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Quadratic Algebra Approach to Relativistic Quantum Smorodinsky-Winternitz Systems

There exist a relation between the Klein-Gordon and the Dirac equations with scalar and vector potentials of equal magnitude (SVPEM) and the Schrodinger equation. We obtain the relativistic energy spectrum for the four Smorodinsky-Winternitz systems from the quasi-Hamiltonian and the quadratic algebras obtained by Daskaloyannis in the non-relativistic context. We point out how results obtained in context of quantum superintegrable systems and their polynomial algebras may be applied to the quantum relativistic case. We also present the symmetry algebra of the Dirac equation for these four systems and show that the quadratic algebra obtained is equivalent to the one obtained from the quasi-Hamiltonian.

The study of quantum systems allowing polynomial integrals of motion using algebraic methods began in the early days of quantum mechanics [21][22][23]. A systematic search for classical and quantum superintegrable systems in two-dimensional Euclidean space with two second-order integrals of motion was presented in [24] and four classes of Hamiltonians were obtained. This search was pursued on spaces of constant and nonconstant curvature [25] and also for systems with third-order integrals of motion [26][27][28][29][30]. Over the years many articles were devoted to superintegrability and for a detailed review we refer the reader to [31]. The study of quantum superintegrable systems by the mean of quadratic algebras and their representations was discussed by many authors [32][33][34][35][36][37][38][39][40][41][42][43][44]. A general quadratic [37] and cubic [28,29] algebras respectively for superintegrable systems with two second order integrals of motion and systems with a second and a third order integrals were studied. Their realizations in terms of deformed oscillator algebras and the finite dimensional unitary representations were obtained. These results were used to obtain the energy spectrum of quantum superintegrable. Systems with higher order polynomial algebras and even infinite families of Hamiltonians with polynomial algebras of arbitrary order were also studied [30,42,43]. Moreover, we presented an algebraic derivation of the energy spectrum of the generalized MICZ-Kepler system in three-dimensional Euclidean space E 3 , its dual the four dimensional singular oscillator in four-dimensional Euclidean space E 4 [44] and the MICZ-Kepler system on the three sphere S 3 using the quadratic algebra approach. We thus pointed out how the quadratic algebra obtained in context of two-dimensional systems can be applied to higher dimensional systems. The quadratic algebra approach was also applied to an exactly solvable position-dependent mass Schrödinger equation in two dimenions [41].
The purpose of this paper is to study Smorodinsky-Winternitz systems in the quantum relativistic context, present an algebraic calculation of the energy spectra and also point out how results obtained in context of non relativistic quantum superintegrable systems and their polynomial algebras can also be applied to study relativistic quantum systems and used to obtain the relativistic energy spectrum.
In Section 2, we recall a relation between the Schrödinger equation and the Dirac and Klein-Gordon equations with scalar and vector potentials of equal magnitude. In Section 3, we also recall results obtained by Daskaloyannis [37] concerning non relativistic quantum superintegrable systems with two second-order integrals of motion and their quadratic algebras. In Section 4, we consider Klein-Gordon and Dirac equations with SVPEM involving the four Smorodinsky-Winternitz systems and obtain the four corresponding quasi-Hamiltonians [1]. We present the four quadratic algebras generated by the integrals of motion of the quasi-Hamiltonians, their realizations in terms of deformed oscillator algebras, together with their finite-dimensional unitary representations and the corresponding relativistic energy spectrum. In Section 5, we use a direct approach [7] and obtain the integrals of motion of the initial Dirac equation for the four Smorodinsky-Winternitz potentials. We also obtain the four quadratic algebras generated by these integrals and show that they are equivalent to those obtained from the quasi-Hamiltonians.

Klein-Gordon equation
The two-dimensional Klein-Gordon equation with equal scalar and vector potentials (SPVEM) can be transformed into a Schrödinger-like equation which is called a quasi-Hamiltonian [1] i.e. a Schrödinger equation without any spin dependence with mass and energy depending of the relativistic energy spectrum. Let us consider the time-independent Klein-Gordon equation We called V s ( r) the scalar potential and V v ( r) the vector potentials. When these two potentials have an equal magnitude 3) The equation (2.3) can be written as wherem andẼ are given by

