Abstract
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,....
Reference7 articles.
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3. Feynman R. P. 1972 Photon-hadron interactions. Reading Massachusetts: Benjamin.
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5. A cohomological description of massive fields
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