Quantum sine-Gordon equation as the massive Thirring model

Abstract

The sine-Gordon equation is the theory of a massless scalar field in one space and one time dimension with interaction density proportional to $cos\ensuremath{\beta}\ensuremath{\phi}$, where $\ensuremath{\beta}$ is a real parameter. I show that if ${\ensuremath{\beta}}^{2}$ exceeds $8\ensuremath{\pi}$, the energy density of the theory is unbounded below; if ${\ensuremath{\beta}}^{2}$ equals $4\ensuremath{\pi}$, the theory is equivalent to the zero-charge sector of the theory of a free massive Fermi field; for other values of $\ensuremath{\beta}$, the theory is equivalent to the zero-charge sector of the massive Thirring model. The sine-Gordon soliton is identified with the fundamental fermion of the Thirring model.