The Thirring model 40 years later

TL;DR

This paper revisits the Thirring model using algebraic quantum field theory, revealing that solutions are fermionic only at specific coupling constants, while otherwise describing a rich spectrum of anyons with complex Hilbert space structure.

Contribution

It demonstrates the existence of fermionic and anyonic solutions in the Thirring model at finite temperature, highlighting the limitations of perturbative expansions in capturing the true solution space.

Findings

Fermionic solutions occur only at specific coupling constants.

Most solutions are anyons with uncountably many types.

The Hilbert space becomes non-separable due to orthogonal anyon sectors.

Abstract

Solutions to the Thirring model are constructed in the framework of algebraic QFT. It is shown that for all positive temperatures there are fermionic solutions only if the coupling constant is $\lambda = \sqrt{2(2n+1)\pi}, n\in \bf N$, otherwise solutions are anyons. Different anyons (which are uncountably many) live in orthogonal spaces, so the whole Hilbert space becomes non-separable and in each of its sectors a different Urgleichung holds. This feature certainly cannot be seen by any power expansion in $\lambda$. Moreover, if the statistic parameter is tied to the coupling constant it is clear that such an expansion is doomed to failure and will never reveal the true structure of the theory. On the basis of the model in question, it is not possible to decide whether fermions or bosons are more fundamental since dressed fermions can be constructed either from bare fermions or directly from the current algebra.