From Unified Field Theory to Quantum Gravity and Back

1.4.1 Ehresmannian Geometry Advantages

Ehresmannian geometry gives the freedom to choose the internal symmetries of a physical theory without being confined to symmetries tied to the tangent bundle of Space-time.

This decoupling allows physics to account for forces like the Strong and Electro-Weak forces responsible for atomic nuclei, the electron orbitals that surround them, and their decay through beta radiation respectively.

Secondarily, the ability to remove redundancy in the description of nature by restricting our attention to Gauge Invariant quantities has turned out to be a powerful tool.

1.4.2 Riemannian Geometry Advantages

In the case of Riemannian and Pseudo-Riemannian geometries, the freedom to consider ad hoc candidates for physical symmetries is radically restricted. Yet there is again a major and a minor advantage.

While the full Riemann curvature tensor is a specific example of the more general Ehresmannian Curvature tensor construction, its decomposition into sub-components has no general analog in Ehresmannian geometry.

Thus the greatest advantage of Einstein and Grossman’s choice of Riemannian geometry is almost certainly to replace the broad freedom lost in bundle choice with the much more restricted freedom to play separately with the Weyl, (Traceless) Ricci, and Scalar components of the full curvature tensor as Einstein did in 1915.

While that may not seem to modern field theoretic tastes like a sensible trade-off given the loss in choice of structure, it appears sufficient for one very particular application of great importance: gravity. Further, properly abstracted, the projection operators onto curvature sub-components may be seen as tensor product decompositions in such a way as to include the Dirac operator on Spinors with its contraction on 1-form valued Spinors.

Viewed in this fashion, the ability to decompose tensor products of representations involving the tangent and cotangent bundles is the clear enticement to work within the metric paradigm.

The way this discussion has been framed by the above is designed to suggest the search for a parallel in the form of a second minor advantage. And here there is room for debate. The other main attribute of Riemannian geometry that suggests itself hides in plain sight under the label of the ‘The Fundamental Theorem’ of Riemannian geometry.

The choice of a Levi-Civita connection made by the metric is certainly convenient, but its use can always be seen as a choice to be made dynamically by allowing more general connections as suggested by Palatini. Thus its main value is that it allows the torsion tensor of any other connection to be calculated by converting the affine space of connections into an honest vector space with the Levi-Civita connection at the origin. However, this has always led to a puzzle: what good is being able to define the torsion tensor of a metric connection if the Levi-Civita connection is the only one that appears to matter in practice? Thus the torsion tensor appears, potentially, to be the solution to a problem which no one has yet thought to ask. As such, its potential advantages have so far been minor with its realized ones closer to non-existent. But it must be considered to be a potential advantage as the natural answer to a question which may some day arise.