mesons predicted by the collinear groupsSU(3)?SU(3)?U(1) andSU(6) _W are derived. Using the hermiticity of the electromagnetic current operator, charge conjugation invariance and invariance under the full Lorentz group andSU(3)-symmetry separately, all electromagnetic matrix elements of these mesons are described by seven independent real form factors. After combining space-time with intrinsic properties of the particles this number is reduced to four by the minimal collinear groupSU(3) ?SU(3) ?U(1) and to three bySU(6) _W which involves more speculative assumptions. In the limit of low momentum transfer the predictions of both models become identical, depending on three real quantities. No disagreement with experiment has been found, as far as a comparison is possible.     

Octonionic Non-Abelian Gauge Theory

We have made an attempt to describe the octonion formulation of Abelian and non-Abelian gauge theory of dyons in terms of 2×2 Zorn vector matrix realization. As such, we have discussed the U(1) _e ×U(1) _m Abelian gauge theory and U(1)×SU(2) electroweak gauge theory and also the SU(2) _e ×SU(2) _m non-Abelian gauge theory in term of 2×2 Zorn vector matrix realization of split octonions. It is shown that SU(2) _e characterizes the usual theory of the Yang Mill’s field (isospin or weak interactions) due to presence of electric charge while the gauge group SU(2) _m may be related to the existence of ’t Hooft-Polyakov monopole in non-Abelian Gauge theory. Accordingly, we have obtained the manifestly covariant field equations and equations of motion.     

Elektromagnetische Übergänge imSU(3)-Modell

A method is developed how to calculate electromagnetic transition probabilities and selection rules forSU(3)-classified wave functions. For this purpose the transition operators of electric multipole radiation are expanded inSU(3)-tensors. The calculations are explicitly carried out forE1- andE2-radiation. ForB(E2)-values in (λ,0)-bands an analytic expression is obtained independent of the special form of the interaction. The results ofSU(3)-theory are compared with those of Hartree-Fock-calculations performed by other authors. An astonishing agreement is found.     

Dynamical symmetries of two-dimensional systems in relativistic quantum mechanics

The two-dimensional Dirac Hamiltonian with equal scalar and vector potentials has been proved commuting with the deformed orbital angular momentum L. When the potential takes the Coulomb form, the system has an SO(3) symmetry, and similarly the harmonic oscillator potential possesses an SU(2) symmetry. The generators of the symmetric groups are derived for these two systems separately. The corresponding energy spectra are yielded naturally from the Casimir operators. Their non-relativistic limits are also discussed.     

Complex and quaternionic analyticity in chiral and gauge theories,I

A comparative and systematic study is made of 2-dimensional CP(n) σ-models and new 4-dimensional HP(n) σ-models and their respective embedded U(1) and Sp(1) holonomic gauge field structures. The central theme is complex versus quaternionic analyticity. A unified formulation is achieved by way of Cartan's method of moving frames adapted to the hypercomplex geometries of the harmonic symmetric spaces $\text{CP(n) ≈ SU(n + 1)}\text{SU(n) × U(1)}$ and $\text{HP(n) ≈ Sp(n + 1)}\text{Sp(n) × Sp(1)}$ respectively. Elements of complex Kähler manifolds are applied to a detailed analysis of the CP(n) σ-model and its instanton sector. Generalization to any Kählerian σ-model is manifest. On the basis of Cauchy-Riemann analyticity, Kählerian models are shown to have an infinite number of local continuity equations. In a parallel manner, new 4-dimensional conformally invariant HP(n) σ-models are constructed. Focus is on the latter's hidden local gauge invariance in their holonomy group Sp(n) × Sp(1) which allows a natural embedding of the Sp(1) ≈ SU(2) pure Yang-Mills theory. The associated quaternionic structure is discussed in light of both quaternionic quantum mechanics and Kählerian geometry. In this chiral setting, the SU(2) Yang-Mills duality equations are cast into quaternionic Cauchy-Riemann equations over S⁴ ≈ HP(1), the conformal spacetime. In analogy to the CP(n) case, their rational solutions are the most general (8n ? 3) parameter instantons where the associated algebraic nonlinear equations of the type of Atiyah, Drinfeld, Hitchin, and Manin are now expressed in a new conformally invariant form. Geometrically, the SU(2) instantons solve the Frenet-Serret equations for quaternionic holomorphic curves; they are conformal maps from HP(1) into HP(n) with n their second Chern index. Fueter's quaternionic analysis is presented, then applied: Fueter functions are particularly suited for the solutions of 't Hooft, of Jackiw, Nohl and Rebbi, and of Witten and Peng, as well as the self-dual finite action per unit time solution of Bogomol'nyi, Prasad and Sommerfield. Generalizing the latter, a new solution with unit Chern index and finite action per unit spacetime cell is found. It is expressed in terms of the quaternionic fourfold quasi-periodic Weierstrass Zeta function. Finally the essence of our method is revealed in terms of universal connections over Stiefel bundles; generalization to real, complex and quaternionic classifying Grassmanian σ-models with their embedded SO(m), SU(m) and Sp(m) gauge fields is outlined in terms of gauge invariant projector valued chiral fields. Other outstanding problems are briefly discussed.     

