Topology Aharonov-Bohm effect

Therefore, this is a statement of our fundamental hypothesis, specifically, that the topology of the vacuum defines the field equations through group and gauge field theory. Prior to the inference and empirical verification of the Aharonov-Bohm effect, there was no such concept in classical electrodynamics, the ether having been denied by Lorentz, Poincare, Einstein, and others. Our development of 0(3) electrodynamics in this chapter, therefore, has a well-defined basis in fundamental topology and empirical data. In the course of the development of... 

The Aharonov-Bohm effect requires topological consideration [1], (i.e., a structured vacuum), and there exist conservation laws of topological origin, the simplest one is given by the sine-Gordon equation, which also appears in the discussion of 0(3) electrodynamics by Evans and Crowell [5]. 

Therefore, the distinction between the topological and dynamical phase has vanished, and the realization has been reached that the phase in optics and electrodynamics is a line integral, related to an area integral over Bt3> by a non-Abelian Stokes theorem, Eq. (553), applied with 0(3) symmetry-covariant derivatives. It is essential to understand that a non-Abelian Stokes theorem must be applied, as in Eq. (553), and not the ordinary Stokes theorem. We have also argued, earlier, how the non-Abelian Stokes explains the Aharonov-Bohm effect without difficulty. 

In the following sections, four topics are addressed (1) the mathematical entities, or waves, called solitons (2) the mathematical entities, called instantons (3) a beam—an electromagnetic wave—that is polarization-modulated over a set sampling interval, and (4) the Aharonov—Bohm effect. Our intention is to show that these entities, waves or effects, can be adequately characterized and differentiated, and thus understood, only by using topological... 

We consider now the Aharonov-Bohm effect as an example of a phenomenon understandable only from topological considerations. Beginning in 1959 Aharonov and Bohm [30] challenged the view that the classical vector potential produces no observable physical effects by proposing two experiments. The one that is most discussed is shown in Fig. 10. A beam of monoenergetic electrons exists from a source at X and is diffracted into two beams by the slits in a wall at Y1 and Y2. The two beams produce an interference pattern at III that is measured. Behind the wall is a solenoid, the B field of which points out of the paper. The absence of a free local magnetic monopole postulate in conventional... 

How does the topology of the situation affect the explanation of an effect A typical previous explanation [44] of the Aharonov-Bohm effect commences with the Lorentz force law ... 

We depart from former treatments in other ways. Commencing with a correct observation that the Aharonov-Bohm effect depends on the topology of the experimental situation and that the situation is not simply connected, a former treatment then erroneously seeks an explanation of the effect in the connectedness of the U(l) gauge symmetry of conventional electromagnetism, but for which (1) the potentials are ambiguously defined, (the U(l) A field is gauge invariant) and (2) in U(l) symmetry V x A = 0 outside the solenoid. 

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