Notes about equivalence between Sine-Gordon theory (free fermion point) and the free fermion theory

TL;DR

This paper investigates the relationship between the Sine-Gordon theory at the free fermion point and free fermion theory, revealing differences in their local integrals of motion and challenging the completeness of the Quantum Inverse Scattering Method.

Contribution

It demonstrates that the equivalence between Sine-Gordon and free fermion theories breaks down at the level of local integrals of motion and constructs these integrals without QISM.

Findings

Differences in local integrals of motion between the theories.

Construction of integrals without QISM.

QISM is not fully complete for these theories.

Abstract

The space of local integrals of motion for the Sine-Gordon theory (the free fermion point) and the theory of free fermions in the light cone coordinates is investigated. Some important differences between the spaces of local integrals of motion of these theories are obtained. The equivalence is broken on the level of the integrals of motion between bosonic and fermionic theories (in the free fermion point). The integrals of motion are constracted without Quantum Inverse Scattering Method (QISM)and the additional quantum integrals of motion are obtaned. So the QISM is not absolutely complete.