Classical and Quantum Superintegrability with Applications (1309.2694v1)

Abstract: A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta. We present a classification of second-order superintegrable systems in two-dimensional Riemannian and pseudo-Riemannian spaces. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order in real Euclidean space $E_2$ are presented and partially solved for the case of third-order integrals. A systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally. The relation between superintegrability and the classification of orthogonal polynomials is analyzed.

• The paper introduces a comprehensive classification of superintegrable systems that exceed the typical integrable framework in both classical and quantum contexts.
• It develops advanced mathematical techniques to resolve third- and fourth-order integrals of motion, addressing the complexity of underlying equations.
• The findings highlight the significance of quadratic algebras and the Stäckel transform in uncovering system symmetries and applications in theoretical physics.

Overview of Classical and Quantum Superintegrability with Applications

The paper "Classical and Quantum Superintegrability with Applications" focuses on the intriguing class of systems known as superintegrable systems, which are characterized by admitting more integrals of motion than degrees of freedom. This property makes them particularly relevant in both classical and quantum mechanics as they exhibit solvability beyond the typical integrable systems. Superintegrable systems are known for their maximal symmetry and the ability to describe a wide range of physical phenomena, from harmonic oscillators to the motion of planets.

Classification of Superintegrable Systems

The classification of superintegrable systems is a core aspect of the paper. These systems are categorized based on the order of the integrals of motion they possess and the dimensions of the space in which they are defined. The primary focus is on systems with second-order integrals in two-dimensional spaces, which have been completely classified, including both classical and quantum scenarios.

Higher-Order Integrals of Motion

The paper extends the discussion to higher-order integrals, predominantly focusing on third-order and fourth-order contexts. This involves developing the necessary mathematical framework to handle the complexity that arises with higher-order terms. For instance, for third-order superintegrability, the determining equations become significantly more complicated and require sophisticated mathematical techniques for resolution.

Quantum vs Classical Systems

A noteworthy element in the classification is the distinction between classical and quantum superintegrable systems. The research highlights how quantum systems can exhibit properties that have no classical counterparts, especially evident in potentials associated with Painlevé transcendents that lead to quantum systems with no classical analogs.

Algebraic Structure and Special Functions

The paper explores the algebraic structures underpinning these systems, often manifesting as quadratic algebras. These algebras and their representations are crucial for understanding the physical properties of the systems, including energy spectra and degeneracies. There is a particular focus on how these structures relate to special functions, like those in the Askey scheme, further illustrating their relevance in mathematical physics.

Role of the Stäckel Transform

The Stäckel transform is presented as an essential method for generating new superintegrable systems from known ones, particularly valuable for revealing the relationship between systems in different coordinate systems and potential forms. This transformation plays a significant role in demonstrating that every nondegenerate superintegrable system is equivalent to a constant curvature space system.

Implications and Applications

From a practical standpoint, the insights gained from the paper of superintegrable systems have implications for theoretical physics and applied mathematics. They help in understanding fundamental symmetries and solving the Schrödinger equation in various potential fields, which is integral to numerous physical applications.

Future Directions

The paper speculates on future developments in the paper of superintegrability. This includes extending the classification to higher-dimensional spaces and potentially uncovering new classes of exactly solvable systems. Furthermore, the development of an algebraic geometric approach to superintegrable systems is suggested as promising for deeper and more comprehensive insights.

Overall, this paper sheds light on the elegant and complex world of superintegrable systems, advancing the understanding of their classification, algebraic structures, and real-world applications.