Quantum theory of the relativistic oscillator from the HOOP (higher-order one-particle) viewpoint

Expanding on earlier work, it is shown how to rewrite the non-relativistic problem of two particles coupled by a spring as one higher-order one-particle (HOOP) problem, which then can be elevated to be Lorentz covariant. The higher-order Lagrangian is then developed into an Ostrogradsky Hamiltonian format, from which canonical commutation rules are invoked and a relativistic wave equation produced. Within the centre-of-momentum frame of reference, the latter is reduced to a radial Schroedinger equation that is solved numerically, with a display of radial eigenfunctions that are compared with non-relativistic ones. The details of this example illustrate conclusively how the HOOP approach to direct-interaction many-particle relativistic dynamics works. (author)