The State Space of Perturbative Quantum Field Theory in Curved Spacetimes

TL;DR

This paper characterizes the space of continuous states for perturbative quantum field theories in curved spacetimes, showing they are precisely those with smooth higher-point functions and Hadamard singularities in the two-point function.

Contribution

It explicitly determines the structure of continuous states in perturbative QFT on curved spacetimes, extending the understanding of state spaces beyond free theories.

Findings

States have smooth truncated n-point functions for n≠2

Two-point functions exhibit Hadamard singularity structure

Positivity of states is crucial in the analysis

Abstract

The space of continuous states of perturbative interacting quantum field theories in globally hyperbolic curved spacetimes is determined. Following Brunetti and Fredenhagen, we first define an abstract algebra of observables which contains the Wick-polynomials of the free field as well as their time-ordered products, and hence, by the well-known rules of perturbative quantum field theory, also the observables (up to finite order) of interest for the interacting quantum field theory. We then determine the space of continuous states on this algebra. Our result is that this space consists precisely of those states whose truncated n-point functions of the free field are smooth for all n not equal to two, and whose two-point function has the singularity of a Hadamard fundamental form. A crucial role in our analysis is played by the positivity property of states. On the technical side, our proof involves functional analytic methods, in particular the methods of microlocal analysis.