Classical and Quantum Superintegrability with Applications
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 secondorder superintegrable systems in twodimensional Riemannian and pseudoRiemannian 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 are presented and partially solved for the case of thirdorder 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.
Contents
 1 Introduction
 2 Background and Definitions
 3 Important Examples
 4 Secondorder systems
 5 Higherorder determining equations on Euclidean space
 6 Higher order classical superintegrable systems
 7 Higher order quantum superintegrable systems
 8 Generalized Stäckel transform
 9 Polynomial algebras and their irreducible representations
 10 Conclusions
List of Figures
1 Introduction
A standard way to gain insight into the behavior of a physical system is to construct a mathematical model of the system, analyze the model, use it to make physical predictions that follow from the model and compare the results with experiment. The models provided by classical and quantum mechanics have been and continue to be spectacularly successful in this regard. However, the systems of ordinary and partial differential equations provided by these models can be very complicated. Usually they cannot be solved analytically and solutions can only be approximated numerically. A relatively few systems, however, can be solved exactly with explicit analytic expressions that predict future behavior, and with adjustable parameters, such as mass or initial position, so that one can can determine the effect on the system of changing these parameters. These are classical and quantum integrable Hamiltonian systems. A special subclass of these systems, called superintegrable, is extremely important for developing insight into physical principles, for they can be solved algebraically as well as analytically, and many of the simpler systems are featured prominently in textbooks. What distinguishes these systems is their symmetry, but often of a much subtler kind than just group symmetry. The symmetries in their totality form quadratic, cubic and other higher order algebras, not necessarily Lie algebras, and are sometimes referred to as ‘hidden symmetries’. Famous examples are the classical harmonic anisotropic oscillator (Lissajous patterns) and Kepler systems (planetary orbits), the quantum Coulomb system (energy levels of the hydrogen atom, leading to the periodic table of the elements) and the quantum isotropic oscillator. The Hohmann transfer, a fundamental procedure for the positioning of satellites and orbital maneuvering of interplanetary spacecraft is based on the superintegrability of the Kepler system.
Superintegrable systems admit the maximum possible symmetry and this forces analytic and algebraic solvability. The special functions of mathematical physics and their properties are closely related to their origin and use in providing explicit solutions for superintegrable systems, for instance via separation of variables in the partial differential equations of mathematical physics. These systems appear in a wide variety of modern physical and mathematical theories, from semiconductors to supersymmetric field theories. As soon as a system is discovered it tends to be implemented as a model, due to the fact that it can be solved explicitly. Perturbations of superintegrable systems are frequently used to study the behavior of more complex systems, e.g., the periodic table is based on perturbations of the superintegrable hydrogen atom system. â¢ The principal research activity in this area involves the discovery, classification and solution of superintegrable systems, and elucidation of their structure, particularly the underlying symmetry algebra structure, as well as application of the results in a wide variety of fields. Superintegrability has deep historical roots, but the modern theory was inaugurated by Smorodinsky, Winternitz and collaborators in 1965 [50, 51, 118] who explored multiseparability in 2 and 3 dimensional Euclidean spaces. Wojciechowski seems to have coined the term ‘superintegrable’ and applied it about 1983 [194]. The earlier terminology was “systems with accidental degeneracy”, going back to Fock and Bargmann [49, 13]. Other terms used in this context were “higher symmetries”, or “dynamical symmetries” [50, 51, 118]. Some explicit solutions of the nbody problem by Calogero [24, 25, 26, 169], dating from the late 1960s were crucial examples for the theory. The technique of coupling constant metamorphosis to map between integrable and superintegrable systems was introduced in the mid1980s [70, 21]. Interest increased greatly due to papers by Evans about 1990 [45, 46, 44] which contained many examples that generalized fundamental solvable quantum mechanical systems in 3 dimensions. About 1995 researchers such as Letourneau and Vinet [113] recognized the very close relationship between QuasiExact Solvability (QES) [186, 187, 58] for quantum systems in one dimension and second order superintegrable systems in two and higher dimensions. Beginning about 2000 the structure theory and classification of second order superintegrable systems has been largely worked out, with explicit theorems that provide a concrete foundation for the observations made in explicit examples (Daskaloyannis, Kalnins, Kress, Miller, Pogosyan, etc.). The quadratic algebras of symmetries of the second order superintegrable systems, and their representation theory has been studied since about 1992 with results by Zhedanov, Daskaloyannis, Kalnins, Kress, Marquette, Miller, Pogosyan, Post, Vinet, Winternitz, etc. New applications of the theory to other branches of physics are appearing, e.g. work by Quesne on variable mass Hamiltonians [158]. More recently, important examples of physically interesting third and fourth order quantum superintegrable systems were announced by Evans and Verrier [188], and by Rodriguez, Tempesta, and Winternitz [166, 167].
