We investigate a generalisation to Lüders’ rule à la Aharonov–Albert in those globally hyperbolic spacetimes which allow unitarily equivalent Hilbert spaces to be defined along Cauchy hypersurfaces, thus relying on the existence of an interaction picture à la Tomonaga–Schwinger. We show that under this rule and under the additional assumptions of the integrability and unitarity of the Tomonaga–Schwinger dynamics and the foliation-independence of rays on acausal Cauchy hypersurfaces, selective quantum measurements satisfy a state-independent anyonic commutation relation over spacelike-separated precompact regions. We highlight that this propagates to positive operator-valued measures, where the commutation is necessarily bosonic. In the instantaneous-measurement idealisation, this implies quantum no-signalling for non-selective measurements. We then examine Sorkin’s impossible measurements and show that immediate contradictions can be averted as long as collapse-inducing measurements are irreversible. These results reaffirm the consistency of the Tomonaga–Schwinger picture of relativistic quantum theory, for which unitarity, integrability and foliation-independence of the states exclude superluminal signalling despite the “instantaneity” of a side-cone measurement collapse rule. We finish by discussing the possibility of extending such results beyond the interaction picture. • Prove Aharonov–Albert side-cone updates are consistent with Einstein causality. • Instantaneous nonselective measurements imply no-superluminal signalling. • Sorkin’s impossible measurements are avoided assuming irreversible measurements.
We investigate a generalisation to Lüders’ rule à la Aharonov–Albert in those globally hyperbolic spacetimes which allow unitarily equivalent Hilbert spaces to be defined along Cauchy hypersurfaces, thus relying on the existence of an interaction picture à la Tomonaga–Schwinger. We show that under this rule and under the additional assumptions of the integrability and unitarity of the Tomonaga–Schwinger dynamics and the foliation-independence of rays on acausal Cauchy hypersurfaces, selective quantum measurements satisfy a state-independent anyonic commutation relation over spacelike-separated precompact regions. We highlight that this propagates to positive operator-valued measures, where the commutation is necessarily bosonic. In the instantaneous-measurement idealisation, this implies quantum no-signalling for non-selective measurements. We then examine Sorkin’s impossible measurements and show that immediate contradictions can be averted as long as collapse-inducing measurements are irreversible. These results reaffirm the consistency of the Tomonaga–Schwinger picture of relativistic quantum theory, for which unitarity, integrability and foliation-independence of the states exclude superluminal signalling despite the “instantaneity” of a side-cone measurement collapse rule. We finish by discussing the possibility of extending such results beyond the interaction picture. • Prove Aharonov–Albert side-cone updates are consistent with Einstein causality. • Instantaneous nonselective measurements imply no-superluminal signalling. • Sorkin’s impossible measurements are avoided assuming irreversible measurements.
The topic of measurement in relativistic quantum field theory is addressed in this article. Some of the long standing problems of this subject are highlighted, including the incompatibility of an instantaneous ``collapse of the wavefunction'' with relativity of simultaneity, and the difficulty of maintaining causality in the rules for measurement highlighted by ``impossible measurement'' scenarios. Thereafter, the issue is considered from the perspective of mathematical physics. To this end, quantum field theory is described in a model-independent, operator algebraic setting, on generic Lorentzian spacetime manifolds. The process of measurement is modelled by a localized dynamical coupling between a quantum field called the ``system'', and another quantum field, called the ``probe''. The result of the dynamical coupling is a scattering map, whereby measurements carried out on the probe can be interpreted as measurements of induced observables on the system. The localization of the dynamical coupling allows it to derive causal relations for the induced observables. It will be discussed how this approach leads to the concept of selective or non-selective system state updates conditioned on the result of probe measurements, which in turn allows it to obtain conditional probabilities for consecutive probe measurements consistent with relativistic causality and general covariance, without the need for a physical collapse of the wavefunction. In particular, the problem of impossible measurements is resolved. Finally, there is a brief discussion of accelerated detectors and other related work.
