We have used a semiclassical method developed earlier to compute the particle spectrum of a field theory in two-dimensional space-time defined by the (sine-Gordon) Lagrangian $\frac{1}{2}{\left({\partial }_{\mu }\phi \right)}^2+\left(\frac{m^4}{\lambda }\right)\left\{cos\right[\left(\frac{\sqrt{\lambda }}{m}\right)\phi ]-1\}$. For weak coupling we find a heavy particle, the soliton, corresponding to a peculiar classical field configuration and an antisoliton. Below the soliton-antisoliton threshold there are a large number of further states. They can be viewed either as soliton-antisoliton bound states or as bound states of $n$ of the usual quanta of the theory. The "elementary particle" $\phi$ is the lowest of these. As the coupling increases, the higher states successively unbind, decaying into soliton-antisoliton pairs. At $\frac{\lambda }{m^2}=4\mathrm{}\pi$, the "elementary particle" unbinds leaving only solitons and antisolitons for $\frac{\lambda }{m^2}>4\mathrm{}\pi$. Comparing our semiclassical results with recent exact results of Coleman and with perturbation theory, we find that the semiclassical calculations are exact. This field theory seems similar to the hydrogen atom for which the Bohr-Sommerfeld quantization rules give the energy levels exactly. We also treat a ${\phi }^4$ theory in weak coupling and carry out a number of calculations which provide nontrivial illustrations of the semiclassical method.

• Received 27 January 1975

DOI:https://doi.org/10.1103/PhysRevD.11.3424