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The Relativistic Oscillator

Hughston, LP; (1982) The Relativistic Oscillator. Proceedings of the Royal Society of London A , 382 pp. 459-466. 10.1098/rspa.1982.0112.

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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] = 4n\Omega+\gamma $, where $\Omega$ and $\gamma$ 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 = 4n\Omega + \gamma $ and s = n, n-2, and so on.

Type: Article
Title: The Relativistic Oscillator
DOI: 10.1098/rspa.1982.0112
Keywords: relativistic quantum mechanics, relativistic harmonic oscillator, entanglement of relativistic systems, Regge trajectories
UCL classification: UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: http://discovery.ucl.ac.uk/id/eprint/1364053
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