New Uncertainty Properties of the Electromagnetic Field

Georges Lemaître

L.D. Faddeev, L.A. Khalfin, I.V. Komarov

If we assume that the most precise measurements of electromagnetic fields using electrons or protons as “test bodies” have been carried out, then by reason of the quantum mechanical uncertainty principle for the coordinate and momentum of the bodies, we get certain bounds on the measurability of the field intensities, namely, the same bounds for measurements both with electrons and protons. In the sense of the quantum mechanical concepts, it means that these fundamental restrictions on the measurability of the electromagnetic fields suggest a reasonable limitation on the possibility to determine precisely the state of the classical field.

A. V. Kozlovskiĭ

It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrödinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.

A. Widom, Terrence D. Clark

Daniel F. Walls, Gerard J Milburn

Peter W. Milonni

Abstract The electromagnetic field is quantized for free space and for dispersive dielectric media. Different quantum states of the field (thermal, laser, squeezed) are discussed and their photon-statistical properties derived. The density matrix operator for the field is introduced along with a detailed discussion of the diagonal representation of the field density operator and its properties. Field commutation relations, correlation functions, and uncertainty relations are derived and discussed. Also discussed are the historically important Bohr-Rosenfeld analysis of field commutation and uncertainty relations; energy-time uncertainty relations; simultaneous (Arthurs-Kelly) measurement of non-commuting observables; and complementarity from the perspective of probability amplitudes and probabilities for distinguishable and indistinguishable processes.