Quantum-field Theories as Representations of a Single∗-Algebra

A. Raab

doi:10.13189/ujpa.2013.010217

Abstract

We show that many well-known quantum field theories emerge as representations of a single $^\ast$-algebra. These include free quantum field theories in flat and curved space-times, lattice quantum field theories, Wightman quantum field theories, and string theories. We prove that such theories can be approximated on lattices, and we give a rigorous definition of the continuum limit of lattice quantum field theories.

Highlights

The Wightman distributions play a fundamental role in Wightman quantum field theories (Wightman QFTs) [1]
The Wightman distributions define a state of a Borchers-Uhlmann (BU) algebra, and the associated Wightman QFT is obtained as a representation of a BU-algebra from that state [2, 3]
The quantum-field algebra, A(M ), is obtained by factoring Free(M ) by a set of relations, in which coefficients of an operator-product expansion (OPE) play a fundamental role, i.e. there exists a ∗-homomorphism, π : Free(M ) → A(M ), which essentially is defined by properties of the OPE coefficients

Summary

Introduction

The Wightman distributions play a fundamental role in Wightman quantum field theories (Wightman QFTs) [1]. The Wightman axioms provide a remarkably successful framework in Minkowski space-time, they cannot be generalized to curved space-times in a straightforward manner This is one of the motivations to study QFT in an algebraic framework, and substantial progress has been achieved with this approach in recent years. We use Q-maps and Q-theories as purely technical devices to construct appropriate ∗-algebras, from which QFTs arise as representations As it is the case for quantum fields in Wightman QFTs, for example, the image of a Q-map is a set of operators in a ∗-algebra, and a Q-theory is the continuous representation of the polynomial algebra generated by the operators in the image of a Q-map.

General setting

Quantum theories from Q-map cores

A universal Q-map

Application to Wightman quantum field theories

Application to free quantum field theories

Application to perturbative quantum field theory in curved spacetimes

Lattice quantum field theory

String theory