Bulk and boundary S-matrices for the SU(N) chain

We consider both closed and open integrable antiferromagnetic chains constructed with the SU(N)-invariant R-matrix. For the closed chain, we extend the analyses of Sutherland and Kulish — Reshetikhin by considering also complex “string” solutions of the Bethe ansatz equations. Such solutions are essential to describe general multiparticle excited states. We also explicitly determine the SU(N) quantum numbers of the states. In particular, the model has particle-like excitations in the fundamental representations [k] of SU(N), with k = 1, …, N − 1. We directly compute the complete two-particle S-matrices for the cases [1] ⊗ [1] and [1] ⊗ [N − 1 ]. For the open chain with diagonal boundary fields, we show that the transfer matrix has the symmetry SU(l) × SU(N−l) × U(1), as well as a new “duality” symmetry which maps l→N−l. With the help of these symmetries, we compute by means of the Bethe ansatz for particles of types [1] and [N − 1 ] the corresponding boundary S-matrices.     

Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in [Zhe Chang, Xin Li, Phys. Lett. B 668 (2008) 453] and [Zhe Chang, Xin Li, Phys. Lett. B 676 (2009) 173], respectively). They consider that “In Finsler manifold, there exists a unique linear connection – the Chern connection … It is torsion freeness and metric compatibility …”. There are well-known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections are not metric compatible. For instance, the Chern's one (being with zero torsion and “weak” compatibility on the base manifold of tangent bundle) is not generally compatible with the metric structure on total space. This results in a number of additional difficulties and sophistication in definition of Finsler spinors and Dirac operators and in additional problems with further generalizations for quantum gravity and noncommutative/string/brane/gauge theories. I conclude that standard physics theories can be generalized naturally by gravitational and matter field equations for the Cartan and/or any other Finsler metric compatible connections. This allows us to construct more realistic models of Finsler spacetimes, anisotropic field interactions and cosmology.     

Letter: Riemannian Metric of the Invariance Quantum Group of the Fermion Algebra

We calculate the bi-invariant metric of FIO(2), the inhomogeneous invariance quantum group of the fermion algebra. We find that this metric is identical to that of the bi-invariant metric of GL(2, R) ⁺ × SU (1, 1). However, the quantum group manifold is restricted to a region of the GL(2, R) manifold.     

The Hermitian Laplace Operator on Nearly Kähler Manifolds

The moduli space ${\mathcal {NK}}The moduli space NK{\mathcal {NK}} of infinitesimal deformations of a nearly K?hler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1, 1) forms (cf. Moroianu et al. in Pacific J Math 235:57–72, 2008). Using the Hermitian Laplace operator and some representation theory, we compute the space NK{\mathcal {NK}} on all 6-dimensional homogeneous nearly K?hler manifolds. It turns out that the nearly K?hler structure is rigid except for the flag manifold F(1, 2) = SU₃/T ², which carries an 8-dimensional moduli space of infinitesimal nearly K?hler deformations, modeled on the Lie algebra \mathfraksu₃{\mathfrak{su}_3} of the isometry group.     

Two-Dimensional Quantum Hamiltonians with Shape Invariance Symmetry

It is shown that the Casimir operator associated with the U(1) Lie derivative defined on the S ²=SU(2)/U(1) base manifold, can be interpreted as Hamiltonians of a pair of scalar particle and scalar anti-particle with opposite charges over the S ² manifold in the presence of a magnetic monopole located at its origin and an external electric field. Using the SU(2) representation, the spectra of these Hamiltonians have been obtained. It is also proved that these Hamiltonians are isospectral and having the shape invariance symmetry, i.e. they are supersymmetric partner of each other. Also the Dirac’s quantization of magnetic charge comes very naturally from the finiteness of the SU(2) representation.     