Superintegrable systems of secondorder, i.e., classical systems where the defining symmetries are secondorder in the momenta and quantum systems where the symmetries are secondorder partial differential operators, have been well studied and there is now a developing structure and classification theory. The classification theory for thirdorder systems that separate in orthogonal coordinate systems, i.e. that also admit a secondorder integral, has begun and many new systems have recently been found, including quantum systems with no classical analog and systems with potentials associated with Painlevé transcendents [62, 63, 126, 120, 183]. These are quantum systems that could not be obtained by quantizing classical ones. Their quantum limits are sometimes free motion. In other cases, the going to 0 limit are singular, in the sense that the quantum potential satisfies partial differential equations in which the leading terms vanish for going to 0. However for nonseparable thirdorder and general higherorder superintegrable systems much less is known. In particular until very recently there were few examples and almost no structure theory and classification theory.
This situation has changed dramatically with the publication of the 2009 paper “An infinite family of solvable and integrable quantum systems on a plane” by F. Tremblay, V.A. Turbiner and P. Winternitz [181, 182]. The authors’ paper had an immediate effect on the active field of classical and quantum superintegrable systems. Their examples and conjectures have led rapidly to new classes of higher order superintegrable systems, thereby reinvigorating research activity and publications in the subject. The authors introduced a family of both classical and quantum mechanical potentials in the plane, parametrized by the constant , conjectured and gave evidence that these systems were both classically and quantum superintegrable for all rational , with integrals of arbitrarily large order. It has now been verified that that the conjectures were correct (Gonera, Kalnins, Kress, Miller, Quesne, Pogosyan, [160, 97, 84, 88, 57]), Higher order superintegrable systems had been thought to be uncommon, but are now seen to be ubiquitous with a clear path to construct families of other candidates at will [11, 30, 87, 88, 124, 121, 123, 164, 148, 151, 31]. Tools are being developed for the verification of classical and quantum superintegrability of higher order that can be applied to a variety of Hamiltonian systems. A structure theory for these systems, classification results and applications are following.
This review is focused on the structure and classification of maximal superintegrable systems and their symmetry algebras, classical and quantum. Earlier reviews exist, including those on the group theory of the hydrogen atom and Coulomb problem [74, 42, 12], oscillators [114, 115, 136] and accidental degeneracy or symmetry in general [133, 127].
There are other interesting approaches to the theory that we don’t address here. In particular there is a geometrical approach to the classical theory, based on foliations, e.g. [48, 131, 137, 177] and those using the methods of differential Galois theory [117, 116] or invariant theory of Killing tensors [1, 2]. Many authors approach classical and quantum superintegrable systems from an external point of view. They use elegant techniques such as Rmatrix theory and coalgebra symmetries to produce superintegrable systems with generators that are embedded in a larger associative algebra with simple structure, such as a Lie enveloping algebra, e.g. [3, 22, 9, 10, 105, 27, 6, 184, 165, 162]. Here we take an internal point of view. The fundamental object for us is the symmetry algebra generated by the system. A useful analogy is differential geometry where a Riemannian space can be considered either as embedded in Euclidean space or as defined intrinsically via a metric.
Let us just list some of the reasons why superintegrable systems are interesting both in classical and quantum physics.

In classical mechanics, superintegrability restricts trajectories to an dimensional subspace of phase space (). For (maximal superintegrability), this implies that all finite trajectories are closed and motion is periodic [137].

At least in principle, the trajectories can be calculated without any calculus.

Bertrand’s theorem [17] states that the only spherically symmetric potentials for which all bounded trajectories are closed are the CoulombKepler system and the harmonic oscillator, hence no other superintegrable systems are spherically symmetric.

The algebra of integrals of motion is a nonAbelian and interesting one. Usually it is a finitely generated polynomial algebra, only exceptionally a finite dimensional Lie algebra or KacMoody algebra [35].

