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arXiv:math-ph/0405062 · Journal of Physics A Mathematical and General

On the existence of local observables in theories with a factorizingS-matrix

Gandalf Lechner

Published Mar 15, 2005• 35 citations
arXivHTMLPDF

Abstract

The scaling limit of the two-dimensional Ising model above the critical temperature is considered as an example for relativistic quantum theories on two-dimensional Minkowski space exhibiting a factorizing S -matrix. In this model, a recently proposed criterion for the existence of local quantum field theories with a prescribed factorizing scattering matrix is verified, thereby establishing a new constructive approach to two-dimensional quantum field theory in a particular example. The existence proof is accomplished by analysing the nuclearity properties of certain specific subsets of fermionic Fock spaces, and yields as a byproduct also a verification of the energy nuclearity condition of Buchholz and Wichmann in models of free fermions in four spacetime dimensions.

DOI
10.1088/0305-4470/38/13/015

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In quantum field theory, the rigorous construction of local observables in the presence of nontrivial interaction is a crucial problem. In a class of integrable quantum field theories, a very abstract existence proof has recently been given by Lechner. We give an explicit characterization of these local observables in terms of the properties of the coefficient functions in an expansion by interacting creators and annihilators. Some results on the operator domains of these local observables are given. Using these, we constructed explicit examples of local observables in the quantum Ising model.
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We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but 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. Our investigation focuses on scalar models without bound states. We establish sufficient criteria for the existence of averaged fields as closable operators, and complete the construction in the specific case of the massive Ising model.
We explore the analytic structure of the non-perturbative S-matrix in arguably the simplest family of massive non-integrable quantum field theories: the Ising field theory (IFT) in two dimensions, which may be viewed as the Ising CFT deformed by its two relevant operators, or equivalently, the scaling limit of the Ising model in a magnetic field. Our strategy is that of collider physics: we employ Hamiltonian truncation method (TFFSA) to extract the scattering phase of the lightest particles in the elastic regime, and combine it with S-matrix bootstrap methods based on unitarity and analyticity assumptions to determine the analytic continuation of the 2 to 2 S-matrix element to the complex s-plane. Focusing primarily on the "high temperature" regime in which the IFT interpolates between that of a weakly coupled massive fermion and the E8 affine Toda theory, we will numerically determine 3-particle amplitudes, follow the evolution of poles and certain resonances of the S-matrix, and exclude the possibility of unknown wide resonances up to reasonably high energies.
Abstract We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but 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. Our investigation focuses on scalar models without bound states. We establish sufficient criteria for the existence of averaged fields as closable operators, and complete the construction in the specific case of the massive Ising model.