We propose a measurement theory for quantum fields based on measurements made with localized nonrelativistic systems that couple covariantly to quantum fields (like the Unruh-DeWitt detector). Concretely, we analyze the positive operator-valued measure induced on the field when an idealized measurement is carried out on the detector after it coupled to the field. Using an information-theoretic approach, we provide a relativistic analogue to the quantum-mechanical L\"uders update rule to update the field state following the measurement on the detector. We argue that this proposal has all the desirable characteristics of a proper measurement theory. In particular it does not suffer from the ``impossible measurements'' problem pointed out by Rafael Sorkin in the 1990s which shows that idealized measurements cannot be used in quantum field theory.
We propose a measurement theory for quantum fields based on measurements made with localized nonrelativistic systems that couple covariantly to quantum fields (like the Unruh-DeWitt detector). Concretely, we analyze the positive operator-valued measure induced on the field when an idealized measurement is carried out on the detector after it coupled to the field. Using an information-theoretic approach, we provide a relativistic analogue to the quantum-mechanical L"uders update rule to update the field state following the measurement on the detector. We argue that this proposal has all the desirable characteristics of a proper measurement theory. In particular it does not suffer from the ``impossible measurements'' problem pointed out by Rafael Sorkin in the 1990s which shows that idealized measurements cannot be used in quantum field theory.
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.
We analyze potential violations of causality in Unruh DeWitt-type detector models in relativistic quantum information. We proceed by first studying the relation between faster-than-light signaling and the causal factorization of the dynamics for multiple detector-field interactions. We show in what way spatially extended nonrelativistic detector models predict superluminal propagation of the field's initial conditions. We draw parallels between this characteristic of detector models, stemming from their nonrelativistic dynamics, and Sorkin's ``impossible measurements on quantum fields'' [R. D. Sorkin, in Directions in General Relativity: Proceedings of the 1993 International Symposium, Maryland (Cambridge University Press, 1993), Vol. 2, pp. 293--305.]. Based on these features, we discuss the validity of measurements in quantum field theory when performed with nonrelativistic particle detectors.
In relativistic Quantum Field Theory (QFT) ideal measurements of certain observables are physically impossible without violating causality. This prompts two questions: i) can a given observable be ideally measured in QFT, and ii) if not, in what sense can it be measured? Here we formulate a necessary and sufficient condition that any measurement, and more generally any state update (quantum operation), must satisfy to respect causality. Our focus is scalar QFT, although our results should be applicable to observables in fermionic QFT. We argue that for unitary `kicks' and operations involving 1-parameter families of Kraus operators, e.g. Gaussian measurements, the only causal observables are smeared fields and the identity - the basic observables in QFT. We provide examples with more complicated operators such as products of smeared fields, and show that the associated state updates are acausal, and hence impossible. Despite this, one can still recover expectation values of such operators, and we show how to do this using only causal measurements of smeared fields.
In relativistic quantum field theory (QFT) ideal measurements of certain observables are physically impossible without violating causality. This prompts two questions: (i) can a given observable be ideally measured in QFT, and (ii) if not, in what sense can it be measured? Here we formulate a necessary and sufficient condition that any measurement, and more generally any state update (quantum operation), must satisfy to respect causality in real scalar QFT. We argue that for unitary `kicks' and operations involving 1-parameter families of Kraus operators, e.g., Gaussian measurements, the only causal observables are smeared fields and the identity---the basic observables in real scalar QFT. We provide examples with more complicated operators such as products of smeared fields, and show that the associated state updates are acausal, and hence impossible. Despite this, one can still recover expectation values of such operators, and we show how to do this using only causal measurements of smeared fields.
We present and investigate two issues within the measurement scheme for QFT presented by Polo-G\'omez et al. [Phys. Rev. D 105, 065003 (2022)]. We point out some discrepancies that arise when the measurement scheme is applied to contextual field states and show that n-point function assignments based on local processing regions sometimes lead to inconsistencies. To solve these issues, we modify the measurement scheme to use nonrelativistic detectors to induce an update rule for algebraic states in the Haag-Kastler formulation of quantum field theory. In this way, n-point functions can be consistently evaluated across any region having a definite causal relation with measurements.
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.