General relativity and general Lorentz-covariance

The principle of general relativity means the principle of generalLorentz-covariance of the physical equations in the language of tetrads and metrical spinors. A generalLorentz-Covariant calculus and the generalLorentz-covariant generalisations of the Ricci calculus and of the spinor calculus are given. The generalLorentz-covariant representation implies theEinstein principle of space-time covariance and allows the geometrisation of gravitational fields according toEinstein's principle of equivalence.     

Infinite-dimensional representations of the Einstein group

The representations of the general linear group GL(4.R) are described. This group corresponds to the space-time transformations discussed in the general theory of relativity. Besides the well-known tensor representations, the group is also characterized by infinite-dimensional representations with integral and half-integral spins. This fact opens up a natural possibility, in principle, of constructing a covariant theory of particle fields. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 5, 309–312 (10 September 1996)     

Einstein<Emphasis Type="Italic">A</Emphasis>-coefficients for rotational transitions in the ring-chain isomer of C<Subscript>5</Subscript>H<Subscript>2</Subscript>

Laboratory formation of four isomers of C⁵H² molecule is reported and detection of the ring-chain isomer (isomer 1) of C₅H₂ in cosmic objects has been suggested. For identification of a molecule in cosmic objects, one of the required input data is EinsteinA-coefficients (radiative transition probabilities) for the molecule. Here, we report EinsteinA-coefficients for electric dipole transitions in the ring-chain isomer of C₅H₂ among the rotational levels of the ground electronic and ground vibrational states up to 21 cm⁻¹.     

Tensor analysis and curvature in quantum space-time

Introducing quantum space-time into physics by means of the transformation language of noncommuting coordinates gives a simple scheme of generalizing the tensor analysis. The general covariance principle for the quantum space-time case is discussed, within which one can obtain the covariant structure of basic tensor quantities and the motion equation for a particle in a gravitational field. Definitions of covariant derivatives and curvature are also generalized in the given case. It turns out that the covariant structure of the Riemann-Christoffel curvature tensor is not preserved in quantum space-time. However, if the curvature tensor _v (z) is redetermined up to the value of theL ² term, then its covariant structure is achieved, and it, in turn, allows us to reconstruct the Einstein equation in quantum space-time.     

On conformal covariance of spinor field equations

It is shown that the equation γ^μ_?μψ (χ) = 0 is covariant with respect to proper conformal transformations if the four-component spinor field ψ(χ) is a Cartan conformal semispinor of the first or second kind. If, instead, ψ is a Dirac spinor, then no less than two such equations are needed in a covariant field theory where the conformal group may be represented linearly. Some consequences are outlined.     

双星放射引力辐射的阻尼力

本文计算了双星的辐射阻尼。按照广义相对论,在广义坐标变换下,星体的运动方程中出现的力是与坐标的选择有关的。不同坐标条件将原则上导致不同的阻尼力。现在人们已经成功地观察到双星辐射阻尼效应。这表明应该找出一个物理的坐标系。在这个坐标系中算出的阻尼力等于观察到的值。我们提出下面物理条件代替通常的坐标条件。这个条件就是不存在时间方向和纵向极化的引力波。这个条件在所要求的近似下决定了我们认为是物理的坐标系。在这个坐标系中得出双星公转周期p的变化为P=-6.5×10^-12(m₁m₂)/M_⊙²((m₁+m₂)/m_⊙)^-1/3。 关键词：     

“Superconducting” causal nets

The world is described as a relativistic quantum neural net with a quantum condensation akin to superconductivity. The sole dynamical variable is an operator representing immediate causal connection. The net enjoys a quantum principle of equivalence implying local LorentzSL(2,C) invariance and causality. The past-future asymmetry of its cell is similar to that of the neutrino. A net phase transition is expected at temperatures on the order of theW mass rather than the Planck mass, and near gravitational singularities.     

