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arXiv:math-ph/0210032 · Journal of Physics A Mathematical and General

Geometric factors in the Bohr-Rosenfeld analysis of the measurability of the electromagnetic field

Vladimir Hnizdo

Published Jan 1, 1999• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/0305-4470/32/12/015

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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.
The electromagnetic field commutation relations are defined in terms of geometric factors that are double averages over two finite four-dimensional space-time regions. The square root of any of the uncertainty relations derived from the aforementioned commutators is taken as a critical field, in the sense that any electromagnetic field much larger than it can be treated as classical. Another critical electromagnetic field associated with the quantum information control of vacuum fluctuations can be chosen as the square root of the mean quadratic fluctuation of each quantity of electromagnetic field, when the number of photons is defined and is equal to zero. Any electromagnetic field expectation value could be measured if it is much greater than the last critical field. This article covers a magnitude order comparison between the critical fields and its consequences for measuring the electromagnetic field information.
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 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.
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.