In the special case of quadratic superintegrability (all integrals of motion are at most quadratic polynomials in the moments), integrability is related to separation of variables in the HamiltonJacobi equation, or SchrÂödinger equation, respectively.

In quantum mechanics, superintegrability leads to an additional degeneracy of energy levels, sometimes called ”accidental degeneracy”. The term was coined by Fock[49] and used by Moshinsky and collaborators [135, 114, 134, 132], though the point of their studies was to show that this degeneracy is certainly no accident.

A conjecture, born out by all known examples, is that all maximally superintegrable systems are exactly solvable [178]. If the conjecture is true, then the energy levels can be calculated algebraically. The wave functions are polynomials (in appropriately chosen variables) multiplied by some gauge factor.

The nonAbelian polynomial algebra of integrals of motion provides energy spectra and information on wave functions. Interesting relations exist between superintegrability and supersymmetry in quantum mechanics
As a comment, let us mention that superintegrability has also been called nonAbelian integrability. From this point of view, infinite dimensional integrable systems (soliton systems) described e.g. by the KortewegdeVries equation, the nonlinear SchrÂdinger equation, the KadomtsevPetviashvili equation, etc. are actually superintegrable. Indeed, the generalized symmetries of these equations form infinite dimensional nonAbelian algebras (the OrlovShulman symmetries) with infinite dimensional Abelian subalgebras of commuting flows[143, 142, 144]
Before we delve into the specifics of superintegrability theory, we give a simplified version of the requisite mathematics and physics governing Hamiltonian dynamical systems in Section 2. Then in Section 3 we study, as examples, the 2D Kepler system and the 2D hydrogen atom in detail, both in Euclidean space and on the 2sphere, as well as the hydrogen atom in 3D Euclidean space with its symmetry. The examples illustrate basic features of superintegrability: complete solvability of the systems via the symmetry algebra, important applications to physics, and relation of superintegrable systems via contraction. These well known systems have been studied literally for centuries, but the pure superintegrability approach has novel features. In Section 4 we sketch the structure and classification theory for secondorder superintegrable systems, the most tractable class of such systems. Section 5 is devoted to the classification of higherorder systems in 2D Euclidean space, where the quantization problem first becomes serious. In Sections 6 and 7 we present examples of higher order classical and quantum systems and tools for studying their structure. Here great strides have been made but, as yet, there is no classification theory. Sections 8 is devoted to the generalized Stäckel transform, an invertible structure preserving transformation of one superintegrable system to another that is basic to the classification theory. Since superintegrability is a concept that distinguishes completely solvable physical systems it should be no surprise that there are profound relations to the theory of special functions. In Sections 9.1 we make that especially clear by showing that the Askey Scheme for orthogonal hypergeometric polynomials can be derived from contractions of 2D secondorder superintegrable systems.
2 Background and Definitions
2.1 Classical mechanics
The Hamiltonian formalism describes dynamics of a physical system in dimensions by relating the time derivatives of the position coordinates and the momenta to a single function on the phase space, the Hamiltonian . A physical system describing the position of a particle at time involves position coordinates , and momentum coordinates, . The phase space of a physical system is described by points , where is the base field, usually or . In its simplest form, the Hamiltonian can be interpreted as the total energy of the system: , where and are kinetic and potential energy, respectively. Explicitly,
(2.1) 
where is a contravariant metric tensor on some real or complex Riemannian manifold. That is , and the metric on the manifold is given by where is the covariant metric tensor, the matrix inverse to . Under a local transformation the contravariant tensor and momenta transform according to
(2.2) 
Here is a scaling parameter that can be interpreted as the mass of the particle. For the Hamiltonian (2.1) the relation between the momenta and the velocities is , so that . Once the velocities are given, the momenta are scaled linearly in . In mechanics the exact value of may be important but for mathematical structure calculations it can be scaled to any nonzero value. To make direct contact with mechanics we may set ; for structure calculations we will usually set . The above formulas show how to rescale for differing values of .
The dynamics of the system are given by Hamilton’s equations, [56, 5]
(2.3) 
Solutions of these equations give the trajectories of the system.
Definition 1
The Poisson bracket of two functions , on the phase space is the function
(2.4) 
The Poisson bracket obeys the following properties, for functions on the phase space and constants.
