Group-theoretical construction of the quantum relativistic harmonic oscillator

We elaborate on a proposal for a quantum relativistic harmonic oscillator, based on a group-theoretical framework. The wave functions in both configuration and Bargmann-Fock-like space are explicitly given and a generalized Bargmann transform is also provided. The energy eigen-functions in configuration space are composed of a general weight function (the vacuum), which leads to the Gaussian one in the c → ∞ limit, a particular power of the vacuum reducing to unity in this limit and a polynomial leading to the corresponding non-relativistic Hermite polynomial as c → ∞.