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 m 1 and m 2 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: M 2 [1— ( m 1 + m 2 ) 2 / M 2 ] [1 — ( m 1 — m 2 ) 2 / M 2 ] = 4 nΩ + 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 m 1 and m 2 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 M 2 = 4 nΩ + γ and s = n , n -2,....