Branes,Quantum Nambu Brackets and the Hydrogen Atom

The Nambu-bracket quantization of the hydrogen atom is worked out as an illustration of the general method. The dynamics of topological open branes is controlled classically by Nambu brackets. Such branes then may be quantized through the consistent quantization of the underlying Nambu brackets: properly defined, the quantum Nambu-brackets comprise an associative structure, although the naive derivation property is mooted through operator entwinement. For superintegrable systems, such as the hydrogen atom, the results coincide with those furnished by Hamiltonian quantization - but the method is not limited to Hamiltonian systems.     

Deformation quantization and Nambu Mechanics

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products. Supported by the European Commission and the Japan Society for the Promotion of Science. NSF grant DMS-95-00557 This article was processed by the author using the L^AT_EX style filepljour1 from Springer-Verlag.     

On superintegrable systems separable in Cartesian coordinates

We continue the study of superintegrable systems of Thompson's type separable in Cartesian coordinates. An additional integral of motion for these systems is the polynomial in momenta of N-th order which is a linear function of angle variables and the polynomial in action variables. Existence of such superintegrable systems is naturally related to the famous Chebyshev theorem on binomial differentials.     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

The Nambu mechanics as a class of singular generalized dynamical formalisms

In a recent work Nambu has proposed ac-number dynamical formalism which can allow an odd numbern of canonical variables. Naturally associated to this new mechanics there exists ann-linear bracket whose study opens interesting possibilities. The purpose of this work is to show that besides this bracket another one which is bilinear and in fact a Lie bracket can also be associated with the Nambu mechanics. For anyn, however, this bracket is singular. In a sense previously used by the present author, this result exhibits the Nambu mechanics as an interesting class of singular generalized dynamical formalisms irrespective of the number of phase space variables. Reasons are given suggesting that such singular formalisms would be, within our context, the only ones capable of describing classical analogues of generalized quantum systems.     

Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -?² plus a quantum deformation of order ?⁴ and ?⁶. The systems inside the classes are transformed using Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.     

Jacobi,Ellipsoidal Coordinates and Superintegrable Systems

Abstract

We describe Jacobi’s method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems to which we then turn. A Schrödinger operator with potential on a Riemannian space is second-order superintegrable if there are 2n ? 1 (classically) functionally independent second-order symmetry operators. (The 2n ? 1 is the maximum possible number of such symmetries.) These systems are of considerable interest in the theory of special functions because they are multiseparable, i.e., variables separate in several coordinate sets and are explicitly solvable in terms of special functions. The interrelationships between separable solutions provides much additional information about the systems. We give an example of a superintegrable system and then present very recent results exhibiting the general structure of superintegrable systems in all real or complex two-dimensional spaces and three-dimensional conformally flat spaces and a complete list of such spaces and potentials in two dimensions.     

Quantized string models

We discuss and compare the Lorentz covariant path integral quantization of the three bose string models, namely, the Nambu, Eguchi and Brink-Di Vecchia-Howe-Polyakov (BDHP) ones. Along with a critical review of the subject with some uncertainties and ambiguities clearly stated, various new results are presented. We work out the form of the BDHP string ansatz for the Wilson average and prove a formal inequivalence of the exact Nambu and BDHP models for any space-time dimension d. The above three models, known to be equivalent on the classical level, are shown to be equivalent in a semiclassical approximation near a minimal surface and also in the leading $\text{1}\text{d}\text{-}\text{approximation}$ for the static $\text{q}$q-potential. We analyse scattering amplitudes predicted by the BDHP string and find that when exactly calculated for d < 26 they are different from the old dual ones, and possess a non-linear spectrum which may be considered as free from tachyons in the ground state.     

Theta functions,root systems and 3-manifold invariants

We describe a semi-abelian version of Witten's theory using the quantization of dimension g tori for a general gauge group G. We derive a family of invariants for closed oriented 3-manifolds which coincide with those defined by Witten for lens spaces and torus bundles.     

Retrospective on Quantization

Quantization is still a central problem of modern physics. One example of an unsolved problem is the quantization of Nambu mechanics. After a brief comment on the role of Harrison cohomology, this review concentrates on the central problem of quantization of QCD and, more generally, quark confinement seen as a problem of quantization. Several suggestions are made, some of them rather extravagant.     

Models of quadratic quantum algebras and their relation to classical superintegrable systems

We show how to construct realizations (models) of quadratic algebras for 2D second order superintegrable systems in terms of differential or difference operators in one variable. We demonstrate how various models of the quantum algebras arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras related to superintegrable systems in n dimensions and are intimately related to multivariable orthogonal polynomials. The text was submitted by the authors in English.     

Completeness of multiseparable superintegrability in two dimensions

For complex Euclidean 2-space and the complex 2-sphere, we have found all classical and quantum superintegrable systems that a polynomial correspond to nondegenerate potentials. These potentials have the property that a polynomial associated with each of them is a quadratic algebra. Further-more, each of these superintegrable systems admits separation of variables in more than one coordinate system. For degenerate superintegrable systems, both properties may be violated.     

Second-order superintegrable quantum systems

A classical (or quantum) superintegrable system on an n-dimensional Riemannian manifold is an integrable Hamiltonian system with potential that admits 2n ? 1 functionally independent constants of the motion that are polynomial in the momenta, the maximum number possible. If these constants of the motion are all quadratic, then the system is second-order superintegrable, the most tractable case and the one we study here. Such systems have remarkable properties: multi-integrability and separability, a quadratic algebra of symmetries whose representation theory yields spectral information about the Schrödinger operator, and deep connections with expansion formulas relating classes of special functions. For n = 2 and for conformally flat spaces when n = 3, we have worked out the structure of the classical systems and shown that the quadratic algebra always closes at order 6. Here, we describe the quantum analogs of these results. We show that, for nondegenerate potentials, each classical system has a unique quantum extension.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

Models for the 3D singular isotropic oscillator quadratic algebra

We give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential—differential or differential—difference and difference—difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in n dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.     

Multiseparability and superintegrability in three dimensions

In complex two-dimensional Euclidean space, the Hamilton-Jacobi or Schrödinger equation with a given “nondegenerate” potential is maximally superintegrable if and only if it is separated in more than one coordinate system. A similar statement for three dimensions is not known. In this paper, a start will be made on this problem by investigating the known separable Hamilton-Jacobi and Schrödinger systems to find those that are superintegrable.     

Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrödinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.     

Lie symmetries and superintegrability in quantum mechanics

Starting from the structure of the higher order Lie symmetries of the Schrödinger equation in the Euclidean plane E2, we establish, in the case of first-and second-order symmetries, the relations between separation of variables and superintegrable systems in quantum mechanics.