Abstract
The Haag–Kastler net of local von Neumann algebras is constructed in the ultraviolet finite regime of the Sine–Gordon model, and its equivalence with the massive Thirring model is proved. In contrast to other authors, we do not add an auxiliary mass term, and we work completely in Lorentzian signature. The construction is based on the functional formalism for perturbative Algebraic Quantum Field Theory together with estimates originally derived within Constructive Quantum Field Theory and adapted to Lorentzian signature. The paper extends previous work by two of us.
Highlights
The corresponding local fields are approached in the so-called form factor program, which, has problems in proving the convergence of the arising series [20,30]
Since the vacuum state of the massless free scalar field is not a regular state on the Weyl algebra of the field, we use a representation introduced by Derezinski and Meissner [12], quite similar to the representation used in early day string theory
The expectation is that the Sine–Gordon theory is massive, and the result on its local quasiequivalence with the massive free theory suggests that the local von Neumann algebras generated by the relative S-matrices in the Derezinski-Meissner representation coincide with those which one would obtain in a vacuum representation of the model that still needs to be constructed
Summary
The free massless field in 2 dimensions is probably the simplest field theory one can think of. According to [1], local quasiequivalence of the representations induced by the vacuum (of mass m) and the Schubert state is equivalent to the following set of conditions (1) the symmetrized scalar products , m,sym and , S,sym , induced by the massive vacuum of mass m > 0 (subscript m) and by the Schubert state (subscript S), respectively, induce the same topology on L(I ), for any compact interval I ⊂ R, and (2) the square roots of the operators that define the respective 2-point functions in terms of e.g.
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