Induced Representations of Quantum Kinematical Algebras and Quantum Mechanics

Unitary representations of kinematical symmetry groups of quantum systems are fundamental in quantum theory. We propose in this paper its generalization to quantum kinematical groups. Using the method, proposed by us in a recent paper (olmo01), to induce representations of quantum bicrossproduct algebras we construct the representations of the family of standard quantum inhomogeneous algebras $U_\lambda(iso_{\omega}(2))$. This family contains the quantum Euclidean, Galilei and Poincar\'e algebras, all of them in (1+1) dimensions. As byproducts we obtain the actions of these quantum algebras on regular co-spaces that are an algebraic generalization of the homogeneous spaces and $q$--Casimir equations which play the role of $q$--Schr\"odinger equations.Comment: LaTeX 2e, 20 page

Integrable mixing of A_{n-1} type vertex models

Given a family of monodromy matrices {T_u; u=0,1,...,K-1} corresponding to integrable anisotropic vertex models of A_{(n_u)-1}-type, we build up a related mixed vertex model by means of glueing the lattices on which they are defined, in such a way that integrability property is preserved. Algebraically, the glueing process is implemented through one dimensional representations of rectangular matrix algebras A(R_p,R_q), namely, the `glueing matrices' zeta_u. Here R_n indicates the Yang-Baxter operator associated to the standard Hopf algebra deformation of the simple Lie algebra A_{n-1}. We show there exists a pseudovacuum subspace with respect to which algebraic Bethe ansatz can be applied. For each pseudovacuum vector we have a set of nested Bethe ansatz equations identical to the ones corresponding to an A_{m-1} quasi-periodic model, with m equal to the minimal range of involved glueing matrices.Comment: REVTeX 28 pages. Here we complete the proof of integrability for mixed vertex models as defined in the first versio

Quantum spin coverings and statistics

SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail.Comment: 15 page

The Branched Polymer Growth Model Revisited

The Branched Polymer Growth Model (BPGM) has been employed to study the kinetic growth of ramified polymers in the presence of impurities. In this article, the BPGM is revisited on the square lattice and a subtle modification in its dynamics is proposed in order to adapt it to a scenario closer to reality and experimentation. This new version of the model is denominated the Adapted Branched Polymer Growth Model (ABPGM). It is shown that the ABPGM preserves the functionalities of the monomers and so recovers the branching probability b as an input parameter which effectively controls the relative incidence of bifurcations. The critical locus separating infinite from finite growth regimes of the ABPGM is obtained in the (b,c) space (where c is the impurity concentration). Unlike the original model, the phase diagram of the ABPGM exhibits a peculiar reentrance.Comment: 8 pages, 10 figures. To be published in PHYSICA

Representations of Quantum Bicrossproduct Algebras

We present a method to construct induced representations of quantum algebras having the structure of bicrossproduct. We apply this procedure to some quantum kinematical algebras in (1+1)--dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum kappa Galilei algebra.Comment: LaTeX 2e, 35 page

Online learning of taskdriven object-based visual attention control

A biologically-motivated computational model for learning task-driven and objectbased visual attention control in interactive environments is proposed. Our model consists of three layers. First, in the early visual processing layer, most salient location of a scene is derived using the biased saliency-based bottom-up model of visual attention. Then a cognitive component in the higher visual processing layer performs an application specific operation like object recognition at the focus of attention. From this information, a state is derived in the decision making and learning layer. Online Learning of Task-driven Object-based Visual Attention Control Ali Borji Top-down attention is learned by the U-TREE Discussions and Conclusions An agent working in an environment receives information momentarily through its visual sensor. It should determine what to look for. For this we use RL to teach the agent simply look for the most task relevant and rewarding entity in the visual scene ( This layer controls both top-down visual attention and motor actions. The learning approach is an extension of the U-TREE algorithm [6] to the visual domain. Attention tree is incrementally built in a quasi-static manner in two phases (iterations): 1) RL-fixed phase and 2) Tree-fixed phase In each Tree-fixed phase, RL algorithm is executed for some episodes by Fig. 1. Proposed model for learning task-driven object-based visual attention control Example scenario: captured scene through the agents' visual sensor undergoes a biased bottom-up saliency detection operation and focus of attention (FOA) is determined. Object at the FOA is recognized (i.e. is either present or not in the scene), then the agent moves in its binary tree in the decision making and leaves. 100% correct policy was achieved. The object at the attended location is recognized by the hierarchical model of object recognition (HMAX) [3] M. Riesenhuber, T. Poggio, Hierarchical models of object recognition in cortex. Nature Neuroscience, 2(1999),11, 1019-1025. Basic saliency-based model of visual attention [1] is revised for the purpose of salient region selection (object detection) at this layer where norm(.) is the Euclidean distance between two points in an image. Saliency is the function which takes as input an image and a weight vector and returns the most salient location. t i is the location of target object in the i-th image. In each Tree-fixed phase, RL algorithm is executed for some episodes by following ε-greedy action selection strategy. In this phase, tree is hold fixed and the derived quadruples (s t , a t , r t+1 , s t+1 ) are only used for updating the Q-table: State discretization occurs in the RL-fixed phase where gathered experiences are used to refine aliased states. An object which minimizes aliasing the most is selected for braking an aliased leaf. Acknowledgement This work was funded by the school of cognitive sciences, IPM, Tehran, IRAN. scene), then the agent moves in its binary tree in the decision making and learning layer. This is done repetitively until it reaches a leaf node which determines its state. The best motor action is this state is performed. Outcome of this action over the world is evaluated by a critic and a reinforcement signal is fed back to the agent to update its internal representations (attention tree) and action selection strategy in a quasi-static manner. Following subsections discuss each layer of the model in detail

