Quantum Mechanics of the Electric Charge and its Connection with the Problem of Interpretation of Quantum Mechanics

Abstract

The orthodox interpretation of quantum mechanics presupposes division of the world into two parts: the observed quantum system and the observer. The observer does not have to be a man, it can be e.g. a computer, but it has to be ``classical{''} i.e. one has to be able to say that a certain variable has a definite value and this statement has to have a clear meaning. In practice ``classical{''} objects are classical only approximately. For example, a pointer of a scientific instrument consists of atoms which means that it is only approximately classical because collective motions of atoms which constitute the pointer reveal certain classical features. Are there genuinely classical objects i.e. objects whose classical behaviour is not a result of approximation but has, so to speak, ontological validity? There are many statements in the literature to the effect that the Coulomb field is such a genuinely classical object. For example, Berestetskii, Lifshitz, and Pitayevskii in their Relativistic Quantum Theory {[}1] which is a part of the celebrated Landau and Lifshitz course of theoretical physics, derive an inequality which says when the electric field is approximately classical, show that for a static field the inequality is trivially satisfied and conclude that ``a static field is always classical{''}. This conclusion seems to be shared by those field theorists who maintain that the asymptotic value of the density of the electric flux at the spatial infinity is a classical superselection parameter, see e.g. {[}2]. This point of view leads, however, to a strange paradox: we know that the electric charge is quantized. Therefore the allegedly classical flux density has to behave like the Bohr-Sommerfeld orbits in the old quantum theory: its spatio-temporal shape is classical but its scale must be quantized to give the quantized value of the total electric charge upon application of the Gauss law. Bohr-Sommerfeld orbits were universally felt intolerable. The present author finds intolerable the repetition of the same idea namely the idea of a classical object whose scale is arbitrarily quantized to save the phenomena. In his talk the author will show that the paradox can be resolved by a more careful analysis of the Berestetskii, Lifshitz, and Pitayevskii inequality.