By S. Weinberg

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**Additional resources for What is Quantum Field Theory, and What Did We Think It Is**

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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.

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

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