(2.5) 
(2.6) 
(2.7) 
(2.8) 
(2.9) 
With the Kronecker delta, coordinates satisfy canonical relations
(2.10) 
Definition 2
A set of coordinate functions is called canonical if the functions satisfy the canonical relations
Canonical coordinates are true coordinates on the phase space, i.e., they can be inverted locally to express as functions of . Furthermore, under this change of coordinates the Poisson bracket (2.4) maintains its form, i.e.,
Any coordinate change of the form , where depends only on and is defined by (2.2), is always canonical.
In terms of the Poisson bracket, we can rewrite Hamilton’s equations as
(2.11) 
For any function , its dynamics along a trajectory is
(2.12) 
Thus will be constant along a trajectory if and only .
Definition 3
If , then is a constant of the motion.
Definition 4
Let be a set of functions defined and locally analytic in some region of a dimensional phase space. We say is functionally independent if the matrix has rank throughout the region. The set is functionally dependent if the rank is strictly less than on the region.
If a set is functionally dependent, then locally there exists a nonzero analytic function of variables such that identically on the region. Conversely, if exists then the rank of the matrix is . Clearly, a set of functions will be functionally dependent.
Definition 5
A system with Hamiltonian is integrable if it admits constants of the motion that are in involution:
(2.13) 
and are functionally independent.
Now suppose is integrable with associated constants of the motion . Up to a canonical change of variables it is possible to assume that . Then by the inverse function theorem we can solve the equations for the momenta to obtain , , where is a vector of constants. For an integrable system, if a particle with position lies on the common intersection of the hypersurfaces for constants , then its momentum is completely determined. Also, if a particle following a trajectory of an integrable system lies on the common intersection of the hypersurfaces at time , where the are constants of the motion, then it lies on the same common intersection for all near . Considering and using the conditions (2.13) and the chain rule it is straightforward to verify . Therefore, there exists a function such that , . Note that , and, in particular, satisfies the HamiltonJacobi equation
(2.14) 
where . By construction and such a solution of the HamiltonJacobi equation depending nontrivially on parameters is called a complete integral. This argument is reversible: a complete integral of (2.14) determines constants of the motion in involution, .
Theorem 1
A system is integrable (2.14) admits a complete integral.
A powerful method for demonstrating that a system is integrable is to exhibit a complete integral by using additive separation of variables.
It is a standard result in classical mechanics that for an integrable system one can integrate Hamilton’s equations and obtain the trajectories, [56, 5]. The Hamiltonian formalism is is well suited to exploiting symmetries of the system and an important tool in laying the framework for quantum mechanics.
2.2 Quantum mechanics
We give a brief introduction to basic principles necessary to understand quantum superintegrable systems. We ignore such issues as the domains of unbounded operators and continuous spectra, and proceed formally. In any case, we will be mainly interested in bound states and their discrete spectra.
In quantum mechanics, physical states are represented as one dimensional subspaces in a complex, projective Hilbert space: two states are equivalent if they differ by a constant multiplicative factor. A standard Euclidean space model for a quantum mechanical bound state system with degrees of freedom is the one where the state vectors are complex square integrable functions on and the transition amplitude between two states is the inner product . Usually, states are normalized: . If is a selfadjoint operator (observable) then
In this model , i.e., multiplication by Cartesian variable , . In analogy to , we have the quantum Hamiltonian on ndimensional Euclidean space,
(2.15) 
Observables correspond to quantities that can be measured. The probability of an arbitrary state, being measured with a given eigenvalue is determined by computing its transition amplitude, , where is an eigenvector of an operator with eigenvalue . As in classical mechanics, we create most of our observables out of the quantities position and momentum. However, in quantum mechanics these quantities correspond to operators. That is, the selfadjoint operator gives the value of the coordinate while the selfadjoint operator gives the th momentum. We can define a bilinear product on the operators by the commutator . This product satisfies the same relations (2.52.8) as the Poisson bracket. The position and momentum operators satisfy the commutation relations,
(2.16) 
where is the reduced Planck constant and . Equations (2.16) define a Heisenberg algebra in a ndimensional space.
