Quantum Fields and Local Measurements

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.

Impossible measurements require impossible apparatus

“…This constitutes a measurement scheme [6] for an induced observable of the system and, importantly, yields an associated state-update rule. Although well established in quantum mechanics, this idea was only recently adapted to QFT in possibly curved spacetimes, thus implementing the concept of a measurement scheme in a local and covariant way [7] (see [8] for a summary). We call this the FV-framework and its elements FVmeasurement schemes.…”

“…In consequence, the idea of a measurement scheme can be implemented in QFT as a local concept. In particular, it was shown in [7,8] how the correspondence between probe observables and induced system observables may be made, and how rules for state update appropriate to selective and non-selective measurements may be described. A non-technical outline of these results now follows.…”

A path integral formulation for particle detectors: the Unruh-DeWitt model as a line defect

“…Operators of this form -non-local operators supported along a line on spacetime -are generically known as line defects [36]. At this point it is interesting to remark that this is an implementation of the ideas featured in [2][3][4]. Indeed, with (4.1) we have found the field observable corresponding to the presence of the detector.…”

“…Instead, we must look for a set of Grassmann variables whose quantization yields an operator algebra Cl 3 , which contains Spin(3) ∼ = SU (2). Recalling that Cl 3 coincides with the even part of Cl 4 [see 31, Theorem 3.7], we can do this by considering the operators constructed with an even number of fields coming from two tuplets (θ,θ) and (η,η), whose quantizations form two anticommuting copies of (3.1).…”

“…Elementary introductions to nonrelativistic quantum mechanics usually get away with this problem by postulating the idealized notion of projective measurements. However, this notion becomes untenable in quantum field theory (QFT) due to the extra requirements of relativistic locality and causality, which are in general violated by projective measurements acting on field states [1][2][3][4].…”