Abstract
The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the “system”, using another, the “probe”. The measurement process involves a dynamical coupling of “system” and “probe” within a bounded spacetime region. The resulting “coupled theory” determines a scattering map on the uncoupled combination of the “system” and “probe” by reference to natural “in” and “out” spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the “in” region, the scattering map determines a completely positive map from “probe” observables in the “out” region to “induced system observables”, thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies–Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both “system” and “probe” are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory.
Highlights
This paper combines ideas and methods from algebraic quantum field theory (AQFT) and quantum measurement theory (QMT) in order to provide improved operational foundations for the measurement theory of relativistic quantum fields in spacetimes
This paper will introduce a generally covariant formalism of measurement schemes adapted to algebraic quantum field theory in curved spacetimes, illustrated by a specific model that can be analysed in detail
We discuss selective and non-selective measurements, introducing the concept of a pre-instrument as the map that sends system states to post-selected states conditioned on the observation of an effect. (The term ‘post-selected’ is used in various different ways in the literature—the precise meaning we have in mind, which amounts to updating the state based on the measurement outcome, will be spelled out in detail.) In particular, we show that at spacelike separation from the coupling region the original and postselected states agree only on observables that are uncorrelated, in the original state, with the system observable induced by the measured probe effect
Summary
This paper combines ideas and methods from algebraic quantum field theory (AQFT) and quantum measurement theory (QMT) in order to provide improved operational foundations for the measurement theory of relativistic quantum fields in (possibly curved) spacetimes. The generators obey various relations that will be spelled out in full later on This way of modelling ‘system’ and probe’, and in particular, their coupling, is in the spirit of the ‘standard model’ of a quantum measurement process as discussed in [18]; it appears in discussions of the Unruh effect taking as ‘probe’ a quantum field on Minkowski spacetime [65] or in a cavity [40]. We show that the scattering morphism satisfies the causal factorization property where multiple couplings are concerned, that the set of induced system observables forms a subalgebra of the algebra of smeared system fields, and that the results replicate those of first order perturbation theory in an appropriate limit.
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