Unified Field Content: The Inhomogeneous Gauge Group and Fermionic Extension

Up until now we have been dealing with finite-dimensional constructions that superficially appear to be geometric (e.g. spinorial), but are actually mostly topological in nature. In this section we focus on the field content of GU which brings us to infinite dimensional algebraic constructions.

5.1 Field Content

In what follows, we endeavor to list the field content of Geometric Unity. The major nuance here is that the fields live on separate spaces which is kept track of by having only two fields on X with Hebrew orthography. All other fields live on Y and carry Greek orthography.

5.1.1 Field Content Native to X

In everything that follows, we will endeavor to separate out all field content native to X by having it appear with Hebrew Letters. There is a single primary field ג native to X which is a section of the metric bundle Y (X) as well as a derived field ℵ = ℵג representing the Levi-Civita connection across all bundles on which it is induced from the metric ג.

5.1.2 Field Content Native to Y

There is a single unified field ω native to Y . Here our interpretation of Unified field is interpreted to mean unified in an algebraic sense of indecomposible. With that said, we should say at the outset that ω is comprised of interlocking sub-sectors:

Of the four sub-component fields, only one, ε is non-linear at the level of topological spaces. Letting ¯ε denote its linearization, the four components fit neatly within the following simple table of tensor products: (Linearized) Field Content on Y:

It is the contention of the author that since the introduction of Special Relativity in 1905, Physicists have become dependent on affine space techniques for their understanding of relativistic mechanics as well as both classical and quantum field theories. While space-time has obviously not been considered flat since Einstein and Grossman first introduced General Relativity in 1913, we are rather more sympathetic to the emphasis on affine space than our frequent irritation with excuse making for Minkowski space techniques might suggest. Simply put, we see affine physics as being central to our understanding of the world and requiring no excuse making, but believe that the culture has chosen the wrong affine space and dimensionality for its emphasis given the presence of gravity.

The simple principle we follow here is that we should implement on the affine space of connections what we are otherwise tempted to do on flattened space-time. To this end we set notation.