Spontaneous collapse models provide a possible, testable solution to the quantum measurement problem, but constructing a relativistic version applicable to quantum field theory remains an important open challenge. Challenges in constructing relativistic collapse models include avoiding superluminal signaling and foliation dependence, controlling energy production, and ensuring vacuum stability. Here, we present a relativistic collapse framework that addresses these problems at the cost of introducing a new space of physical states, whose complete characterization is left open. A modified, history-dependent Aharonov--Albert prescription describes instantaneous nonlocal state updates on arbitrary spacelike foliations while preserving foliation independence. To ensure vacuum stability, we modify the standard unitary evolution through a prescription that we name dynamical vacuum subtraction. The resulting dynamical map is linear and trace-preserving but not positive on the full set of standard density operators. We establish no-signaling by proving microscopic parameter independence, and identify conditions under which standard nonrelativistic collapse dynamics are recovered. Finally, we apply our framework to a free massive scalar field, obtaining finite particle-number and energy-increase rates at leading order in the long-time regime, together with further promising evidence for the consistency of the framework.
Abstract Arguments by Sorkin (Impossible measurements on quantum fields. In: Directions in general relativity: proceedings of the 1993 International Symposium, Maryland, vol 2, pp 293–305, 1993) and Borsten et al. (Phys Rev D 104(2), 2021. https://doi.org/10.1103/PhysRevD.104.025012 ) establish that a natural extension of quantum measurement theory from non-relativistic quantum mechanics to relativistic quantum theory leads to the unacceptable consequence that expectation values in one region depend on which unitary operation is performed in a spacelike separated region. Sorkin [1] labels such scenarios ‘impossible measurements’. We explicitly present these arguments as a no-go result with the logical form of a reductio argument and investigate the consequences for measurement in quantum field theory (QFT). Sorkin-type impossible measurement scenarios clearly illustrate the moral that Microcausality is not by itself sufficient to rule out superluminal signalling in relativistic quantum theories that use Lüders’ rule. We review three different approaches to formulating an account of measurement for QFT and analyze their responses to the ‘impossible measurements’ problem. Two of the approaches are: a measurement theory based on detector models proposed in Polo-Gómez et al. (Phys Rev D, 2022. https://doi.org/10.1103/physrevd.105.065003 ) and a measurement framework for algebraic QFT proposed in Fewster and Verch (Commun Math Phys 378(2):851–889, 2020). Of particular interest for foundations of QFT is that they share common features that may hold general morals about how to represent measurement in QFT. These morals are about the role that dynamics plays in eliminating ‘impossible measurements’, the abandonment of the operational interpretation of local algebras A(O) A ( O ) as representing possible operations carried out in region O , and the interpretation of state update rules. Finally, we examine the form that the ‘impossible measurements’ problem takes in histories-based approaches and we discuss the remaining challenges.
In this essay, we explore the limits imposed by the impossibility of superluminal signalling on the class of physically realisable quantum operations, focusing on the difference in approaches one can take towards this problem in Hilbert space quantum theory and algebraic quantum field theory (AQFT). We review a recent proposal by Fewster and Verch for a generally covariant measurement scheme in AQFT, which we call the FV scheme; along the way, we argue that the framework as set up in its original proposal can be greatly simplified without losing generality or physical motivation. We then discuss the problem, introduced by Sorkin, that naive generalisations of the notion of operation to the relativistic context can lead to signalling between spacelike observers by introducing a third observer. We consider approaches to tackle this problem in Hilbert space quantum theory and argue that these are sometimes ambiguous, since the physical (im)possibility of operations depends in part on the specific spacetime embodiment of the quantum system under consideration, which lies outside the Hilbert space formalism. In the AQFT context, we review and generalise a recent result showing that operations induced by the FV measurement scheme do not enable superluminal signalling. We connect this result to Hilbert space by introducing a concrete model of AQFT, which we call the `hybrid model' as it provides a way to explicitly embed Hilbert space quantum theory into spacetime using the formalism of AQFT. Finally, we show that in this model, under suitable necessary and sufficient conditions, the converse result also holds: every operation that does not enable superluminal signalling can be implemented in the FV framework.
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" 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 to describe local operations, 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 plays an essential role in the factorisation identity we prove for the field Kraus operators.
We provide a framework and explicit construction for the regularized measurement of a large class of spacetime-localized observables in bosonic quantum field theory. The measurements fully satisfy relativistic causality and causal transparency, i.e., avoid unphysical superluminal signaling. We show explicitly how the measurement of time-extended observables back-reacts on itself and induces correlations between other measurements in its causal future. Our framework is fully compositional in spacetime and extends previous results on the measurement of instantaneous observables.