Towards an Explicit Construction of Local Observables in Integrable Quantum Field Theories

Abstract

We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. Rather than exhibiting their n-point functions and verifying the Wightman axioms, we aim to establish them as closed operators affiliated with a net of local von Neumann algebras, which is defined indirectly via wedge-local quantities. We also investigate whether these fields have the Reeh-Schlieder property, and in which sense they generate the net of algebras. For these results, we establish sufficient criteria that do not depend on details of the interaction, i.e., on the two-particle scattering matrix, and we complete the construction in the case of the massive Ising model.

On the status of pointlike fields in integrable QFTs

“…It is at this point that convergence issues of an infinite series show up in the construction; (4) Show that Φ(g) is affiliated with A(O). This will follow from locality (2) and closability (3) by an abstract argument.…”

“…In the remainder of this section, we will comment briefly on each of these four aspects. Details of the construction can be found in [3].…”

“…A main difficulty in establishing (20) are the kinematical poles of the form factors: They mean that the kernels F m+n (θ + i0, η + iπ − i0) are boundary values of meromorphic functions at a pole, i.e., the T are singular integral operators of non-convolution type. Estimating the norms F m+n (α) m×n therefore requires considerable technical effort; see [3,Sec. 5.2] for the techniques used.…”

“…This turns out to be a consequence of locality (relative to the wedge-local field) and closability. The argument is roughly as follows; details can be found in [3,Sec. 3].…”

“…The present paper contains a brief and mostly non-technical summary of these results; the reader is referred to [3] for details.…”