Homotopy and path integrals in the time dependent Aharonov-Bohm effect

Foundations of Physics 41 (9):1462-1474 (2011)

For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the so-called holonomies are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function.
Keywords Aharonov-Bohm effect  Time-dependence  Homotopy  Path integral  Essential self-adjointness  Gauge theory  Vector potential  Holonomies
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DOI 10.1007/s10701-011-9559-y
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