Poisson Cohomology of SU (2)-Covariant “Necklace” Poisson Structures on S 2

Abstract

We compute the Poisson cohomology of the one-parameter family of SU(2)-covariant Poisson structures on the homogeneous space S ²=?P ¹=SU(2)/U(1), where SU(2) is endowed with its standard Poisson–Lie group structure, thus extending the result of Ginzburg [2] on the Bruhat–Poisson structure which is a member of this family. In particular, we compute several invariants of these structures, such as the modular class and the Liouville class. As a corollary of our computation, we deduce that these structures are nontrivial deformations of each other in the direction of the standard rotation-invariant symplectic structure on S ²; another corollary is that these structures do not admit smooth rescaling.     

Changes in Vocal Fold Morphology During Singing Over Two Octaves

ObjectiveVocal folds are widely assumed to only elongate to raise vocal pitch. However, the mechanisms seem to be more complex and involve both elongation and tensioning of the vocal folds in series. The aim of the present study was to show that changes in vocal fold morphology depend on vocal fold elongation and tensioning during singing.Study designThis was a prospective study.MethodsForty-nine professional female singers (25 sopranos, 24 altos) were recruited and three-dimensional laryngeal images analyzed in a coronal view derived from high-resolution computed tomography scans obtained at the mean speaking fundamental frequency (ƒ₀) and one (2ƒ₀) and two octaves (4ƒ₀) above ƒ₀.ResultsThe vocal fold angle, defined by a tangent above and below the vocal folds, was 58° at ƒ₀, 47° at 2ƒ₀, and 59° at 4ƒ₀.ConclusionThe decreased caudomedial angle of the vocal fold from ƒ₀ to 2ƒ₀ (change in muscle belly from “;fat” to “thin”) and increased angle from 2ƒ₀ to 4ƒ₀ (from “thin” to “fat”) strongly supports the hypothesis that the vocal folds elongate and then tension when singing from ƒ₀ to 4ƒ₀. This is the first study to show this relationship in vivo.     

Duality and quantum-algebra symmetry of the AN−1(1) open spin chain with diagonal boundary fields

We show that the transfer matrix of the A_N−1⁽¹⁾ open spin chair with diagonal boundary fields has the symmetry U_q(SU(l)) × U_q(SU(N−l)) × U(1), as well as a “duality” symmetry which maps l ↔ N − l. We exploit these symmetries to compute exact boundary S-matrices in the regime with q real.     

Non‐Ideal Dust Structures in Cryogenic Complex Plasmas

The possibility of the formation of dust structures in cryogenic environment at 4.2‐77 K was proved experimentally in the previous researches of cryogenic complex (dusty) plasma [1–5]. It was revealed from the experiments, among others, that the dust structures with high concentration of dust particles can be formed, in which interparticle distance is comparable with particle size ‐ super dense dusty plasma structures. Such structures had exotic properties such as globular (spherical) form, free boundaries, etc. In the present work new results on the experimental investigations of new phenomenon of spheroidizing ‐ process of the dust structure transition to compact globular shape at cryogenic temperatures ‐ were presented. Possible nature of such phenomenon is discussed (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental Consequences of the Asymptotically Free SU(5)-Model

The experimental consequences of the asymptotically free SU(5)-theory are presented which make it possible to judge about its reliability. The phenomenology of the τ-lepton is analysed with special attention paid to the so-called exotic processes which must solve whether “light” or “heavy” the ν_τ-neutrino should be. These exotic processes are completely forbidden within standard SU(5) schemes. The problem of the finite mass of the e, μ-neutrino which is a four-component Dirac spinor interacting via the Zel'dovitch-Mahmound-Konopinski scheme is discussed. The finiteness of such neutrino mass is a peculiarity of the SU(5) theory under discussion, although its value is not fixed here in a unique way. The version of the theory wherein m_ν ≦ 10⁻³ eV is of especial interest since in this case the proton instability and the finite e, μ-neutrino mass are found to be explained in a common way as manifestations of the existence of superheavy leptoquark bosons in the theory. The possibility is pointed out to explain within the SU(5) theory under consideration the existing experimental situations concerning the search for parity violating effects in atoms. It is shown that the present SU(5) theory encounters no difficulty in this respect and can recommend itself to any experimental set of data, whereas the standard schemes of SU(5) unification cannot. Cosmological consequences of the asymptotically free SU(5) theory are briefly discussed and the latter is found to rather reasonably explain a number of facts about the very early evolution of the Universe.     

