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Physical Review

Field and Charge Measurements in Quantum Electrodynamics

Niels Henrik David Bohr, L. Rosenfeld

Published Jun 15, 1950• 202 citations
Publisher

Abstract

A survey is given of the problem of measurability in quantum electrodynamics and it is shown that it is possible in principle, by the use of idealized measuring arrangements, to achieve full conformity with the interpretation of the formalism as regards the determination of field and charge quantities.
DOI
10.1103/physrev.78.794

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This chapter discusses quantum measurements of relativistic systems, such as quantum fields. The quantum theory of fields can be constructed with the help of a functional integral with integration over field configurations. It is natural that a description of measurements of quantum fields can be achieved by the restriction of such integrals. The main point is that different definitions of electromagnetic field strength lead to different conclusions about the existence or nonexistence of an absolute quantum restriction on the measurability of the field strength. To analyse the measurement process in quantum mechanics, one can use a simple model of the sensitive element of the instrument, sometimes called a 'meter'. The purely classical part of the instrument is not then explicitly considered. The coupling of the apparatus to the measured system is described by an interaction Lagrangian that is proportional to the product of the measured quantity and the coordinate of the meter.
The Aharonov--Bohm (AB) phase for a closed interferometric loop is widely regarded as gauge invariant. This conclusion is usually obtained by treating the electromagnetic source as a fixed background that is unaffected by the test charge. We investigate whether this conclusion remains valid when backreaction is included within the standard gauge-fixed formulation of quantum electrodynamics (QED). Our analysis indicates that, when the interfering branches become entangled with distinct electromagnetic field configurations, changes of the gauge-fixing condition can induce a non-vanishing shift of the total phase. This result gives rise to an apparent paradox: the total phase predicted by the QED formalism appears gauge dependent, whereas experiments robustly exhibit a definite interference shift. We examine two possible resolutions of this paradox. The standard approach attributes the observed interference only to a gauge-invariant component of the total phase and treats the remaining gauge-dependent contribution as physically irrelevant. We argue, however, that the QED formalism itself predicts the total phase governing the interference outcome and does not provide a dynamical prescription that extracts a gauge-invariant component while discarding the remaining terms. We then explore an alternative perspective in which the electromagnetic potential AμA_\muAμ​ is regarded as a physical field whose configuration is determined by the Lorenz condition together with appropriate causal and boundary conditions. From this viewpoint, the total phase can be interpreted as a well-defined physical quantity within the QED formalism. We conclude by discussing a possible experimental test based on direct measurements of open-path relative phases, which may help clarify the ontological status of electromagnetic potentials.
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