Unsharp Degrees of Freedom and the Generating of Symmetries

In quantum theory, real degrees of freedom are usually described by operators which are self-adjoint. There are, however, exceptions to the rule. This is because, in infinite dimensional Hilbert spaces, an operator is not necessarily self-adjoint even if its expectation values are real. Instead, the operator may be merely symmetric. Such operators are not diagonalizable - and as a consequence they describe real degrees of freedom which display a form of "unsharpness" or "fuzzyness". For example, there are indications that this type of operators could arise with the description of space-time at the string or at the Planck scale, where some form of unsharpness or fuzzyness has long been conjectured. A priori, however, a potential problem with merely symmetric operators is the fact that, unlike self-adjoint operators, they do not generate unitaries - at least not straightforwardly. Here, we show for a large class of these operators that they do generate unitaries in a well defined way, and that these operators even generate the entire unitary group of the Hilbert space. This shows that merely symmetric operators, in addition to describing unsharp physical entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g. within a fundamental theory of quantum gravity.Comment: 23 pages, LaTe

Local D=4 Field Theory on κ\kappa--Deformed Minkowski Space

We describe the local D=4 field theory on $\kappa$--deformed Minkowski space as nonlocal relativistic field theory on standard Minkowski space--time. For simplicity the case of $\kappa$-deformed scalar field $\phi$ with the interaction $\lambda \phi^{4}$ is considered, and the $\kappa$--deformed interaction vertex is described. It appears that fundamental mass parameter $\kappa$ plays a role of regularizing imaginary Pauli--Villars mass in $\kappa$--deformed propagator.Comment: revtex, 2 figures.The text has been enlarged by two pages, mostly the explicite description of local scalar field on $kappa$-deformed Minkowski space has been extended. One figure adde

Once again about quantum deformations of D=4 Lorentz algebra: twistings of q-deformation

This paper together with the previous one (arXiv:hep-th/0604146) presents the detailed description of all quantum deformations of D=4 Lorentz algebra as Hopf algebra in terms of complex and real generators. We describe here in detail two quantum deformations of the D=4 Lorentz algebra o(3,1) obtained by twisting of the standard q-deformation U_{q}(o(3,1)). For the first twisted q-deformation an Abelian twist depending on Cartan generators of o(3,1) is used. The second example of twisting provides a quantum deformation of Cremmer-Gervais type for the Lorentz algebra. For completeness we describe also twisting of the Lorentz algebra by standard Jordanian twist. By twist quantization techniques we obtain for these deformations new explicit formulae for the deformed coproducts and antipodes of the o(3,1)-generators.Comment: 17 page

Quantum Symmetries and Marginal Deformations

We study the symmetries of the N=1 exactly marginal deformations of N=4 Super Yang-Mills theory. For generic values of the parameters, these deformations are known to break the SU(3) part of the R-symmetry group down to a discrete subgroup. However, a closer look from the perspective of quantum groups reveals that the Lagrangian is in fact invariant under a certain Hopf algebra which is a non-standard quantum deformation of the algebra of functions on SU(3). Our discussion is motivated by the desire to better understand why these theories have significant differences from N=4 SYM regarding the planar integrability (or rather lack thereof) of the spin chains encoding their spectrum. However, our construction works at the level of the classical Lagrangian, without relying on the language of spin chains. Our approach might eventually provide a better understanding of the finiteness properties of these theories as well as help in the construction of their AdS/CFT duals.Comment: 1+40 pages. v2: minor clarifications and references added. v3: Added an appendix, fixed minor typo