Q-deformed Models Research Articles

Extended RC Impedance and Relaxation Models for Dissipative Electrochemical Capacitors

Electrochemical capacitors are a class of energy devices in which complex mechanisms of accumulation and dissipation of electric energy take place when connected to a charging or discharging power system. Reliably modeling their frequency-domain and time-domain behaviors is crucial for their proper design and integration in engineering applications, knowing that electrochemical capacitors in general exhibit anomalous tendency that cannot be adequately captured with traditional integer-order-based models. In this study we first review some of the widely used fractional-oder models for the description of impedance and relaxation functions of dissipative resistive-capacitive system, namely the Cole-Cole, Davidson-Cole, and Havriliak-Negami models. We then propose and derive new q-deformed models based on modified evolution equations for the charge or voltage when the device is discharged into a parallel resistive load. We verify our results on anomalous spectral impedance response and time-domain relaxation data for voltage and charge obtained from a commercial supercapacitor.

On Faithful Matrix Representations of q-Deformed Models in Quantum Optics

Consider the q -deformed Lie algebra, t q : K ^ 1 , K ^ 2 q = 1 − q K ^ 1 K ^ 2 , K ^ 3 , K ^ 1 q = s K ^ 3 , K ^ 1 , K ^ 4 q = s K ^ 4 , K ^ 3 , K ^ 2 q = t K ^ 3 , K ^ 2 , K ^ 4 q = t K ^ 4 , and K ^ 4 , K ^ 3 q = r K ^ 1 , where r , s , t ∈ ℝ − 0 , subject to the physical properties: K ^ 1 and K ^ 2 are real diagonal operators, and K ^ 3 = K ^ 4 † , ( † is for Hermitian conjugation). The q -deformed Lie algebra, t q is introduced as a generalized model of the Tavis–Cummings model (Tavis and Cummings 1968, Bashir and Sebawe Abdalla 1995), namely, K ^ 1 , K ^ 2 = 0 , K ^ 1 , K ^ 3 = − 2 K ^ 3 , K ^ 1 , K ^ 4 = 2 K ^ 4 , K ^ 2 , K ^ 3 = K ^ 3 , K ^ 2 , K ^ 4 = K ^ 4 , and K ^ 4 , K ^ 3 = K ^ 1 , which is subject to the physical properties K ^ 1 and K ^ 2 are real diagonal operators, and K ^ 3 = K ^ 4 † . Faithful matrix representations of the least degree of t q are discussed, and conditions are given to guarantee the existence of the faithful representations.

On matrix models and their q-deformations

Motivated by the BPS/CFT correspondence, we explore the similarities be- tween the classical β-deformed Hermitean matrix model and the q-deformed matrix models associated to 3d mathcal{N} = 2 supersymmetric gauge theories on D2×qS1 and {S}_b^3 by matching parameters of the theories. The novel results that we obtain are the correlators for the models, together with an additional result in the classical case consisting of the W -algebra representation of the generating function. Furthermore, we also obtain surprisingly simple expressions for the expectation values of characters which generalize previously known results.

Effects of bosonic and fermionic q-deformation on the entropic gravity

In this paper, we study thermodynamical contributions to the theory of gravity under the q-deformed boson and fermion gas models. According to Verlinde’s proposal, the law of gravity is not based on a fundamental interaction but it emerges as an entropic force from the changes of entropy associated with the information on the holographic screen. In addition, Strominger shows that the extremal quantum black holes obey neither boson nor fermion statistics, but they obey deformed statistic. Using these notions, we find q-deformed entropy and temperature functions. We also present the contributions that come from the q-deformed model to the Poisson equation, Newton’s law of gravity and Einstein’s field equations.

Quantum deformation of two four-dimensional spin foam models

We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the Engle, Pereira, Rovelli and Livine (EPRL) model. The q-deformed models are based on the representation theory of two copies of \documentclass[12pt]{minimal}\begin{document}$U_q(\mathfrak {su}(2))$\end{document}Uq(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models, we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties, and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.

Q-deformation and semidualization in 3D quantum gravity

We explore in detail the role in euclidean 3D quantum gravity of quantum Born reciprocity or ‘semidualization’. The latter is an algebraic operation defined using quantum group methods that interchanges position and momentum. Using this we are able to clarify the structural relationships between the effective noncommutative geometries that have been discussed in the context of 3D gravity. We show that the spin model based on D(U(su2)) for quantum gravity without cosmological constant is the semidual of a quantum particle on a 3-sphere, while the bicrossproduct (DSR) model based on is the semidual of a quantum particle on hyperbolic space. We show further how the different models are all specific limits of q-deformed models with , where mp is the Planck mass and Λ is the cosmological constant, and argue that semidualization interchanges mp ↔ lc, where lc is the cosmological length scale . We investigate the physics of semidualization by studying representation theory. In both the spin model and its semidual we show that irreducible representations have a physical picture as solutions of a respectively noncommutative/curved wave equation. We explain, moreover, that the q-deformed model, at a certain algebraic level, is self-dual under semidualization.

Q-deformed spin foam models of quantum gravity

We numerically study Barrett–Crane models of Riemannian quantum gravity. We have extended the existing numerical techniques to handle q-deformed models and arbitrary spacetime triangulations. We present and interpret expectation values of a few selected observables for each model, including a spin–spin correlation function which gives insight into the behaviour of the models. We find the surprising result that, as the deformation parameter q goes to 1 through roots of unity, the limit is discontinuous.

An algebraic construction of generalized coherent states associated with q-deformed models for primary shape-invariant systems

Generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show that this generalized formalism is able (a) to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system, (b) to reproduce, as particular cases, results already known for shape-invariant systems (such as standard harmonic oscillator and Pöschl–Teller potentials as well as quantum deformed harmonic oscillator models) and (c) point to a formalism that provides a unified description of the different kind of coherent states for quantum systems, deformed or not deformed.

Q-Boson in Quantum Integrable Systems

q-bosonic realization of the underlying Yang-Baxter algebra is identified for a series of quantum integrable systems, including some new models like two-mode q-bosonic model leading to a coupled two-component derivative NLS model, wide range of q-deformed matter-radiation models, q-anyon model etc. Result on a new exactly solvable interacting anyon gas, linked to q-anyons on the lattice is reported.

TWO-PARAMETER QUANTUM-DEFORMED DUAL AMPLITUDE

A four-point dual amplitude, with nonlinear trajectories, is constructed from a two-parameter deformation of the harmonic oscillator algebra. The earlier versions of the q-deformed dual models are reproduced as limiting cases of the model introduced in the present work.

Extension of q-deformed analysis and q-deformed models of classical mechanics

The author presents the generalization of the q-derivative and investigates its properties. The formula of conjugation for scalar products with respect to the Lebesgue and discrete measures is obtained. This formalism is applied to classical mechanical systems depending on functional derivatives. He derives the deformed Euler-Lagrange equations and deformed Poisson bracket by an assumption that it determines the functional evolution of the systems. It appears that in such models a Hamiltonian analogue is no longer conserved in time. For a few examples the explicit forms of constants of evolution are given.