A Perturbative Approach to the Relativistic Harmonic Oscillator and Unitarity

Abstract

A quantum realization of the Relativistic Harmonic Oscillator is realized in terms of the spatial variable x and d dx (the minimal canonical representation). The eigenstates of the Hamiltonian operator are found (at lower order) by using a perturbation expansion in the constant c −1 . Unlike the Foldy-Wouthuysen transformed version of the relativistic hydrogen atom, conventional perturbation theory cannot be applied and a perturbation of the scalar product itself is required.

Group approach to the quantization of the Pöschl–Teller dynamics

“…To describe the quantum SL(2, R) symmetry, physically realized as a quantum relativistic harmonic oscillator [9,10,11] we can dilate (as the opposite to contract) the algebra (17) with an extra term in the r.h.s. so that it contract to the Poincaré algebra in the limit ω → 0 and to the non-relativistic harmonic oscillator, with angular frequency ω, in the c → ∞ limit.…”

“…This viewpoint has been demonstrated in many finite-and infinitedimensional cases by applying a Group Approach to Quantization developed since the original paper [8], where the quantum free Galilean particle and the harmonic oscillator were derived. Then, this algorithm has been applied to less elementary groups as those associated with relativistic particles, in particular the relativistic harmonic oscillator [9,10,11], field theories in curved space-times, non-linear σ-models, the Virasoro group and others concerning conformal symmetry and quantum gravity (see, for instance [12,13,14]). …”

Newton-Equivalent Hamiltonians for the Harmonic Oscillator

“…[21] and in the related papers Refs. [22][23][24], from which further work along similar lines can be traced. The two associated Hamiltonians and their eigenfunctions are quite different from ours.…”

Quantum Field Theory in a Symmetric Curved Space From a Second Quantization on a Group

“…Here, the functions ∆ (±) P (g ′ ,g) play the role of propagators (central matrices of the cocycle). The propagators in two different parametrizations ofG (2) , corresponding to two different polarization subalgebras P 1 and P 2 ofG L (or UG L ), are related through polarizationchanging operators (15) as follows:…”