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arXiv:2307.08524 · Foundations of Physics

Eliminating the ‘Impossible’: Recent Progress on Local Measurement Theory for Quantum Field Theory

Maria Papageorgiou, Doreen Fraser

Published May 8, 2024• 24 citations
arXivHTMLPDF

Abstract

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 $${\mathcal {A}}(O)$$ 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.

DOI
10.1007/s10701-024-00756-8

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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){\mathcal {A}}(O)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.
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.
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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 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 nnn-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, nnn-point functions can be consistently evaluated across any region having a definite causal relation with measurements.
As noted by Sorkin, regarding quantum instruments whose Kraus operators are localizable within some spacetime region as operations accessible therein leads to superluminal communication. This so-called Sorkin paradox can be resolved by further constraining the set of allowed local operations in quantum field theory (QFT). In this spirit, Fewster and Verch proposed a framework for local QFT operations that generalizes non-relativistic quantum measurement theory and does not lead to Sorkin-like paradoxes. Shortly afterwards, Jubb (and later Oeckl) identified the minimal conditions that QFT instruments must satisfy to be compatible with Einstein's causality. In this work, we study both approaches in the quantum field theory of the free scalar field. First, we prove that a very wide class of causal instruments is FV-realizable: namely, those whose measurement channels are random displacements of the field operators. As we show, this class allows implementing arbitrary instruments in a heralded, probabilistic way, as well as non-demolition measurements deterministically. Second, we construct examples of causal channels that do not admit an approximate FV realization. Some of such causal, not FV-realizable channels violate basic physical principles, so they should not be part of any measurement theory for QFT. Third, we investigate the difficulty of characterizing the set of QFT channels that can be generated through the composition of several FV schemes. In this regard, we find a countable family of simple QFT channels for which no Turing machine can discriminate between channels within or far away from the implementable set.