The Relativistic Oscillator

This paper addresses the problem of the quantization of the relativistic simple harmonic oscillator. The oscillator consists of a pair of scalar particles of masses m1 and m2 moving under the influence of a potential that is linear in the squared magnitude of the spatial separation of the particles. A novel feature of the model is that the potential is an operator, this being necessary to render the notion of spatial separation for a pair of particles meaningful in the context of relativistic quantum theory. The state of the oscillator is characterized (as in the non-relativistic theory) by the excitation number n and the total spin s. The total mass M of the system is quantized, and the main result of the paper is to derive a formula for the allowable mass-levels, namely: M2[1-(m1+m2)2/M2][1-(m1-m2)2/M2]=4nΩ+γ, where Ω and γ are constants (with dimensions of mass squared) which determine the strength and zero-point energy of the oscillator, respectively. A striking feature of this formula is that when m1 and m2 are both small compared with M (for example, for 'light quarks' combining to form meson states) the allowable states of the system lie on linear Regge trajectories, with M2 = 4nΩ+γ and s = n, n-2, ... .