The time evolution of quantum mechanical states is determined by : The dynamics is given by the timedependent Schrödinger Equation,
(2.17) 
Using the Hamiltonian, we can define the time evolution operator as a one parameter group of unitary operators determined by the selfadjoint operator . A state at an arbitrary time will be given in terms of the state at time by . If a state at time is an eigenvector for the Hamiltonian, say , then the time evolution operator acts on an eigenvector by a (time dependent) scalar function, . The eigenvalue problem
(2.18) 
is called the timeindependent Schrödinger Equation. If we can find a basis of eigenvectors for , we can expand an arbitrary vector in terms of this energy basis to get the time evolution of the state. Finding eigenvalues and eigenvectors for Hamiltonians is a fundamental task in quantum mechanics.
The expected value of an observable in a state obeys the relation
(2.19) 
This is in analogy with the classical relation (2.12). If , then we can choose a basis for the Hilbert space which is a set of simultaneous eigenfunctions of and , [189]. It is important to determine a complete set of commuting observables to specify the system. This idea leads naturally to quantum integrability.
Definition 6
A quantum mechanical system in dimensions is integrable if there exist integrals of motion, , that satisfy the following conditions.

They are well defined Hermitian operators in the enveloping algebra of the Heisenberg algebra or convergent series in the basis vectors ,

They are algebraically independent in the sense that no Jordan polynomials formed entirely out of anticommutators in vanishes identically

the integrals commute pairwise.
The extension of these ideas from classical mechanics to quantum mechanics is straightforward, except for functional independence. There is no agreedupon operator equivalence for this concept. In this review we use algebraic independence of a set of operators . We say the operators are algebraically independent if there is no nonzero Jordan polynomial that vanishes identically. That is, there is no symmetrized polynomial in noncommuting variables such that . By symmetrized we mean that if the monomial appears in it occurs in the symmetric form via the anticommutator . Similarly third order products are symmetrized sums of 6 monomials, etc.
If the classical Hamiltonian admits a constant of the motion we also want to find a corresponding operator , expressible in terms of operators , that commutes with . However, in quantum mechanics the position and momentum operators do not commute so the analogy with the function isn’t clear. We usually take symmetrized products of any terms with mixed position and momentum, though there are other conventions. The quantization problem of determining suitable operator counterparts to classical Hamiltonians and constants of the motion is very difficult, sometimes impossible to solve but, for many of the simpler superintegrable systems the solution of the quantization problem is unique and straightforward.
More generally we can consider quantum systems on a Riemannian manifold, in analogy with the classical systems (2.1):
(2.20) 
where is the LaplaceBeltrami operator on the manifold [33],(Volume I). Inner products are computed using the volume maeasure . In the special case of Cartesian coordinates in Euclidean space, expression (2.20) reduces to (2.15).
We often consider the timeindependent Schrödinger equation for functions on complex Riemannian manifolds with complex coordinates . In these cases we are going beyond quantum mechanics and Hilbert spaces and considering formal eigenvalue problems. However many concepts carry over, such as symmetry operators, integrability, superintegrability, relations to special functions, etc. While for direct physical application it is important to specify the mass and to retain the Planck constant , for many mathematical computations we can rescale these constants, without loss of generality. We rewrite the Schrödinger eigenvalue equation as
(2.21) 
or where are rescaled potential and energy eigenvalue, respectively. This is appropriate because many of the superintegrable systems we consider have potentials of the form where is an arbitrary parameter. Thus, we will make the most convenient choice of , usually or , with the understanding that the result can be scaled to reinsert and obtain any desired . In cases where doesn’t admit an arbitrary multiplicative factor or the quantization problem can’t be solved we will have to retain the original formulation with given by (2.20). We shall see in Section 5 that in quantum mechanics integrable and superintegrable systems exist that have no classical counterparts. The limit corresponds to free motion, or this limit may be singular. This occurs when integrals of motion of order three or higher are involved. Thus, for the case of third and higherorder integrals of motion, it is best to keep explicitly in all formulas in order to be able to pass to the classical limit.
2.3 Superintegrability
Let us first focus on the explicit solvability properties of classical systems and operator systems . For this review, we will be focusing mainly on classical integrable and superintegrable systems that are polynomial in the momenta both for the purposes of quantization as well as simply for the fact that these are the most well studied. We thus define polynomial integrability and superintegrability for classical systems. The quantum analogy of these systems will then be integrals which are finiteorder differential operators and hence we define quantum integrability and superintegrability of finiteorder. For most of the review will will omit the the adjective “polynomial” or “of finiteorder” and just speak of integrability and superintegrability.
