The operator algebra of the quantum relativistic oscillator

Abstract

The operator algebras of a new family of relativistic geometric models of the relativistic oscillator [1] are studied. It is shown that, generally, the operator of number of quanta and the pair of the shift operators of each model are the generators of a non-unitary representation of the so(1, 2) algebra, except a special case when this algebra becomes the standard one of the non-relativistic harmonic oscillator.Pacs: 04.62.+v, 03.65.Ge

THE LEADER OPERATORS OF THE (d+1)-DIMENSIONAL RELATIVISTIC ROTATING OSCILLATORS

“…We hope that further investigations should enlighten this delicate point. Concluding we can say that the results presented here are in accordance with our previous results we have obtained for the relativistic (1 + 1)-dimensional Pöschl-Teller problems [2,3]. One can convince ourselves that the operators shifting the quantum number n (of the pairs with m = 2, 3) have similar forms to those of the leader operators of the models with d = 1 [2].…”

“…This means that these models can be studied using the method of supersymmetry and shape invariance [10] with minimal changes requested by the specific form of the Klein-Gordon equation [3,7]. In this way we have shown that all the (1+1)-dimensional Pöschl-Teller models have the same dynamical algebra, so(1, 2), formed by a pair of leader operators and the operator of the number of oscillation quanta [2,3,4]. In the case of any dimensions we have studied the supersymmetry of the relativistic Pöschl-Teller radial problems but without to write down the leader operators [7].…”

A relativistic supersymmetry and some algebraic consequences

“…6 These have the advantage of the special natural frames in which the metrics are conformally flat and, consequently, the relativistic scalar product coincides with the usual one. 7 In these frames the models with countable energy spectra of our family of (1ϩ1) models appear as the relativistic correspondents of the well-known nonrelativistic Pöschl-Teller systems. 8,9 They will be referred as relativistic Pöschl-Teller models ͑RPTMs͒.…”

“…We must specify that in the frame (t,x) the scalar product of the functions defined on D is just the usual one. 7 This means that here the definition of the Hermitian conjugation is also the usual one ͑e.g., ‫ץ‬ x ϩ ϭϪ‫ץ‬ x ͒. The Klein-Gordon equation of a test particle of the mass m and energy E has the form…”

“…The major difference is that the Hamiltonian of the nonrelativistic theory is replaced here by the operator ⌬, which is linearly dependent on the squared energy operator. 7 For this reason, the role of the energies of the nonrelativistic theory of supersymmetry is played now by the differences ͑10͒. Moreover, the homogeneous structure of the Klein-Gordon equation induces some kind of regularities which lead to a very simple parametrization of the theory in such a manner that for any pair of models with superpartner potentials we have ⌬kϭ1.…”