Lectures on measurement in quantum field theory

TL;DR

The work addresses the lack of a covariant, causal measurement theory for quantum field theory, including curved spacetimes. It adopts the Fewster–Verch framework within algebraic QFT to model measurements as local couplings between a system and a probe in a compact spacetime zone, yielding an induced system observable and state-update rules that respect causality and composition. The approach provides concrete machinery for local measurements (via $\varepsilon_\sigma$, $\eta_\sigma$, and the scattering map $\Theta$), proves consistency under causal factorisation, and resolves Sorkin’s impossible-measurement paradox within this setting, while offering asymptotic measurement schemes in weak coupling. Taken together, the framework offers an operational, covariant protocol for measuring local QFT observables and informs interpretations of quantum states and updates in curved spacetimes, with applications to RFHGO models and beyond.

Abstract

These lectures present a brief introduction to measurement theory for QFT in possibly curved spacetimes introduced by the author and R. Verch [Comm. Math. Phys. 378 (2020) 851-889]. Topics include: a brief introduction to algebraic QFT, measurement schemes in QFT, state updates, multiple measurements and the resolution of Sorkin's "impossible measurement" problem. Examples using suitable theories based on Green hyperbolic operators are given, and the interpretational significance of the framework is briefly considered. The basic style is to give details relating to QFT while taking for granted various facts from the theory of globally hyperbolic spacetimes.

Figures (5)

• Figure 1: Illustration of $J^\pm(S)$, $J(S)$ and an example of a causal hull. Light rays are at $45^\circ$. The set $S$ is causally convex, i.e., equal to its causal hull.
• Figure 2: Illustrating the proof of Theorem \ref{['thm:timeslice']}. The red line represents a Cauchy surface within a region $O$ (not shown) that contains the region between Cauchy surfaces $\Sigma^\pm$. The support of $\chi E^+f$ is contained within the dashed boundary and is compact. In this instance $(1-\chi)E^-f=0$.
• Figure 3: Diagrams showing the 'in' region $M^-$ and 'out' region $M^+$ of a compact coupling zone $K$.
• Figure 4: $O_A$ and $O_C$ are causally disjoint, though $O_A$ can influence $O_B$, and $O_B$ can influence $O_C$. In a globally hyperbolic spacetime it follows that there is a region $N$ in $B$'s 'in' region, containing $O_C$ in its Cauchy development but causally disjoint from $O_A$.
• Figure 5: Illustration of the sets appearing in Lemma \ref{['lem:Theta_on_Lplus']}. $L^-$ lies in the 'in' region $M^-=M\setminus J^+(K)$ of coupling zone $K$, while $L^+$ lies in the 'out' region $M^-=M\setminus J^+(K)$ and is also in the Cauchy development of $L^-$.

Theorems & Definitions (27)

• Theorem 1.1
• proof
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6