On the covariance of the Fokker-Planck equation

We study the covariance of the Fokker-Planck equation under general gross variables transformations by means of Riemann differential geometry using an affine connection Γ^?_μν unsymmetric in their lower indices and without assuming that the covariant derivative of the diffusion tensor D^μν_;? be zero. We come to the conclusion that to achieve our aim we only need the value of the contraction Γ^?_?ν, all other components of the affine connection remaining completely arbitrary. We argue, therefore, that the most economic way of presenting the covariance of the Fokker-Planck equation is by means of exterior differential calculus. As an application we study physical systems under detailed balance showing that for them the irreversible part of the contravariant drift vector, that is then uniquely determined, is related only to a symmetric tensor while its reversible component is exclusively related to an antisymmetric tensor. A criticism of a compact Fokker-Planck equation is also included.     

Covariant,algebraic, and operator spinors

We deal with three different definitions for spinors: (I) thecovariant definition, where a particular kind ofcovariant spinor (c-spinor) is a set of complex variables defined by its transformations under a particular spin group; (II) theideal definition, where a particular kind of algebraic spinor (e-spinor) is defined as an element of a lateral ideal defined by the idempotente in an appropriated real Clifford algebra _p,q (whene is primitive we writea-spinor instead ofe-spinor); (III) the operator definition where a particular kind of operator spinor (o-spinor) is a Clifford number in an appropriate Clifford algebra _p,q determining a set of tensors by bilinear mappings. By introducing the concept of spinorial metric in the space of minimal ideals ofa-spinors, we prove that forp+q5 there exists an equivalence from the group-theoretic point of view among covariant and algebraic spinors. We also study in which senseo-spinors are equivalent toc-spinors. Our approach contain the following important physical cases: Pauli, Dirac, Majorana, dotted, and undotted two-component spinors (Weyl spinors). Moreover, the explicit representation of thesec-spinors asa-spinors permits us to obtain a new approach for the spinor structure of space-time and to represent Dirac and Maxwell equations in the Clifford and spin-Clifford bundles over space-time.     

Einstein A-coefficients for rotational transitions in the HN2O+

EinsteinA-values for the electric dipole transitions between the rotational levels up to 540 cm⁻¹ andJ=11 in the ground vibrational state of the protonated N₂O (i.e., HN₂O⁺) are calculated. The coefficients are used to compute the mean radiative lifetimes of the levels. TheseA-values can be used for analysing the spectra from astronomical objects, if observed.     

Direct gauging of the poincaré group. III. Interactions with internal symmetries

The problem of gauging matter fields with a Poincaré invariant action functional that admits anr parameter, semisimple groupG(r) of internal symmetries is considered. A minimal replacement operator for the total groupP ₁₀×G(r) is obtained, together with the requisite compensating 1-forms for both the Poincaré and theG(r) sectors. A basis forP ₁₀×G(r)-invariant Lagrangian densities for the free fields is obtained. Restriction to Lagrangian densities that are at most quadratic in the associated curvature and torsion fields eliminates active coupling between theP ₁₀ free field Lagrangian and theG(r) free field Lagrangian, although there is passive coupling that arises through the requirement of tensorial covariance under general coordinate transformations generated by the local action of the translation group. A superposition principle, modulo passive coupling, thus holds for quadratic free field Lagrangian for the total group:L _TOT=L _P +L _G(r) . Field equations for the matter fields and the compensating fields of theG(r) sector are obtained. Both share the passive coupling toP ₁₀ that is required in order to achieve tensorial covariance, but only the matter fields couple directly to the Poincaré fields and only to the Lorentz sector. For weak Poincaré fields, the field equations for the matter fields and the compensating fields of the internal symmetries go over into the standard field equations of gauge theory for an internal symmetry group.     

