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 $cos\beta \varphi$, where $\beta$ is a real parameter. I show that if ${\beta }^2$ exceeds $8\pi$, the energy density of the theory is unbounded below; if ${\beta }^2$ equals $4\pi$, the theory is equivalent to the zero-charge sector of the theory of a free massive Fermi field; for other values of $\beta$, 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.