Similarly, we define superintegrability as having more than integrals of motion. There is thus a notion of minimal and maximal superintegrability. We will focus in this review on maximal superintegrability and often omit the adjective maximal. Maximal superintegrability requires that there exists integrals of motion, one of which (the Hamiltonian) commutes with all of the others. In this sense, maximal superintegrability coincides with noncommutative integrability [137, 131] which requires integrals, of which commute with all of the integrals.
Classical superintegrability and the order of integrable systems
Definition 7
A Hamiltonian system is (polynomially) integrable if it is integrable and the constants of the motion are each polynomials in the momenta globally defined (except possibly for singularities on lower dimensional manifolds).
Systems with symmetry beyond polynomial integrability are (polynomially) superintegrable; those with maximum possible symmetry are maximally (polynomially) superintegrable.
Definition 8
A classical Hamiltonian system in dimensions is (polynomially) superintegrable if it admits with functionally independent constants of the motion that are polynomial in the momenta and are globally defined except possibly for singularities on a lower dimensional manifold. It is minimally (polynomially) superintegrable if and maximally (polynomially) superintegrable if .
Every constant of the motion , polynomial or not, is a solution of the equation where is the Hamiltonian. This is a linear homogeneous first order partial differential equation for in variables. It is a wellknown result that every solution of such equations can be expressed as a function of functionally independent solutions, [33]. Thus there always exist independent functions, locally defined, in involution with the Hamiltonian, the largest possible number. However, it is rare to find such functions that are globally defined and polynomial in the momenta. Thus superintegrable systems are very special.
Most authors require a superintegrable system to be integrable. We have not done so here because we know of no proof (or counter example) that every superintegrable system in our sense is necessarily integrable. It is a theorem that at most functionally independent constants of the motion can be in mutual involution, [5]. However, several distinct subsets of the polynomial constants of the motion for a superintegrable system could be in involution. In that case the system is multiintegrable.
Another feature of superintegrable systems is that the classical orbits traced out by the trajectories can be determined algebraically, without the need for integration. Along any trajectory each of the symmetries is constant: , . Each equation determines a dimensional hypersurface in the dimensional phase space, and the trajectory must lie in that hypersurface. Thus the trajectory lies in the common intersection of independent hypersurfaces; hence it must be a curve. An important property of real superintegrable systems is loosely stated as “all bounded trajectories are periodic” [179]. The formal proof is in [137].
The polynomial constants of motion for a system with Hamiltonian H are elements of the (polynomial) Poisson algebra of the system. Stated as a lemma we have:
Lemma 1
Let be a Hamiltonian with constants of the motion . Then , and are also constants of the motion.
More generally, any set of polynomial constants of the motion will generate a symmetry algebra , a subalgebra of , simply by taking all possible finite combinations of scalar multiples, sums, products and Poission brackets of the generators. (Since is always a constant of the motion, we will always require that must belong to the symmetry algebra generated by .) We will be particularly interested in finding sets of generators for which . The defining constants of the motion of a polynomially integrable system do not generate a very interesting symmetry algebra, because all Poisson brackets of the generators vanish. However, for the generators of a polynomial superintegrable system the brackets cannot all vanish and the symmetry algebra has nontrivial structure.
The order of a polynomial constant of the motion is its order as a polynomial in the momenta. (Note that the order is an intrinsic property of a symmetry: It doesn’t change under a transformation from position coordinates to coordinates .) Here has order 2. The order of a set of generators , is the maximum order of the generators, excluding .
Let be a symmetry algebra of a Hamiltonian system generated by the set . Clearly, many different sets can generate the same symmetry algebra. Among all these there will be at least one set of generators for which is a minimum. Here is unique, although is not. We define the order of to be .
Extension to quantum systems
The extension of superintegrability to quantum systems is relatively straightforward. We state our definitions for systems determined by Schrödinger operators of the form (2.20) in dimensions.
Definition 9
A quantum system is integrable (of finiteorder) if it is integrable and the integrals of motion are finiteorder differential operators.
Definition 10
A quantum system in dimensions is superintegrable (of finiteorder) if it admits algebraically independent finiteorder partial differential operators in the variables globally defined (except for singularities on lower dimensional manifolds), such that =0. It is minimally superintegrable (of finiteorder) if and maximally superintegrable (of finiteorder) if .
