Michael Atiyah’s synopsis of history of gauge theory brings out the salient problems that led to it and worth repeating: “Gauge theory first appeared in physics in an early attempt by H. Weyl to unify general relativity and electro-magnetism. Weyl had noticed the conformal invariance of the Maxwell’s equations and sought to exploit this fact by interpreting the Maxwell field as the distortion of relativistic length produced by moving around a closed path. Weyl’s interpretation was disputed by Einstein and never generally accepted. However after the advent of quantum mechanics with its all-important complex wave functions it became clear that phase rather than scale was the correct concept for Maxwell’s equations, or in modern language, that the gauge group was the circle rather than the multiplicative numbers. Unfortunately, while scale changes could be fitted into Einstein’s theory by replacing the metric with a conformal structure, there was no room for phase to be incorporated into general relativity. Rather the gauge theory had to be superimposed as an additional structure on space-time and the unification sought by Weyl then disappeared.”

The importance of this account of history is that it gives some perspective in the modification of general relativity that results in an background. Technically the modification is to minimize an Einstein-Hilbert action not over all metrics on a fixed three-dimensional manifold but over all hypersurfaces of $S^4(1/h)$. In order to be completely consistent with classical general relativity, one considers Lorentzian metrics on but the issues are clearer in the space-only analysis. For any hypersurface of and any point there is a unique normal geodesic which is a circle of length . Here we can see the room to fit electromagnetism in terms of scale but interestingly we are not dealing with the total space of a circle-bundle. In other words electromagnetism may be fitted geometrically here with gravity (which involves choice of a hypersurface) but the resulting electromagnetic theory will not necessarily be invariant under a circle-action because there is not always a away to deform along these circles without changing the metric on .

Now Kaluza-Klein theories have been considering products with with a circle of microscopic radius for a long period. Based on the strong bias towards three macroscopic space dimensions on apparently empirical grounds one would reject an universe but in the latter case the homogeneity of the the directions can allow an interpretation of the experienced three macroscopic spatial dimensions in terms of four fundamental spatial dimensions.

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