The construction of vector-like supersymmetric gauge models of weak and electromagnetic interactions

Despite great efforts and partial successes the situation with respect to spontaneously broken supersymmetric unified gauge models of weak, electromagnetic and strong interactions has remained quite unsatisfactory up to now. Starting from the most simple SU(2) × U(1) cases we exploit possible extensions. This naturally leads to a consideration of vector-like models with—in the first instance—a larger number of multiplets. Although the later can be made massive without spoiling the conservation of fermion number, the additional massive fermions only show parity conserving interactions with all the intermediate vector fields. Therefore models with larger gauge groups are considered: SU(2) × SU(2) × U(1) with two quartets, SU(3) × U(1) with four triplets, and finally SU(3) × SU(2) × U(1) with two sextets of matter fields. None of these can be accepted yet as a true model for physical particles, but it is shown how different negative features in the simple theories may be avoided in the more complicated ones. Thus our results may be considered as an encouraging starting point for investigations of larger gauge groups in supersymmetric models.     

Fractional charges and the renormalization group in theSU(4)×O(4) string model

We study the renormalization group equations of the gauge couplings in theSU(4)×O(4)～SU(4)×SU(2) _L ×SU(2_ _R string model, derived in the context of the free fermionic formulation of the four dimensional superstring. We calculate the effective string unification scale taking into account string threshold corrections and we consider the consequences of then _L andn _R fractionally charged states, sitting in the (1, 2, 1) and (1, 1, 2) representations correspondingly, of the gauge symmetry of the model. Some of these states become massive at a very high scale, when a number of singlet fields acquire vev's. However, many of them (the precise number depends on the specific choice of the flat direction) remain in the massless spectrum. We consider various cases and find that, for specific choices of flat directions, the physical parameters of the model, like the grand unification scale and the low energy parameters sin²θ _W and α₃, depend only on the differencen _?=n_L-n_R. We study more general cases where remnants of the exotic doublets remain below theSU(4) breaking scale. We also solve the coupled differential system of the renormalization group equations for the gauge and the Yukawa couplings and estimate the range of the top quark mass which is found to lie in the range 140 GeV<m _t<190GeV.     

SO(N) supergravity and unification of all fundamental forces

From the group theoretical arguments, we find that among allSO(N) supergravitiesN=10 is the minimal supersymmetry group which unifies all fundamental forces of weak, electromagnetic, strong and gravitational interactions. The (super)symmetry is broken throughSO(10)→SU(3)?SU(2)?U(1)→SU(3)?U(1). All observed particles of the low energy physics (three generations of quarks and leptons, γ,Z, W ^± and gluons) and graviton can be minimally accomodated with the correctSU(3)?SU(2)?U(1) quantum numbers. Some characteristic predictions, which can be checked in the coming high energy experiments, are briefly discussed.     

Semi-classical spectrum of the homogeneous sine-Gordon theories

The semi-classical spectrum of the homogeneous sine-Gordon theories associated with an arbitrary compact simple Lie group G is obtained and shown to be entirely given by solitons. These theories describe quantum integrable massive perturbations of Gepner's G-parafermions whose classical equations-of-motion are non-abelian affine Toda equations. One-soliton solutions are constructed by embeddings of the SU(2) complex sine-Gordon soliton in the regular SU(2) subgroups of G. The resulting spectrum exhibits both stable and unstable particles, which is a peculiar feature shared with the spectrum of monopoles and dyons in N = 2 and N = 4 supersymmetric gauge theories.     

The quantum 2-sphere as a complex quantum manifold

We describe the quantum sphere of Podles for c = 0 by means of a stereographic projection which is analogous to that which exibits the classical sphere as a complex manifold. We show that the algebra of functions and the differential calculus on the sphere are covariant under the coaction of fractional transformations with SU _q(2) coefficients as well as under the action of SU _q(2) vector fields. Going to the classical limit we obtain the Poisson sphere. Finally, we study the invariant integration of functions on the sphere and find its relation with the translationally invariant integration on the complex quantum plane.     

