Wavefront sets in algebraic quantum field theory

TL;DR

This paper generalizes the concept of wavefront sets to states in algebraic quantum field theory, introducing the asymptotic correlation spectrum and exploring its properties and implications for the singularities of quantum fields.

Contribution

It introduces the asymptotic correlation spectrum as a new wavefront set generalization for algebraic QFT states and analyzes its properties and relation to Wightman distributions.

Findings

The asymptotic correlation spectrum is well-defined for physical states.

Connections between the asymptotic correlation spectrum and wavefront sets of Wightman distributions are established.

Certain spacetime points are shown to be in the singular support of 2n-point distributions.

Abstract

The investigation of wavefront sets of n-point distributions in quantum field theory has recently acquired some attention stimulated by results obtained with the help of concepts from microlocal analysis in quantum field theory in curved spacetime. In the present paper, the notion of wavefront set of a distribution is generalized so as to be applicable to states and linear functionals on nets of operator algebras carrying a covariant action of the translation group in arbitrary dimension. In the case where one is given a quantum field theory in the operator algebraic framework, this generalized notion of wavefront set, called "asymptotic correlation spectrum", is further investigated and several of its properties for physical states are derived. We also investigate the connection between the asymptotic correlation spectrum of a physical state and the wavefront sets of the corresponding Wightman distributions if there is a Wightman field affiliated to the local operator algebras. Finally we present a new result (generalizing known facts) which shows that certain spacetime points must be contained in the singular supports of the 2n-point distributions of a non-trivial Wightman field.