Dirac equation
The two-dimensional time-independent Dirac equation with a scalar and a vector potentials has the following form [2][3][4][5][6][7][8][9][10][11][12][13][14] [cα · P + β( In the Pauli-Dirac representation ψ( r) = φ( r) ξ( r) , we obtain the following set of coupled equations for the spinor components When the scalar and the vector potentials are of equal magnitude the equation (2.7) and (2.8) reduce to the following system of two equations From these two equations given by (2.9), we obtain that the component φ( r) satisfy the Schrödinger-like equation (2.4) using the parameters given by equation (2.5). The same results are also valid for systems with scalar and vector potentials with opposite sign (pseudo-spin symmetry). It was also shown that the condition which originate the spin and pseudospin symmetries (i.e. SPVEM) in the Dirac equations are the same that produce equivalent energy spectra of relativistic spin-1 2 and spin-0 particles in the presence of vector and scalar potentials [20].
Considering the case of a Hamiltonian H allowing two second order integrals A and B (i.e. [H, A] = [H, B] = 0), the most general quadratic algebra generated by these integrals is given by the following commutation relations [37] : The structure constants γ, δ, β, ǫ, ζ and z of the quadratic algebra given by A deformed oscillator algebras is given by the following equations : There are realizations of the quadratic algebras (3.1) in a deformed oscillator They are two cases.
The Casimir operator K can be written in terms of the Hamiltonian only.
We have a energy dependent Fock space of dimension p+1 if The Fock space is defined by H|Ẽ, n >= E|E, n >, N|E, n >= n|E, n > b|E, 0 >= 0.

Quadratic algebras and quasi-Hamiltonians
The relation between Klein-Gordon and Dirac equations and the Schrödinger equation discussed in Section 2 allows to study relativistic systems from the quadratic algebra approach using the corresponding quasi-Hamiltonian.
In the case of a quasi-HamiltonianH allowing two second order integrals A and B (i.e. [H, A] = [H, B] = 0) the most general quadratic algebra is also given by (3.1) and the structure constant are now polynomials of the quasi-Hamiltonian.
The energy spectrumẼ of the quasi-Hamiltonian can be calculated from the constraints given by the equation (3.6). The equation (2.5) relate the energy spectrum of the quasi-Hamiltonian and the relativistic energy spectrum (E). Thus, the problem of obtaining the relativistic energy spectrum of the initial Dirac or Klein-Gordon equation can done algebraically from the quadratic algebra approach.

Smorodinsky-Winternitz systems
The four Smorodinsky-Winternitz systems [24] have the following form : These systems for the two-dimensional Euclidean space allows two second order integrals of motion [24]. They possess many properties quantum mechanics such the multiseparability and exact solvability [45]. Over the years many articles were devoted to these systems [45][46][47][48][49]. These systems were studied using the path integral approach [46] and their coherent states obtained recently [47]. Moreover, it was shown by Daskaloyannis that the integrals of motion for these four systems generate a quadratic algebra [37].

Quadratic algebras and relativistic energy spectum
As discussed in the Section 3, we can use the quadratic algebra approach [37] and the realizations of quadratic algebras as deformed oscillator algebras introduced in context of non relativistic quantum mechanics to obtain the relativistic energy spectrum of the Dirac or Klein-Gordon equations with SVPEM.
Let us present these quadratic algebras generated by the integrals of motion of the four quasi-Hamiltonians.

Case 2:
The structure function and the energy and structure function are given by (with u = 1 2 ) From the (2.5) and (4.5) we obtain the relativistic energy spectrum E r Case 3: The structure function is given by (with u = 1 2 (2 + ǫ 1k1 + ǫ 2k2 )) From the equation (2.5) and (4.8) we obtain the relativistic energy spectrum E r (E r −mc 2 )2m 2 (2(p+1)+ǫ 1 ( Er (4.9) Case 4: The structure function is given by (with u = ǫμ (4.11) From the equation (2.5) and (4.11) we obtain the relativistic energy spectrum E r

Symmetry algebras of Dirac equations
In Section 4, we obtained the relativistic energy spectrum of the Klein-Gordon and Dirac equations with SVPEM for the four Smorodinsky-Winternitz systems using the corresponding quasi-Hamiltonians and the quadratic algebras approach. In this Section, we will obtain the symmetry algebra directly of the Dirac equation given by equation (2.6) for these four systems and show that these quadratic algebras are the same than those obtained for the quasi-Hamiltonians.
Let us consider as in Ref.7 integrals of the following form The condition [Q d , H] = 0 give the following constraints where the components of A d and B d are closely related to the integrals of motion of the quasi-Hamiltonians and thus of the quantum non relativistic systems.

Conclusion
The main result is the application of the quadratic algebra approach in the quantum relativistic context. We studied Klein-Gordon and Dirac with scalar and vector potentials of equal magnitude involving the four Smorodinsky-Winternitz systems using the quadratic algebras of the corresponding quasi-Hamiltonians. We thus obtained the relativistic energy spectrum for these systems. We also applied directly the quadratic algebra approach to the initial Dirac equation for these four potentials. We also showed that quadratic algebras generated by the integrals of the quasi-Hamiltonians and the integrals of initial Dirac equations coincide.
We pointed out also how the study of quantum superintegrable systems and their polynomial algebras is also interesting in regard of relativistic quantum mechanics.