Covariant Harmonic Oscillator Formalism

The relativistic wave function quest originated with Schrödinger’s attempt to formulate his wave mechanics using the relativistic wave equation. The resulting Klein–Gordon equation has a negative energy problem and difficulties with a probability interpretation. The first problem was later solved by the second quantization procedure, but not the probability interpretation. Dirac’s equation for electrons, successful for electron properties in the static limit, has the same relativistic feature as the Klein–Gordon equation. In 1945, Dirac suggested the use of normalizable relativistic harmonic oscillator wave functions. Yukawa in 1953 constructed relativistic harmonic oscillator wave functions which led to infinite-component wave functions, but suggested a subsidiary condition involving the four-momentum of the particle was needed. Feynman et al. advocated the use of relativistic harmonic oscillators instead of Feynman diagrams for studying hadron structures and interactions. The basic problem facing any relativistic harmonic oscillator equation is the negative-energy spectrum due to time-like excitations. Eliminating time-like excitations was thought to lead to a violation of probability conservation. Harmonic oscillator wave functions without time-like excitations, but with a probability interpretation can be constructed. These form the vector spaces for unitary irreducible representations of the Poincaré group. We discuss this in detail here.