Anyons and the Bose-Fermi duality in the finite-temperature Thirring model

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 $λ=\sqrt{2(2n+1)π}, n\in {\bf N}$. These fermions are inequivalent and only for $n=1$ they are canonical fields. In the general case solutions are anyons. Different anyons (which are uncountably many) live in orthogonal spaces and obey dynamical equations (of the type of Heisenberg's "Urgleichung") characterized by the corresponding values of the statistic parameter. Thus statistic parameter turns out to be related to the coupling constant $λ$ and the whole Hilbert space becomes non-separable with a different "Urgleichung" satisfied in each of its sectors. This feature certainly cannot be seen by any power expansion in $λ$. Moreover, since the latter is tied to the statistic parameter, it is clear that such an expansion is doomed to failure and will never reveal the true structure of the theory. The correlation functions in the temperature state for the canonical dressed fermions are shown by us to coincide with the ones for bare fields, that is in agreement with the uniqueness of the $τ$-KMS state over the CAR algebra ($τ$ being the shift automorphism). Also the $α$-anyon two-point function is evaluated and for scalar field it reproduces the result that is known from the literature.