Rigorous Formulation of a 2D Conformal Model in the Fock–Krein Space: Construction of the Global Op J ^*-Algebra of Fields and Currents

Abstract

We use the formalism of the 2D massless scalar field model in an indefinite space of the Fock–Krein type as a basis for constructing a rigorous formulation of 2D quantum conformal theories. We show that the sought construction is a several-stage procedure whose central block is the construction of a new type of representation of the Virasoro algebra. We develop the first stage of this procedure, which is to construct a special global algebra of fields and currents generated by exponential generators. We obtain a system of commutation relations for the Wick-squared currents used in the definition of the Virasoro generators. We prove the existence of Wick exponentials of the current given by operator-valued generalized functions; the sought global algebra is rigorously defined as the algebra of current and field, Wick and normal exponentials on a common dense invariant domain in a Fock–Krein space.