Boson-fermion correspondence and quantum integrable and dispersionless models

This paper is devoted to a detailed description of the notion of boson-fermion correspondence introduced by Coleman and Mandelstam and to applications of this correspondence to integrable and related models. An explicit formulation of this correspondence in terms of massless fermionic fields is given, and properties of the resulting scalar field are studied. It is shown that this field is a well-defined operator-valued distribution on the fermionic Fock space. At the same time, this is a non-Weyl field, and its correlation functions do not exist. Further, realizing a bosonic field as a current of massless (chiral) fermions, we derive a hierarchy of quantum polynomial self-interactions of this field determined by the condition that the corresponding evolution equations of the fermionic fields are linear. It is proved that all the equations of this hierarchy are completely integrable and admit unique global solutions; however, in the classical limit this hierarchy reduces to the dispersionless KdV hierarchy. An application of our construction to the quantization of generic completely integrable interactions is shown by examples of the KdV and mKdV equations for which the quantization procedure of the Gardner–Zakharov–Faddeev bracket is carried out. It is shown that in both cases the corresponding Hamiltonians are sums of two well-defined operators which are bilinear and diagonal with respect to either fermion or boson (current) creation–annihilation operators. As a result, the quantization procedure needs no spatial cut-off and can be carried out on the whole axis of the spatial variable. It is shown that, in the framework of our approach, soliton states exist in the Hilbert space, and the soliton parameters are quantized.