Thirring Model in Terms of Currents: Solution and Light-Cone Expansions

Abstract

Exact expansions of operator products, in terms of $c$-number functions singular on the light cone and regular operators, are exhibited explicitly in the Thirring model. For the products ${\psi }_1\mathrm{}\left(x\right)\mathrm{}{\psi }_1^{†}\left(x^{\prime }\right)$ and ${\psi }_2\mathrm{}\left(x\right)\mathrm{} {\psi }_2^{†}\left(x^{\prime }\right)$ of fermion fields the expansion reduces to one term only, with the $c$-number function having a singularity on the light cone which depends on the coupling constant, and the regular operator depending only on the currents, which are free. The resulting formula allows one to calculate all Wightman functions in terms of current matrix elements and thereby provides a simple and complete solution to the Thirring model. The different charge sectors are realized as inequivalent irreducible representation spaces of the canonical current commutation relations, on which the charged field $\psi$ acts as an intertwining operator.