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Physics Essays

Is There a Flaw in the Seminal Bohr‐Rosenfeld Paper on the Measurement Problem of the Free Quantum Electromagnetic Field?

Luis Gerardo Pedraza Saavedra

Published Mar 1, 2000• 2 citations
Publisher

Abstract

In 1933, Bohr and Rosenfeld published a seminal paper in which they claimed to solve the measurement problem of the free quantum electromagnetic field. In 1951, however, Corinaldesi discovered, in the analysis of the above-mentioned authors, an error in the order of magnitude of a certain critical field. This paper shows, through a comparative analysis between the results of Bohr and Rosenfeld and Corinaldesi, that the error discovered by the latter does not affect the conclusions of the former. It is also possible to consider as a deficiency of their work the fact that Bohr and Rosenfeld never responded to the mistake found by Corinaldesi. It will be shown here that in fact Bohr and Rosenfeld studied a measurement situation, the problem of which was equal to the one that would be originated if the correction of Corinaldesi were taken into account.

DOI
10.4006/1.3025421

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In 1933, Bohr and Rosenfeld published a seminal paper in which they claimed to solve the measurement problem of the free quantum electromagnetic field. In 1951, however, Corinaldesi discovered, in the analysis of the above-mentioned authors, an error in the order of magnitude of a certain critical field. This paper shows, through a comparative analysis between the results of Bohr and Rosenfeld and Corinaldesi, that the error discovered by the latter does not affect the conclusions of the former. It is also possible to consider as a deficiency of their work the fact that Bohr and Rosenfeld never responded to the mistake found by Corinaldesi. It will be shown here that in fact Bohr and Rosenfeld studied a measurement situation, the problem of which was equal to the one that would be originated if the correction of Corinaldesi were taken into account.
In the 1933 Bohr and Rosenfeld’s famous article on electromagnetic field measurements they did not consider the measurement of a magnetic component in a given space‐time region, nor the measurement of two parallel magnetic components, nor the measurement of two non‐parallel magnetic components, nor the parallel measurement of one magnetic and one electric component, nor the non‐parallel measurement of one electric and one magnetic component, nor the non‐parallel measurement of one magnetic and one electric component in two different space‐time regions. This article covers the first one of these pending matters.
In 1933, Bohr and Rosenfeld published a famous article, “On the question of the measurability of electromagnetic field quantities”, which by many accounts became a milestone. Historiography, though, does not seem to have really grasped its impact. In this brief paper I will highlight how the work by Bohr and Rosenfeld had an important reception story also outside of QED, more specifically in quantum gravity attempts: in the USSR, at the hands of Bronstein (mid-1930s), and in the US, at the hands of Wheeler (mid-1950s).
Bohr and Rosenfeld showed in 1933 that the order of magnitude of precision, in the measurement of the free quantum electromagnetic components imposed by their uncertainty relations, could be estimated with thought experiments. The ingenious article by Bohr and Rosenfeld describes the complicated measuring devices that would be needed, but does not include a graphic representation of these. Neither does it show the measurement of a magnetic component in a given space-time region, nor the measurement of two parallel magnetic components, nor the measurement of two non-parallel magnetic components, nor the parallel measurement of one magnetic and one electric component, nor the non-parallel measurement of one electric and one magnetic component, nor the non-parallel measurement of one magnetic and one electric component in two different space-time regions. This article covers these pending matters and the graphics of cases studied by them.
The precision with which the amplitude of the free electromagnetic field can be measured locally in QED is evaluated by analyzing a well-known gedanken experiment originally proposed by Bohr and Rosenfeld (BR). The analysis is performed by applying standard theoretical techniques familiar in quantum optics. The main result obtained for the precision is significantly different from the generally accepted Bohr-Rosenfeld result. This leads to questioning the widely accepted notion of the compensating field, fostered by these authors. A misconception at the origin of this notion is pointed out by a careful investigation of the self-force acting on the apparatus designed to measure the field. The correct expression for this self-force is found to be at variance with that proposed by Bohr and Rosenfeld and generally accepted. It is argued that, as a consequence of this new expression and in contrast with the generally accepted view, no compensating force of nonelectromagnetic nature is required in order to perform measurements of the quantum field amplitude with any desired accuracy. It is shown that the only limitations to the precision of the measurement, in the BR gedanken experiment, arise from the time-energy uncertainty principle, as well as from the finite dimensions of the measuring apparatus.
It is argued that the findings of a recent reanalysis by Compagno and Persico [Phys. Rev. A 57, 1595 (1998)] of the Bohr-Rosenfeld procedure for the measurement of a single space-time-averaged component of the electromagnetic field are incorrect when the field measurement time is shorter than that required for light to traverse the measurement's test body. To this end, the time-averaged ``self-force'' on the test body, assumed for simplicity to be of a spherical shape, is evaluated in terms of a one-dimensional quadrature for the general trajectory allowed for the test body by Compagno and Persico, and in closed form for the limiting steplike trajectory used by Bohr and Rosenfeld.
It is shown in two examples that to measure the momentum of a particle with great accuracy, allowing for large uncertainties in the position, it is still necessary to consider additional quantum effects. For the case of measurements by electromagnetic means, the uncertainties in the measurability of field strengths and of charge-current densities should be considered. Within the framework of the Bohr-Rosenfeld analysis, uncertainties in the field measurements do not limit the accuracy of momentum measurements. What happens when the Bohr-Rosenfeld approximations are not made is an open question.