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To get an idea of the difference between trivial and non-trivial bundles, it is helpful to bear in mind the contrast between a cylinder and a M¨obius strip—the surface formed by rotating one end of a rectangle through two right angles out of the plane of the rectangle, and then gluing it to the other end to form a closed, twisted ribbon. Each is locally like the product of two spaces: a circle S and a line segment L with labeled end-points. But while one can smoothly map images of L onto ‘‘vertical’’ lines in the side of a cylinder all the way around, any map of images of L onto ”vertical” lines in the side of a M¨obius strip must incorporate discontinuities where arbitrarily close points on its edge are images of distinct end-points of L.

It is closely analogous to the Riemannian curvature of a spatial or space-time manifold. Fμν derives from in just the same way that Aμ derives from ω, namely as σ∗ = iFμν , where σ∗ is the pull-back of corresponding to the section σ. 7 of the electromagnetic field strength. But this is not the only way to understand gauge transformations and gauge symmetry in the fiber bundle formulation. It is not the way Trautman understands these notions, and it does not mesh well with the emphasis placed by Wu and Yang on the Dirac phase factor as offering just the right description of electromagnetism.

A choice of section σ for P determines how s represents the wave-function , and how ∇s represents Dμ , as well as how the connection is represented by Aμ on M, the manifold representing space (-time). γ˜ represents a horizontal lift of curve γ on M. As explained in appendix B, A# is the fundamental vector on P: its value in a section is directly related to Aμ . 2 a fiber bundle formulation 17 by Aμ = (ϕ, −A) or Aμ = (ϕ , −A ) respectively. In this way, a choice of potential for electromagnetism goes along with a choice of phase at each point for the particle’s wave-function, and this simultaneous choice corresponds to a choice of section for the principal bundle P.

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An extensible model of the electron by Dirac

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