Noncommutativity in (2+1)-dimensions and the Lorentz group

In this article we considered models of particles living in a three-dimensional space-time with a nonstandard noncommutativity induced by shifting canonical coordinates and momenta with generators of a unitary irreducible representation of the Lorentz group. The Hilbert space gets the structure of a direct product with the representation space, where we are able to construct operators which realize the algebra of Lorentz transformations. We study the modified Landau problem for both Schr\"odinger and Dirac particles, whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of the dynamical variables from the ones of the usual problem in the normal space. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters. We find no constraint between the parameters referring to no-commutativity in coordinates and momenta but they rather play similar roles. Since the representation space of the unitary irreducible representations SL(2,R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.Comment: PACS: 03.65.-w; 11.30.Cp; 02.40.Gh, 19 pages, no figures. Version to appear in Physical Review

Extended Fermion Representation of Multi-Charge 1/2-BPS Operators in AdS/CFT -- Towards Field Theory of D-Branes --

We extend the fermion representation of single-charge 1/2-BPS operators in the four-dimensional N=4 super Yang-Mills theory to general (multi-charge) 1/2-BPS operators such that all six directions of scalar fields play roles on an equal footing. This enables us to construct a field-theorectic representation for a second-quantized system of spherical D3-branes in the 1/2-BPS sector. The Fock space of D3-branes is characterized by a novel exclusion principle (called `Dexclusion' principle), and also by a nonlocality which is consistent with the spacetime uncertainty relation. The Dexclusion principle is realized by composites of two operators, obeying the usual canonical anticommutation relation and the Cuntz algebra, respectively. The nonlocality appears as a consequence of a superselction rule associated with a symmetry which is related to the scale invariance of the super Yang-Mills theory. The entropy of the so-called superstars, with multiple charges, which have been proposed to be geometries corresponding to the condensation of giant gravitons is discussed from our viewpoint and is argued to be consistent with the Dexclusion principle. Our construction may be regarded as a first step towards a possible new framework of general D-brane field theory.Comment: 43 pages, 4 figures; version 2, corrected typos and added reference

Integral forms of Kac-Moody groups and Eisenstein series in low dimensional supergravity theories

Kac-Moody groups $G$ over $\mathbb{R}$ have been conjectured to occur as symmetry groups of supergravities in dimensions less than 3, and their integer forms $G(\mathbb{Z})$ are conjecturally U-duality groups. Mathematical descriptions of $G(\mathbb{Z})$, due to Tits, are functorial and not amenable to computation or applications. We construct Kac-Moody groups over $\mathbb{R}$ and $\mathbb{Z}$ using an analog of Chevalley's constructions in finite dimensions and Garland's constructions in the affine case. We extend a construction of Eisenstein series on finite dimensional semisimple algebraic groups using representation theory, which appeared in the context of superstring theory, to general Kac-Moody groups. This coincides with a generalization of Garland's Eisenstein series on affine Kac-Moody groups to general Kac-Moody groups and includes Eisenstein series on $E_{10}$ and $E_{11}$. For finite dimensional groups, Eisenstein series encode the quantum corrections in string theory and supergravity theories. Their Kac-Moody analogs will likely also play an important part in string theory, though their roles are not yet understood

Accurate Prediction Of Vibronic Levels And Branching Ratios For Laser-coolable Linear Polyatomic Molecules: The Construction Of The Quasidiabatic Hamiltonian

The vibronic structures of the low-lying electronic states in linear polyatomic molecules, which are utilized to construct closed optical cycling, play crucial roles in the laser-cooling processes. The construction of a multi-state K\"oeppel-Domcke-Cederbaum (KDC) quasidiabatic Hamiltonian with spin-orbit coupling, linear vibronic coupling, and Renner-Teller effects taken into account is reported aiming to obtain accurate vibronic levels and wave functions for laser-coolable triatomic molecules. The parameters for this KDC Hamiltonian were obtained from relativistic equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) calculations. Discrete variable representation (DVR) calculations were then carried out to obtain the vibronic levels and wave functions. The accuracy of the present parametrization for the KDC Hamiltonian is demonstrated with calculations for vibronic levels of the $X^2\Sigma$ and $A^2\Pi$ states of the SrOH molecule

CML: the commonKADS conceptual modelling language

We present a structured language for the specification of knowledge models according to the CommonKADS methodology. This language is called CML (Conceptual Modelling Language) and provides both a structured textual notation and a diagrammatic notation for expertise models. The use of our CML is illustrated by a variety of examples taken from the VT elevator design system

A comparative analysis of business process modelling techniques

Business process modelling is an increasingly popular research area for both organisations and academia due to its usefulness in facilitating human understanding and communication. Several modelling techniques have been proposed and used to capture the characteristics of business processes. However, available techniques view business processes from different perspectives and have different features and capabilities. Furthermore, to date limited guidelines exist for selecting appropriate modelling techniques based on the characteristics of the problem and its requirements. This paper presents a comparative analysis of some popular business process modelling techniques. The comparative framework is based on five criteria: flexibility, ease of use, understandability, simulation support and scope. The study highlights some of the major paradigmatic differences between the techniques. The proposed framework can serve as the basis for evaluating further modelling techniques and generating selection procedures

Geometric representations for minimalist grammars

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations.Comment: 43 pages, 4 figure