Note that, unlike in the case of classical superintegrability, there is no proof that is indeed the maximal number of possible symmetry operators. However, there are no counterexamples known to the authors, that the maximum possible number of algebraically independent symmetry operators for a Hamiltonian of form (2.20) is .
In analogy with the classical case the symmetry operators for quantum Hamiltonian form the symmetry algebra of the quantum system, closed under scalar multiplication, multiplication and the commutator:
Lemma 2
Let be a Hamiltonian with symmetries , and be scalars. Then , and are also symmetries.
We will use the term symmetries and integrals of motion interchangeably.
Any set of symmetry operators will generate a symmetry algebra , a subalgebra of , simply by taking all possible finite combinations of scalar multiples, sums, products and commutators of the generators. (Since is always a symmetry, we will always require that belongs to the symmetry algebra generated by .) We will be particularly interested in finding sets of generators for which . However, for the generators of a superintegrable system the commutators cannot all vanish and the symmetry algebra has nonabelian structure, [189].
The order of a symmetry is its order as a linear differential operator. The order of a set of generators , is the maximum order of the generators, excluding . Let be a symmetry algebra of a Hamiltonian system generated by the set . Many different sets of symmetry operators can generate the same symmetry algebra. Among all these there will be a set for which is a minimum. We define the order of to be .
3 Important Examples
We present some simple but important examples of superintegrable systems and show, for classical systems, how the trajectories can be determined geometrically (without solving Newton’s or Hamilton’s equations) and, for quantum systems, how the energy spectrum can be determined algebraically (without solving the Schrödinger equation), simply by exploiting the structure of the symmetry algebra. We treat twodimensional versions of the Kepler and hydrogen atom systems in flat space and in positive constant curvature space and the hydrogen atom in threedimensional Euclidean space.
3.1 The classical Kepler system
The Kepler problem is a specific case of the two body problem for which one of the bodies is stationary relative to the other and the bodies interact according to an inverse square law. The motion of two isolated bodies satisfies this condition to good approximation if one is significantly more massive than the other. Kepler specifically investigated the motion of the planets around the sun and stated three laws of planetary motion: 1) Planetary orbits are planar ellipses with the Sun positioned at a focus. 2) A planetary orbit sweeps out equal areas in equal time. 3) The square of the period of an orbit is proportional to the cube of the length of the semimajor axis of the ellipse.
We will see that the precise mathematical statements of these laws are recovered simply via superintegrability analysis. Since planetary orbits lie in a plane we can write the Hamiltonian system in two dimensional Euclidean space with Cartesian coordinates:
(3.1) 
Hamilton’s equations of motion are
leading to Newton’s equations . We shall not solve these equations to obtain time dependence of the trajectories, but rather show how superintegrability alone implies Kepler’s laws and determines the orbits. Here and so three constants of the motion are required for superintegrability. Kepler’s second law is a statement of the conservation of angular momentum. The conserved quantity is which can be verified by checking that . Angular momentum is conserved in any Hamiltonian system with potential that depends only on the radial distance . However, the gravitational potential (along with the isotropic oscillator potential) is special in that it admits a third constant of the motion, see the Bertrand Theorem, [17, 56]. Consider the 2vector
(3.2) 
in the plane. One can check that , so the components of are constants of the motion and the classical Kepler system is superintegrable. The quantity is called the LaplaceRungeLenz vector. We will see that for an elliptical orbit or a hyperbolic trajectory the LaplaceRungeLenz vector is directed along the axis formed by the origin and the perihelion (point of closest approach) of the trajectory to the origin. The perihelion is timeinvariant in the Kepler problem, so the direction in which points must be a constant of the motion. The length squared of the Laplace vector is determined by the energy and angular momentum:
(3.3) 
We have found 4 constants of the motion and only can be functionally independent. The functional dependence is given by (3.3). We can use the symmetries, to generate the symmetry algebra. The remaining nonzero Poisson brackets are
(3.4) 
(The first equations show that transforms as a 2vector under rotations about the origin.) The structure equations do not define a Lie algebra, due to the quadratic term term . They, together with the Casimir (3.3), define a quadratic algebra, a Lie algebra only if is restricted to a constant energy. An alternative is to consider as a “loop parameter” and then the operators generate a twisted KacMoody algebra [35, 36].