Automorphic Functions,Poincaré Series,and Conformal Fields on Riemann Surfaces

Poincaré series and automorphic functions for SU(1, 1) and a discrete subgroup Γ are studied with harmonic analysis. We consider automorphic functions on the open unit circle with general “spin label” m and their decomposition into irreducible automorphic functions by means of the Plancherel formula. These automorphic functions are bijectively mapped onto automorphic distributions on the boundary of the unit circle by meam of the Poisson kernel. The exponent of convergence of Poincaré series is expressed in representation theory language. The results are applied to two-point functions of conformal fields.     

Vector boson scattering ate + e − colliders as a test ofW ±,Z 0, γ self-interactions

We investigate the potential of a high-energye ⁺ e ^? collider (e.g., CLIC,E _e ⁺=E _e ^?=1 TeV) for determining trilinear and quadrilinear vector boson self-interactions in various vector boson scattering processes which can be measured in reactions of the type \(e^ + e^ - \to (e^ + e^ - ,ve^ - ,\bar ve^ + ,v\bar v) + VV'\) . Our analysis is based upon a recently suggested single parameter Lagrangian model for vector boson self-interactions incorporating globalSU(2) weak isospin symmetry broken by electromagnetism and a minimal increase of vector boson scattering tree amplitudes as a function of the energy. The results are compared with theSU(2) _L ×U(1) _Y predictions for the cases of a light and a heavy Higgs boson. We find that the crosssection for the production of a vector boson pair,VV′, is very sensitively dependent on the magnitude of the single free parameter, κ, the anomalous magnetic dipole moment of theW ^±. The cross-section changes by approximately one order of magnitude, even near production threshold, if κ is varied by one unit around theSU(2) _L ×U(1) _Y value of κ=1.     

Spectral analysis and the Haar functional on the quantumSU (2) group

The Haar functional on the quantumSU(2) group is the analogue of invariant integration on the groupSU(2). If restricted to a subalgebra generated by a self-adjoint element the Haar functional can be expressed as an integral with a continuous measure or with a discrete measure or by a combination of both. These results by Woronowicz and Koornwinder have been proved by using the corepresentation theory of the quantumSU(2) group and Schur's orthogonality relations for matrix elements of irreducible unitary corepresentations. These results are proved here by using a spectral analysis of the generator of the subalgebra. The spectral measures can be described in terms of the orthogonality measures of orthogonal polynomials by using the theory of Jacobi matrices.     

Obtaining non-Abelian field theories via the Faddeev-Jackiw symplectic formalism

In this Letter we construct non-Abelian field theories employing the Faddeev-Jackiw symplectic formalism. The original Abelian fields were modified in order to introduce the non-Abelian algebra. We construct the SU(2) and SU(2)⊗U(1) Yang-Mills theories having as starting point the U(1) Maxwell electromagnetic theory.     

Die Klassifikation nach inneren Symmetrien beim quantenmechanischen Mehrkörperproblem

For systems ofn identical particles with harmonic two-body interaction a method is derived which allows for anyn the construction of a complete orthonormal set of functions that are translationally invariant and are classified according to energy, permutational symmetry,SU (3), angular momentum, and parity. Moshinsky's method for the determination of translationally invariant states with definite permutational symmetry for harmonic two-body interaction is briefly reviewed and is extended to additionalSU (3)-classification. His method, however, is seen to be restricted to the casesn=3,n=4.     

Quadrupole perturbation of the harmonic oscillator potential

TheSU(3) symmetry of the harmonic oscillator potential is used to calculate exactly the second order energy correction, caused by quadrupole deformations. This can be done in a purely algebraic manner, using the Dalgarno-Schwartz formulation of perturbation theory. Some connections of this method with group theory are discussed and an extension to more general situations is proposed.     

A model for the u-d mass difference

Extending the conditions which lead to natural symmetries we show that m_d?m_u can be made positive in an SU(2)_L × SU(2)_R × U(1) gauge model for the weak and electromagnetic interactions.     

Phase portrait ofSU(5) grand unified model

The one-loop effective potential of theSU(5) model is investigated both in high and low temperature approximation. We find the regions of values of coupling constants and temperatures where theSU(5), theSU(4)×U(1) and theSU(3)×SU(2)×U(1) symmetric states are metastable. A general method of such an investigation is proposed. We observe that the domain structure of the Universe with the simultaneous existence of the gauge symmetriesSU(4)×U(1) andSU(3)×SU(2)×U(1) in different domains could take place.     

利用SUq(2)量子代数的q变形振子实现讨论SUq(2)相干态

利用SU_q(2)量子代数的q变形振子实现构造出SU_q(2)的相干态。证明SU_q(2)代数的表示基是正交的,并讨论了它的相干态的归一性、完闭性。指出SU_q(2)相干态的相干性受q参数影响较大,它比通常的SU(2)相干态更具有一般性。 关键词：