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International Journal of Modern Physics B

A FORMAL PSEUDO REALISTIC MEASUREMENT OF ELECTROMAGNETIC FIELD UNCERTAINTY RELATIONS

Luis Gerardo Pedraza Saavedra

Published May 20, 2006• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217979206034030

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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.
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.
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.
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).
The unequal time commutators of the field operators of the free electromagnetic field, and their physical interpretation in terms of the simultaneous measurability of the EM field components at different space time points, have long been known since the work of Jordan and Pauli, and Bohr and Rosenfeld. The behaviour of these commutators can be understood in terms of the field of uncertain strength unavoidably generated as a consequence of a measurement of field strength in some region in space time. This field propagates freely in space, and if present in some other region in space time at which another field measurement is made, will affect the outcome of this measurement in an uncontrollable way. In the presence of matter, these unequal time commutators are modified, with readily understandable consequences for the measurability properties of the field. When calculating these commutators, the matter can be treated in two different ways. First of all is the case in which the matter occurs in bulk as in a macroscopic dielectric, with the quantized description of the EM field obtained by imposing classical boundary conditions on the EM field. The second case is that in which the matter is treated quantum mechanically. In both cases, the modified commutators contain terms whose physical significance can be understood as representing the effect of the field generated by a field measurement being scattered by the matter present. Thus one measurement can affect another either through the disturbing field propagating freely from one space time region to the other, or through the scattering of this field. In this paper, an outline is given of the general kinds of results obtained, how they can be obtained in each case, and where the results have played a role in determining the behaviour of certain quantum systems including the Glauber model of a photodetector, and in the theory of frequency filtered photon detection.