Distinguishing Three-Dimensional Lens Spaces L(7,1) and L(7,2) by Means of Classical Pentagon Equation

Abstract

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a “2 tetrahedra → 3 tetrahedra” local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.     

Exotic smoothness,noncommutative geometry,and particle physics

We investigate how exotic differential structures may reveal themselves in particle physics. The analysis is based on A. Connes' construction of the standard model. It is shown that, if one of the copies of the spacetime manifold is equipped with an exotic differential structure, a compact object of geometric origin may exist even if the spacetime is topologically trivial. Possible implications are discussed. AnSU(3) ⊗SU(2) ⊗U(1) gauge model is constructed. This model may not be realistic, but it shows what kind of physical phenomena might be expected due to the existence of exotic differential structures on the spacetime manifold.     

On some hydrodynamical aspects of quantum mechanics

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the complex polynomial (i.e. Borel-Weil) realization of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schrödinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly determine the critical points of the pressure of this “Schrödinger fluid”, together with its vorticity, which turns out to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.     

La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes

K.D. Kirchberg [Ki1] gave a lower bound for the first eigenvalue of the Dirac operator on a spin compact Kähler manifoldM of odd complex dimension with positive scalar curvature. We prove that manifolds of real dimension 8l+6 satisfying the limiting case are twistor space (cf. [Sa]) of quaternionic Kähler manifold with positive scalar curvature and that the only manifold of real dimension 8l+2 satisfying the limiting case is the complex projective spaceCP ^4l+1.     

The geometry of a covariant expansion in chiral theories with nucleons

We deal with the geometrical foundations of a covariant scheme, which is developed to give a co-ordinate independent perturbation expansion ofSU(2)×SU(2) chiral theories with pions and nucleons. The pion fields play the role of local co-ordinates on a 3-D manifold with constant curvature in isospace. The presence of the nucleon isospinor field forces us to deal with the problem of endowing the manifold with a spin structure. In this way the nucleon isospinor is accommodated in the fiber space of the principal fiber bundle of the tangent bundle of the manifold.     

Quotient space solutions of 11-dimensional supergravity

All the Freund-Rubin type solutions of the 11-dimensional supergravity with a simply connected quotient spaceG/H as the compact 7-dimensional manifold are found. Their geometries depend only on the imbedding ofHG and the Riemannian structure ofG/H. In particular, SU₃×SU₂×U₁/SU₂×U₁×U₁, Einstein solutions are discussed in detail.     

Quaternion Octonion Reformulation of Quantum Chromodynamics

We have made an attempt to develop the quaternionic formulation of Yang–Mill’s field equations and octonion reformulation of quantum chromo dynamics (QCD). Starting with the Lagrangian density, we have discussed the field equations of SU(2) and SU(3) gauge fields for both cases of global and local gauge symmetries. It has been shown that the three quaternion units explain the structure of Yang–Mill’s field while the seven octonion units provide the consistent structure of SU(3) _C gauge symmetry of quantum chromo dynamics.     

Nonlinear local supersymmetric SU(5)-model

We construct a realistic nonlinear supersymmetric SU(5) model without superpartner particles in curved space and investigate the Higgs sector in the limit of flat space. We need at least an adjoint-, a quintuplet- and an antiquintuplet-Higgs multiplet. In order to obtain a realistic breakdown of SU(5) to SU(3) × U(1) at tree level we have to modify the vacuum structure of the adjoint representation in comparison to other SU(5)-models. This model requires an “elegant” fine-tuning. We determine the mass spectrum of the electroweak Higgs sector.     

Search for rotational fine structure in the nucleus 184Hg

One possible explanation for the anomalous isotope shifts of ^183,185Hg is the sudden occurrence of permanent quadrupole deformations. The α-particle decay of 25 s ¹⁸⁸Pb, however, does not show any rotational fine structure. It is interesting to consider the possibility that the nuclei ^183–185Hg have “bubble” shape.