Now suppose we have a specific solution of Hamilton’s equations with angular momentum and energy . Because of the radial symmetry we are free to choose the coordinate axes of the Cartesian coordinates centered at in any orientation we wish. We choose peripatetic coordinates such that the Laplace vector corresponding to the solution is pointed in the direction of the positive axis. In these coordinates we have , and . Then the first two structure equations simplify to , . The original expression for allows us to write:
which after rearrangement and squaring becomes
(3.5) 
As is well known from second year calculus, these are conic sections in the plane; our trajectories are ellipses, parabolas, and hyperbolas, depending on the discriminant of the equation, i.e., depending on the constants of the motion, . There are special cases of circles, stationary points, and straight lines. The three general cases are presented in Figure 1.
Returning to Kepler’s laws, the first is now obvious. The only closed trajectories, or orbits, are elliptical. Kepler’s second law is a statement of conservation of angular momentum. Indeed, introduce polar coordinates such that , note that along the trajectory
The area traced out from time to time is . Differentiating with respect to time: , so the rate is constant. Note that Kepler’s third law is only valid for closed trajectories: ellipses. We may write the period of such an orbit in terms of the constants of the motion. Explicit evaluation for , , yields as the area of the ellipse. Kepler’s third law follows easily from equation (3.5) and the simple calculus expression for the area of an ellipse.
3.2 A Kepler analogue on the 2sphere
There are analogs of the Kepler problem on spaces of nonzero constant curvature that are also superintegrable. We consider a 2sphere analog and show that superintegrability yields information about the trajectories, and that in a particular limit we recover the Euclidean space problem. It is convenient to consider the 2sphere as a two dimensional surface embedded in Euclidean 3space. Let be standard Cartesian coordinates. Then the equation
(3.6) 
defines the unit sphere. The embedding phase space is now six dimensional with conjugate momenta . The phase space for motion on the 2sphere will be a four dimensional submanifold of this Euclidean phase space. One of the constraints is (3.6). Since the tangent vector to any trajectory constrained to the sphere is orthogonal to the normal vector, we have the additional phase space constraint . The Hamiltonian is
(3.7) 
with and , where , are obtained by cyclic permutations of . If the universe has constant positive curvature, this would be a possible model for planetary motion about the Sun, [172]. Note that the are angular momentum generators, although are not constants of the motion. Due to the embedding, we have
so we can use the usual Euclidean Poisson bracket for our computations if at the end we restrict to the unit sphere. (Note that we have here normalized the parameters .) The Hamilton equations for the trajectories in phase space are , , . The classical basis for the constants is
(3.8) 
The structure and Casimir relations are
(3.9) 
(3.10) 
Here, transforms as a vector with respect to rotations about the axis. The Casimir relation expresses the square of the length of this vector in terms of the other constants of the motion: , , where . We choose the coordinate system so that the vector points in the direction of the positive axis: . Then
(3.11) 
Substituting the first three equations into the fourth, we obtain:
(3.12) 
For fixed values of the constants of the motion equation (3.12) describes a cone. Thus the orbit lies on the intersection of this cone with the unit sphere , a conic section. This is the spherical geometry analog of Kepler’s first law. A convenient way to view the trajectories is to project them onto the plane: . The projected points describe a curve in the unit disc . This curve is defined by
(3.13) 
The plus sign corresponds to the projection of the trajectory from the northern hemisphere, the minus sign to projection from the southern hemisphere.
Notice that the potential has an attractive singularity at the north pole: and a repulsive singularity at the south pole. In polar coordinates , the equation of the projected orbit on the plane is given by
(3.14) 
Theorem 2
For nonzero angular momentum, the projection sweeps out area in the plane at a constant rate with respect to the origin (0,0).
Theorem 3
For nonzero angular momentum, the period of the orbit is
(3.15) 
Contraction to the Euclidean space Kepler problem
The sphere model can be considered as describing a 2dimensional bounded “universe” of radius . Suppose an observer is situated “near” the attractive north pole. This observer uses a system of units with unit length where . The observer is unable to detect that she is on a 2sphere; to her the universe appears flat. In her units, the coordinates are
(3.16) 
Here, is so small that to the observer, it appears that the universe is the plane with local Cartesian coordinates . We define new constants by