Multi-indexed (q-)Racah Polynomials

As the second stage of the project multi-indexed orthogonal polynomials, we present, in the framework of `discrete quantum mechanics' with real shifts in one dimension, the multi-indexed (q-)Racah polynomials. They are obtained from the (q-)Racah polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of `virtual state' vectors, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier. The virtual state vectors are the `solutions' of the matrix Schr\"odinger equation with negative `eigenvalues', except for one of the two boundary points.Comment: 29 pages. The type II (q-)Racah polynomials are deleted because they can be obtained from the type I polynomials. To appear in J.Phys.

Constrained Dynamics of an Anomalous (g/neq2)(g/neq 2) Relativistic Spinning Particle in Electromagnetic Background

In this paper we have considered the dynamics of an anomalous ($g\neq 2$) charged relativistic spinning particle in the presence of an external electromagnetic field. The constraint analysis is done and the complete set of Dirac brackets are provided that generate the canonical Lorentz algebra and dynamics through Hamiltonian equations of motion. The spin-induced effective curvature of spacetime and its possible connection with Analogue Gravity models are commented upon.Comment: 10 pages Latex, minor corrections and changes in ref., slightly enlarged version, to appear in EPJ

From Feynman Proof of Maxwell Equations to Noncommutative Quantum Mechanics

In 1990, Dyson published a proof due to Feynman of the Maxwell equations assuming only the commutation relations between position and velocity. With this minimal assumption, Feynman never supposed the existence of Hamiltonian or Lagrangian formalism. In the present communication, we review the study of a relativistic particle using ``Feynman brackets.'' We show that Poincar\'e's magnetic angular momentum and Dirac magnetic monopole are the consequences of the structure of the Lorentz Lie algebra defined by the Feynman's brackets. Then, we extend these ideas to the dual momentum space by considering noncommutative quantum mechanics. In this context, we show that the noncommutativity of the coordinates is responsible for a new effect called the spin Hall effect. We also show its relation with the Berry phase notion. As a practical application, we found an unusual spin-orbit contribution of a nonrelativistic particle that could be experimentally tested. Another practical application is the Berry phase effect on the propagation of light in inhomogeneous media.Comment: Presented at the 3rd Feynman Festival (Collage Park, Maryland, U.S.A., August 2006

On the linear forms of the Schrodinger equation

Generalizing the linearisation procedure used by Dirac and later by L\'evy-Leblond, we derive the first-order non-relativistic wave equations for particles of spin 1 and spin 3/2 starting from the Schrodinger equation

Quantum isotonic nonlinear oscillator as a Hermitian counterpart of Swanson Hamiltonian and pseudo-supersymmetry

Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian Hamiltonian H_{-}=\omega(\xi^{\dag} \xi+\1/2)+\alpha \xi^{2}+\beta \xi^{\dag 2}, where $\alpha \neq \beta$ and $\xi$ is a first order differential operator, to obtain the partner potentials $V_{+}(x)$ and $V_{-}(x)$ which are new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian equivalents of the non-Hermitian partner Hamiltonians $H_{\pm}$. We have provided an algebraic way to obtain the spectrum and wavefunctions of a nonlinear isotonic oscillator. The solutions of $V_{-}(x)$ which are Hermitian counterparts of Swanson Hamiltonian are obtained under some parameter restrictions that are found. Also, we have checked that if the intertwining operator satisfies $\eta_{1} H_{-}=H_{+} \eta_{1}$, where $\eta_{1}=\rho^{-1} \mathcal{A} \rho$ and $\mathcal{A}$ is the first order differential operator, which factorizes Hermitian equivalents of $H_{\pm}$.Comment: 11 page

Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics

In recent years, one of the most interesting developments in quantum mechanics has been the construction of new exactly solvable potentials connected with the appearance of families of exceptional orthogonal polynomials (EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre and Hermite) classical orthogonal polynomials, which start with a constant, the EOP families begin with some polynomial of degree greater than or equal to one, but still form complete, orthogonal sets with respect to some positive-definite measure. We show how they may appear in the bound-state wavefunctions of some rational extensions of well-known exactly solvable quantum potentials. Such rational extensions are most easily constructed in the framework of supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class of translationally shape invariant potentials. We review the most recent results in this field, which use higher-order SUSYQM. We also comment on some recent re-examinations of the shape invariance condition, which are independent of the EOP construction problem.Comment: 21 pages, no figure; communication at the Symposium Symmetries in Science XV, July 31-August 5, 2011, Bregenz, Austri

One-Dimensional Fermi liquids

I attempt to give a pedagogical overview of the progress which has occurred during the past decade in the description of one-dimensional correlated fermions. Fermi liquid theory based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence and because of charge-spin separation. It is replaced by a Luttinger liquid whose elementary excitations are collective charge and spin modes, based on the exactly solvable Luttinger model. I review this model and various solutions with emphasis on bosonization (and its equivalence to conformal field theory), and its physical properties. The notion of a Luttinger liquid implies that all gapless 1D systems share these properties at low energies. Chapters 1 and 2 of the article contain an introduction and a discussion of the breakdown of Fermi liquid theory. Chapter 3 describes in detail the solution of the Luttinger model both by bosonization and by Green's functions methods and summarizes the properties of the model, expressed thorugh correlation functions. The relation to conformal field theory is discussed. Chapter 4 of the article introduces the notion of a Luttinger liquid. It describes in much detail the various mappings applied to realistic models of 1D correlated fermions, onto the Luttinger model, as well as important corrections to the Luttinger model properties discussed in Ch.3. Chapter 5 describes situations where the Luttinger liquid is not a stable fixed point, and where spin or charge gaps open in at least one channel. Chapter 6 discusses multi-band and multichain problems, in particular the stability of a Luttinger liquid with respect to interchain hopping. Ch. 7 gives a brief summary of experimental efforts to uncover Luttinger liquid correlations in quasi-1D materials.Comment: uuencoded Latex files and postscript figures, one Readme-file approx 160 pages + 13 figures; to be published by Reports on Progress in Physic

Extended Krein-Adler theorem for the translationally shape invariant potentials

Considering successive extensions of primary translationally shape invariant potentials, we enlarge the Krein-Adler theorem to mixed chains of state adding and state-deleting Darboux-Backlund transformations. It allows us to establish novel bi-linear Wronskian and determinantal identities for classical orthogonal polynomials