How can we generalize harmonic oscillator to relativistic case ?
One way (see here, here, and here) is simply to write a relativistic Lagrangian/Hamiltonian with a quadratic potential. Such an oscillator is "harmonic" in the sense that we call a quadratic potential "harmonic". However, it is arguably not Harmonic in terms of the equations-of-motion and/or symmetry between position and momentum (which plays important role in QM, e.g., in the context of coherent states.) In this sense, there might be multiple generalizations to relativistic and quantum relativistic case, depending on what property of the non-relativistic oscillator one would like to preserve.
