Covariant Harmonic Oscillators and the Quark Model

An attempt is made to give a physical interpretation to the phenomenological wave function of Yukawa, which gives a correct nucleon form factor in the symmetric quark model. This wave function is first compared with the Bethe-Salpeter wave function. It is shown that they have similar Lorentz-contraction properties in the high-momentum limit. A hyperplane harmonic oscillator is then introduced. It is shown that the Yukawa wave function, which is defined over the entire four-dimensional Euclidean space, can be interpreted in terms of the three-dimensional hyperplane oscillators. It is shown further that this wave function satisfies a Lorentz-invariant differential equation from which excited harmonic-oscillator states can be constructed, and from which a gauge-invariant electromagnetic interaction can be generated.