Factorisation conditions and causality for local measurements in QFT

TL;DR

The paper addresses the challenge of defining physically implementable, local measurements in quantum field theory (QFT) without enabling superluminal signalling. By adopting Bogoliubov’s local S-matrix framework and a hierarchy of factorisation conditions (notably continuous additivity and the Hammerstein property), the authors derive general no-signalling criteria for local operations and connect these to the induced Kraus operators on the field via probe-based dilations. They present a concrete exactly solvable model where a local pointer linearly couples to the smeared field, yielding explicit Kraus operators and a retarded-propagator bound that fixes the sharpness of local measurements; this binds measurement accuracy to causal propagation and links to previous abstract results on non-ideal measurements. The work clarifies when local measurements in QFT can be meaningfully defined and dilated, and suggests a relativistic analogue of Stinespring’s dilation with causality constraints, with potential implications for relativistic dilation theorems and the FV framework. Overall, the paper provides operational, locality-aware criteria that reconcile measurement theory with relativistic causality in QFT, and establishes concrete mechanisms by which one can perform or rule out local measurements in a relativistic setting.

Abstract

Quantum operations that are perfectly admissible in non-relativistic quantum theory can enable signalling between spacelike separated regions when naively imported into quantum field theory (QFT). Prominent examples of such "impossible measurements", in the sense of Sorkin, include certain unitary kicks and projective measurements. It is generally accepted that only those quantum operations whose physical implementation arises from a fully relativistically covariant interaction, between the quantum field and a suitable probe, should be regarded as admissible. While this idea has been realised at the level of abstract algebraic QFT, or via particular measurement models, there is still no general set of operational criteria characterising which measurements are physically implementable. In this work we adopt the local S-matrix formalism, and make use of a hierarchy of factorisation conditions that exclude both superluminal signalling and retrocausality, thereby providing such a criterion. Realising the local S-matrices through explicit interactions between smeared field operators and a pointer degree of freedom, we further derive local causality conditions for the induced Kraus operators, which guarantee the absence of signalling in "impossible measurement" scenarios. Finally, we show that the accuracy with which local field observables can be measured is fundamentally limited by the retarded propagator of the field, which also plays an essential role in a factorisation identity we prove for the field Kraus operators.

Figures (8)

• Figure 1: Bob's lab (white) is partly in the future of Alice's (red) and partly in the past of Charlie's (blue), while Alice and Charlie are strictly spacelike. Relativistic causality then requires that no signals can be sent from Alice to Charlie, i.e. there must be a no-signalling condition on the experimental setup. Here we have labelled the labs by the supports of the functions $f,g,h$, but the overall causal structure is common to all Sorkin scenarios.
• Figure 2: The region $R_\text{in}=\mathcal{M}\backslash J^+(R)\; (R_\text{out}=\mathcal{M}\backslash J^-(R))$ (where $\mathcal{M}$ is the spacetime, and $J^\pm(R)$ refer to the red and blue regions respectively) for a region $R$, given by the spacelike white regions and the blue (red) region. The out region is the set of all points $\mathsf x$ for which $\mathsf x\not\preccurlyeq \text{R}$, and likewise the in region is the set of all points $\mathsf y$ such that $\text{R}\not\preccurlyeq \mathsf y$.
• Figure 3: Continuous additivity of an $S$-matrix corresponds to selection of a (partial) Cauchy surface $\Sigma$ that bisects the support of $f$. The grey region above (below) $\Sigma$ is $J^+(\Sigma), \; (J^+(\Sigma))$.
• Figure 4: A Sorkin scenario where the dynamics of the interior region can be split as given in Eq \ref{['eq:continuousfactorisation']}. Note that $\Sigma$ is chosen such that $\mathop{\mathrm{supp}}\nolimits{f_-}$ is spacelike to A and $\mathop{\mathrm{supp}}\nolimits{f_+}$ is spacelike to C.
• Figure 5: The spacetime factorisation of $f$ into $f_+$ (red) $f_0$ (white) and $f_-$ (blue), where we note the disjoint and time ordered supports of $f_\pm$, and the support overlaps of $f_0$ with $f_\pm$. Key to the proof is that the support of $f_0$ is spacelike separated from both A and C.