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By S. Weinberg

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Appendix B shows that in the case of a principal fiber bundle P(M, G), this is provided by a geometric object called a Lie-algebra-valued one-form ω on P. Moreover, each section maps ω onto a Lie-algebra-valued one-form iA on (an open set of) M, where A is a covector field that may be identified with the (electro)magnetic (four-)vector potential! But while ω is a geometric object that is quite independent of sections, it corresponds to a particular A only relative to a given section. Consequently, a change of section induces a 5 A vector bundle always has a global section, whether or not it is trivial.

How is this phenomenon to be explained? At first sight, it appears that the magnetic field inside the solenoid must have some kind of non-local effect on the electrons, since B is zero everywhere outside the solenoid, in the region through which they must pass on their way from the slits to the screen. But Aharonov and Bohm denied that the effect was non-local, claiming instead that it arose from a purely local interaction with the magnetic vector potential A (or more generally the electromagnetic potential Aμ ).

4 In general, the structure group G of a fiber bundle is a continuous group of transformations that map every Fm onto itself. In a principal fiber bundle P(M, G), the typical fiber F is just the structure group G itself. 3. A fiber bundle (E, π, M, F, G). The total space E is represented by the M¨obius strip and the fibers by lines isomorphic to the typical fiber F (with element f ). The projection map π sends the fiber π−1 (m) into m ∈ M. The base space M is covered by coordinate patches. The local trivialization χγ untwists the fibers and reveals the total space as a local product Uγ × F.

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What is Quantum Field Theory, and What Did We Think It Is by S. Weinberg


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