Fun and frustration with quarkonium in a 1+1 dimension

H.

Galić

, “,” , – ().

A. Z.

Capri

and

R.

Ferrari

, “,” , – ().

Since (1) is also the solution of the static massless Klein–Gordon equation with a point source in $1+1$ dimension, namely, $d2V/dx2=2gδ(x),$ and thus represents the massless [Lorentz] scalar field, say, due to a point quark. The reader may compare it with the solution of the static massive Klein–Gordon equation with a point source in $3+1$ dimension, namely, $(∇2−μ2)φ=Gδ(3)(x),$ given on pp. 8 and 9 of J. J. Sakurai, Advanced Quantum Mechanics (Addison–Wesley, Reading, MA, 1967).

R.

Jackiw

and

C.

Rebbi

, “,” , – ();

R.

Jackiw

and

J. R.

Schrieffer

, “,” , – ().

It would be interesting to derive this result analytically. But that is as of now an open problem. It is hoped that this short note may stimulate students and teachers alike to try to obtain an analytic proof.

However, we must mention here for the sake of completeness that field theoretic models in a $1+1$ dimension have been constructed, wherein one obtains a meson spectrum;

see, for example,

G.

’t Hooft

, “,” , – ().