Causality and dispersion relations and the role of the S-matrix in the ongoing research

TL;DR

This paper discusses the role of the S-matrix in quantum field theory, emphasizing the importance of modular localization and thermal properties, and critiques pure S-matrix approaches while proposing a new constructive framework.

Contribution

It introduces the role of modular localization and thermal properties in understanding the S-matrix, proposing a new nonperturbative approach to QFT that moves beyond pure S-matrix formulations.

Findings

Critiques the shortcomings of pure S-matrix approaches.

Highlights the importance of modular localization and thermal KMS condition.

Proposes a new constructive framework incorporating these concepts.

Abstract

The adaptation of the Kramers-Kronig dispersion relations to the causal localization structure of QFT led to an important project in particle physics, the only one with a successful closure. The same cannot be said about the subsequent attempts to formulate particle physics as a pure S-matrix project. The feasibility of a pure S-matrix approach are critically analyzed and their serious shortcomings are highlighted. Whereas the conceptual/mathematical demands of renormalized perturbation theory are modest and misunderstandings could easily be corrected, the correct understanding about the origin of the crossing property requires the use of the mathematical theory of modular localization and its relation to the thermal KMS condition. These new concepts, which combine localization, vacuum polarization and thermal properties under the roof of modular theory, will be explained and their potential use in a new constructive (nonperturbative) approach to QFT will be indicated. The S-matrix still plays a predominant role but, different from Heisenberg's and Mandelstam's proposals, the new project is not a pure S-matrix approach. The S-matrix plays a new role as a "relative modular invariant"..