Gauge theory is about the geometry of internal spaces

Abstract

In general relativity, the strong equivalence principle is underpinned by a geometrical interpretation of fields on spacetime: all fields and bodies probe the same geometry. This geometric interpretation implies that the parallel transport of all spacetime tensors and spinors is dictated by a single affine connection. Can something similar be said about gauge theory? Agreed, in gauge theory different symmetry groups rule the interactions of different types of charges, so we cannot expect to find the same kind of universality found in the gravitational case. Nonetheless, the parallel transport of all the fields that are charged under the same symmetry group is dictated by a single 'gauge connection', and they all transform jointly under a gauge transformation. Is this kind of 'restricted universality' as geometrically underpinned as in general relativity? Here I argue that it is. The key difference is that the gauge geometry concerns 'internal', as opposed to 'external', spaces. The gauge symmetry of the standard model is thus understood as merely the automorphism group of an internal geometric structure -- $C^3\otimes C^2\otimes C^1$ endowed with an orientation and canonical inner product -- in the same way as spacetime symmetries (such as Poincare transformations), are understood as the automorphism group of an external geometric structure (respectively, a Minkowski metric). And the Ehresmann connection can then be understood as determining parallelism for this internal geometry.