One-dimensionalXY model: Ergodic properties and hydrodynamic limit

We prove theorems on convergence to a stationary state in the course of time for the one-dimensionalXY model and its generalizations. The key point is the well-known Jordan-Wigner transformation, which maps theXY dynamics onto a group of Bogoliubov transformations on the CARC ^*-algebra overZ ¹. The role of stationary states for Bogoliubov transformations is played by quasifree states and for theXY model by their inverse images with respect to the Jordan-Wigner transformation. The hydrodynamic limit for the one-dimensionalXY model is also considered. By using the Jordan-Wigner transformation one reduces the problem to that of constructing the hydrodynamic limit for the group of Bogoliubov transformations. As a result, we obtain an independent motion of normal modes, which is described by a hyperbolic linear differential equation of second order. For theXX model this equation reduces to a first-order transfer equation.     

I=0 toI=1 transition form factors

A method is proposed to extend the hard scattering picture of Brodsky and Lepage to transitions between hadrons with orbital angular momentuml=0 andl=1. The use of covariant spin wave functions turns out to be very helpful in formulating that method. As a first application we construct a light-cone wave function of the nucleon resonance N^*(1535) in the quark-diquark picture. Using this wave function and the extended hard scattering picture, theN-N ^* transition form factors are calculated at large momentum transfer and the results compared to experimental data. As a further application of our method we briefly discuss the- a₁ form factors in an appendix.Supported by the Deutsche Forschungsgemeinschaft.     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

The hidden geometry of the quantum Euclidean space

We briefly describe how to introduce the basic notions of noncommutative differential geometry on the 3-dim quantum space covariant under the quantum group of rotations SO _q(3).     

De Rham complex with ternary differential on the quantum group SL h (2)

In this work, we construct the de Rham complex with the differential operator d satisfying theQ-Leibniz rule, whereQ is a complex number, and the condition d³=0 (ternary differential) on theh-deformed quantum plane in dimension two following the general formalism elaborated in [Bazunova et al.: Czech. J. Phys.51 (2001) 1226]. Then we construct the de Rham complex on the group SL _h (2) which preserves theh-deformed quantum plane by the differentiation of RTT-relations, and, introducing the Maurer-Cartan forms, get an analog of the Maurer-Cartan equation. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Beyond the relativistic point particle: A reciprocally invariant system and its generalisation

We investigate a reciprocally invariant system proposed by Low and Govaerts et al., whose action contains both the orthogonal and the symplectic forms and is invariant under global O(2,4)∩Sp(2,4) transformations. We find that the general solution to the classical equations of motion has no linear term in the evolution parameter, τ, but only the oscillatory terms, and therefore cannot represent a particle propagating in spacetime. As a remedy, we consider a generalisation of the action by adopting a procedure similar to that of Bars et al., who introduced the concept of a τ derivative that is covariant under local Sp(2) transformations between the phase space variables x^μ(τ) and p^μ(τ). This system, in particular, is similar to a rigid particle whose action contains the extrinsic curvature of the world line, which turns out to be helical in spacetime. Another possible generalisation is the introduction of a symplectic potential proposed by Montesinos. We show how the latter approach is related to Kaluza–Klein theories and to the concept of Clifford space, a manifold whose tangent space at any point is Clifford algebra Cl(8), a promising framework for the unification of particles and forces.     

Mirror Symmetric SU(3)-Structure Manifolds with NS Fluxes

When string theory is compactified on a six-dimensional manifold with a nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely geometrical composite NS form associated with lack of integrability of the complex structure on the mirror side. Considering a general class of T³-fibered geometries admitting SU(3) structure, we find an exchange of pure spinors (e^iJ and ) in dual geometries under fiberwise T–duality, and study the transformations of the NS flux and the components of intrinsic torsion. A complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with NS flux.     

Operator coproduct-realization of quantum group transformations in two-dimensional gravity I

A simple connection between the universalR matrix ofU _q(sl(2)) (for spins 1/2 andJ) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct action on the covariant operators. It allows us to derive the expected quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of theU _q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extendedU _q(sl(2)) algebra, which includes the Liouville zero-mode momentum in a non-trivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry related to the presence of both of the screening charges of 2D gravity.Partially supported by the EC contracts CHRXCT920069 and CHRXCT920035.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.