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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Formal Description of Measurements in Local Quantum Field Theory

K. -E. Hellwig, K. Kraus

Published Jan 15, 1970• 108 citations
Publisher

Abstract

Ideal measurements in quantum mechanics can be described formally as instantaneous changes of the state vector or, more generally, of the density matrix $W$. By its lack of covariance, this formalism is not very adequate for relativistic quantum theories such as local field theory. For the latter, we propose a modified formalism, according to which a field measurement in a space-time region $C$ changes the field state $W$ in the future and side cone of $C$. This proposal is justified by physical considerations.

DOI
10.1103/physrevd.1.566

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Ideal measurements in quantum mechanics can be described formally as instantaneous changes of the state vector or, more generally, of the density matrix WWW. By its lack of covariance, this formalism is not very adequate for relativistic quantum theories such as local field theory. For the latter, we propose a modified formalism, according to which a field measurement in a space-time region CCC changes the field state WWW in the future and side cone of CCC. This proposal is justified by physical considerations.
This chapter considers a more realistic variation of the measuring procedure, taking into account the fact that the measured particles become entangled with the measuring apparatus at some finite time. The results are also generalized to quantum field theory, where quantum states are represented by wave functions depending on an infinite number of space-time coordinates. Thus, to make Bohmian mechanics covariant, one needs first to reformulate the standard quantum mechanics (QM) in a covariant way such that time is treated on an equal footing with space. Another physically interesting situation is when the entanglement with the environment takes the form, where are coherent states. In general, in the Bohmian interpretation all the other quantities may have a distribution totally different from those predicted by purely probabilistic QM. The space probability density should be generalized to the space-time probability density.
The measurement of nonlocal properties of relativistic quantum-mechanical systems, the compatibility (or lack of it) of two or more such measurements, and various other of their characteristics are considered; and with these notions in hand an old problem (which is to produce a covariant description of the state reduction associated with the measuring process) is attacked, and succumbs. The solution requires us to depart (not as we did in part I of the present work, but in an entirely different direction) from the usual picture of the time evolution of quantum states.
It is shown that the attempt to extend the notion of ideal measurement to quantum field theory leads to a conflict with locality, because (for most observables) the state vector reduction associated with an ideal measurement acts to transmit information faster than light. Two examples of such information-transfer are given, first in the quantum mechanics of a pair of coupled subsystems, and then for the free scalar field in flat spacetime. It is argued that this problem leaves the Hilbert space formulation of quantum field theory with no definite measurement theory, removing whatever advantages it may have seemed to possess vis a vis the sum-over-histories approach, and reinforcing the view that a sum-over-histories framework is the most promising one for quantum gravity. INTRODUCTION: IDEAL MEASUREMENTS AND QUANTUM FIELD THEORY Whatever may be its philosophical limitations, the textbook interpretation of nonrelativistic quantum mechanics is probably adequate to provide the quantum formalism with all the predictive power required for laboratory applications. It is also self-consistent in the sense that there exist idealized models of measurements which allow the system-observer boundary to be displaced arbitrarily far in the direction of the observer. And the associated “transformation theory” possesses a certain formal beauty, seemingly realizing the “complementarity principle” in terms of the unitary equivalence of all orthonormal bases.
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 study the instantaneity of the state-reduction process in relativistic quantum mechanics. The conclusion of various authors that this instantaneity will restrict the set of relativistic observables to purely local ones (i.e., that the measurement of any nonlocal property of a system at a well-defined time would give rise to violations of relativistic causality) is found to be erroneous, and experiments (of a kind not encountered before in measurement theory) are described whereby certain nonlocal properties of some simple physical systems can be measured at a well-defined time without violating causality. The attempts of certain authors to reconcile the reduction process with the covariance of the relativistic quantum state are considered and found wanting, and it is argued that the covariance of relativistic quantum theories resides exclusively in the experimental probabilities, and not in the underlying quantum states. The problem of nonlocal measurement is considered in general: distinctions (which are not to be met with in the nonrelativistic case) arise in relativistic quantum mechanics between what can be measured for fermions and what can be measured for bosons, between what can be measured for individual systems and what can be measured for ensembles, and between what kinds of states can be verified by measurement and what kinds of states can be prepared by measurement; and these pose difficult questions about the nature of measurement itself.
In its usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turn, again gives time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics? We observe that the preferred role of time in quantum theory is the consequence of an idealization: that measurements are instantaneous. Canonical quantum theory can be given a covariant form by dropping this idealization. States prepared by noninstantaneous measurements are described by ``spacetime smeared states.'' The theory can be formulated in terms of these states, without making any reference to a special time variable. The quantum dynamics is expressed in terms of the propagator, an object covariantly defined on the extended configuration space.