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AIP conference proceedings

Measurement of electrical field amplitude by a needle-like detector

G. Compagno

Published Jan 1, 2000• 0 citations
Publisher

Abstract

The Bohr-Rosenfeld measurement theory of the electromagnetic field amplitude is applied to a needle-like detector. The expression of the time average of the self-force due to the measurement process is explicitly obtained. It is shown that the appearance of divergences does not allow the approximations required by the Bohr-Rosenfeld theory.

DOI
10.1063/1.1303334

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The Bohr-Rosenfeld measurement theory of the electromagnetic field amplitude is applied to a needle-like detector. The expression of the time average of the self-force due to the measurement process is explicitly obtained. It is shown that the appearance of divergences does not allow the approximations required by the Bohr-Rosenfeld theory.
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.
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.
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.
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.
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.
The geometric factors in the field commutators and spring constants of the measurement devices in the famous analysis of the measurability of the electromagnetic field by Bohr and Rosenfeld are calculated using a Fourier-Bessel method for the evaluation of folding integrals, which enables one to obtain the general geometric factors as a Fourier-Bessel series. When the space regions over which the factors are defined are spherical, the Fourier-Bessel series terms are given by elementary functions, and using the standard Fourier-integral method of calculating folding integrals, the geometric factors can be evaluated in terms of manageable closed-form expressions.