A C*-algebraic Approach to Interacting Quantum Field Theories

Detlev Buchholz,Klaus Fredenhagen

doi:10.1007/s00220-020-03700-9

Abstract

A novel C*-algebraic framework is presented for relativistic quantum field theories, fixed by a Lagrangean. It combines the postulates of local quantum physics, encoded in the Haag–Kastler axioms, with insights gained in the perturbative approach to quantum field theory. Key ingredients are an appropriate version of Bogolubov’s relative S-operators and a reformulation of the Schwinger–Dyson equations. These are used to define for any classical relativistic Lagrangean of a scalar field a non-trivial local net of C*-algebras, encoding the resulting interactions at the quantum level. The construction works in any number of space-time dimensions. It reduces the longstanding existence problem of interacting quantum field theories in physical spacetime to the question of whether the C*-algebras so constructed admit suitable states, such as stable ground and equilibrium states. The method is illustrated on the example of a non-interacting field and it is shown how to pass from it within the algebra to interacting theories by relying on a rigorous local version of the interaction picture.

Highlights

Quantum field theory aims to reconcile the principles of quantum physics, governing the microcosmos, with those of relativistic causality, regulating all physical processes
It combines the postulates of local quantum physics, encoded in the Haag–Kastler axioms, with insights gained in the perturbative approach to quantum field theory
Whereas quantum mechanics quickly matured into a meaningful theory with solid mathematical foundations, the consolidation of quantum field theory took several decades and, as a matter of fact, has not yet come to a fully satisfactory end

Summary

Introduction

Quantum field theory aims to reconcile the principles of quantum physics, governing the microcosmos, with those of relativistic causality, regulating all physical processes. Disregarding examples in a low dimensional model world or non-interacting theories, these constructive attempts have not yet succeeded in establishing the existence of quantum field theories in real spacetime, which comply with all basic constraints put forward in the general framework [22, 32] It is the aim of the present article to combine these two attempts. The existence of vacuum and thermal equilibrium states in physical spacetime has been established in interacting theories in the perturbative approach to the S-operators [15,16,17] These encouraging results do not yet settle the problem for the dynamical algebras in the present C*-algebraic setting. In the “Appendix”, the dynamical relations used in our approach are derived from the Schwinger–Dyson equations

Classical Field Theory

F2 time F1

Non-interacting Theories

Interacting Theories

Summary and Outlook