The second development is the Aharonov- Bohm effect.18 A charged particle traveling outside a solenoid in which there is a magnetic field should not be affected, since the magnetic field is confined within the solenoid. Aharonov and Bohm predicted that the particle would be sensitive to whether there is a current in the solenoid, a prediction that was subsequently verified. A simplistic ontological interpretation, along the lines given above would claim that what is really real is the vector potential, A, the field Maxwell interpreted as representing a momentum, from which E and B may be derived. Then one might reinterpret measurements as really measuring E and A . This, however, would lead to ontological indeterminism, since measurements determine A only up to a gauge transformation. There is a reasonable consensus that the Aharonov-Bohm effect cannot be properly analyzed within the framework of classical electrodynamics. So, our final consideration concerns the status of quantum field theory.
We are not treating quantum field theory as such, merely considering its significance in determining the nature and limits of classical electrodynamics. Two points seem pertinent. First, the relativistic formulation of Maxwell's equations may be derived by setting up a Lagrangian field density for a matter field, applying the principle of local gauge invariance to the Lagrangian and its first derivatives and deducing a vector field, A, which obeys the relativistic formulation of Maxwell's equations.19 The second point is the idea of effective field theory. Both classical electrodynamics and quantum field theory may be thought of as successive low energy approximations to an ultimate theory that may be quite different, e.g., string theory or brane theory. (Weinberg, 1995, p. xx) If the real local action occurs at the level of unification of strong, weak, and electromagnetic interactions (10-17 cm), or at the level of quantum gravity (10-31 cm), then at the enormously larger levels proper to classical electrodynamics, any effective theory will look like a field theory. The value assigned to a field at a point is an idealization on the same level as the value assigned to an infinitesimal. Classical electrodynamics has a developed way of representing fields and assigning field values, but it does not have or presuppose an ontology of fields.
1.4 Bohr's Classical Crisis.
By 1900 the program of atomistic mechanism was clearly in trouble. For present purposes it is helpful to distinguish the two components: mechanistic foundations and atomic hypotheses. Though there was no minimally adequate atomic theory, anticipated explanations of the properties and activities of bodies in terms of their ultimate constituents had a unifying role in physics. Atomism, however, was in trouble. Thomson (by then Lord Kelvin) abandoned his vortex model of the atoms and suggested new models to accommodate electrical forces as basic.20 Spectroscopy, kinetic theory, and chemistry suggested different models of atoms with radical differences in the number of degrees of freedom. The large number of internal atomic degrees of freedom that spectroscopy required seemed incompatible with the equipartition of energy. The sustained attempts to give all of physics a mechanistic foundation had effectively foundered. Neither
18 This was introduced in Aharonov, 1959. Belot, 1998, interprets its general significance in a contemporary context.
19 Alternatively, it is possible to assume massless spin one particles, apply the principle of local gauge invariance and deduce the form of the matter field. See Weinberg, 1995, pp. 339-343.
20 The vortex model could not explain mass or handle dissipation. After Kelvin accepted electrons (which he called electrions) as atomic constituents he devised new atomic models. See Larmor, Kelvin, Papers, Vol. 6, pp. 204-243.