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

Notes about equivalence between the Sine-Gordon theory (free fermion point) and the free fermion theory. 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 obtaned. 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 constructed without Quantum Inverse Scattering Method (QISM) and the additional quantum integrals of motion are obtained.So the QISM is not absolutely complete. We have obtained equivalence between the Sine-Gordon theory (the free fermion point) and free fermions by comparing the expressions commuting with the Hamil-tonians of these theories in a space of integrated expressions of local densities. Equivalence of these theories has been investigated by comparing the expressions commuting with the Hamiltonians in a space of non integrated expressions of local densities. We have obtained some important differences in this space. In the subsequent paper we will investigate this equivalence between arbitrary values of coupling constants. In our opinion, we must present quantum integrals of motion if we quantize the integrable models. Quantum equivalence of the theories must be checked also by using quantum integrals of motion. The procedure of quanti-zation for an integrable theory, which has no quantum integrals of motion, must not be cousistent [1,2]. We have obtained that the S-G theory (the free fermions point) and the theory of massive fermions are not equivalent on the level of the quantum integrals of motion. So the question about massiveness of the S-G theory at this point is open. Our construction of quantum integrals of motion does not really use the QISM. Up to now in the literature (there are many articles and even textbook [5]) there is no evidence of that QISM gives all the integrals of motion. 2. Classical IM for free fermions Let us consider the classical model of free fermions in the light-cone coordinates. The equations of motion have the form i∂ − ψ 1 = mψ 2 , i∂ + ψ 2 = mψ 1 ,