11 Observed Field Content

One of the features that arises when doing away with the primary nature of space-time and replacing a single metric space with a tension between two separate but related spaces linked by metrics, is that we find ourselves in the novel situation where must relate fields that are native to different spaces. The principal means of doing this is via the pull back operation.

To interpret differential equations governing fields native to Y back on X means pulling back not only bundles but so-called Jets or Sprays of sections on Y . But, to begin with, we can simply analyze the zeroth order of the activity on Y by pulling back bundles via the ג

∗ operation derived from making a metric observation of Y by X. To begin with, let us detach from the spaces TxX to general metric vector spaces W in even dimensions to say a few words about direct sums and tensor products of defining and spin representations for Spin(W).

11.1 Fermionic Quantum Numbers as Reply to Rabi’s question.

To begin with, there is a simple rule for tensor products of defining representations and spinors whereby the tensor product W ⊗ /SW = /SW ⊕ /RW (11.1) breaks into a piece representing the action of gamma matrices as spinor endomorphisms and a second piece giving the pure Rarita-Schwinger spin 3/2 representation corresponding to the sum of the highest weights of the factors.

We note further, that spinor representations carry the property of the exponential in that they take in direct sums as input and return products of the spinors of the summands as output.

W = U ⊕ V /S(W) = /S(U ⊕ V ) = /S(U) ⊗ /S(V ) (11.2)

Both of these are likely to be well known to physicists. Some what less familiar is that the Rarita-Schwinger representation has slightly odd behavior when applied to direct sums of vector spaces.

with an odd re-appearance of a final term which has purely spinorial with no 3/2 spin Rarita-Schwinger component.

To apply the above to our situation we recognize that ζ represents a spinor valued 1-form and ν a spinor on Y with U representing the Horizontal and V the Vertical normal bundle Nג to the metric as an embedding ג : X −→ Y. (11.4)

Even at zero-level before the introduction of higher Jets, the pull back of ν, ζ and their host bundles is potentially of considerable interest.

11.2 The Three Family Problem in GU and Imposter Generations.

‘Who ordered that?’ -Isidore Rabi on the Muon

We have had to restrict ourselves to a world without auxiliary internal quantum numbers as essentially everything has been generated endogenously from X4.

This leaves the question of why we appear to see a rich offering of repeating internal Fermionic quantum numbers.

In fact we will make two likely to be controversial claims in this section that may appear to fly in the face of experimental observation. The first is that we do not believe that nature has simply repeated herself three times albeit at different mass scales. While we do believe that a second copy of Fermionic matter matches this description, we believe that a third family is merely effectively identical to the other two and, presumably, only at low energy.

Secondly, while we are often told that the discovery of parity violation in beta decay found in the 1950s by Chien-Shiung Wu following theories of Yang and Lee, proves that nature is intrinsically chiral, we will again hazard the guess that it is merely effectively chiral so that at a deeper level it remains intrinsically balanced between left and right. To see this more clearly we will decompose our Fermionic sector under the decompositions of ג

To this, our rolled up Fermionic complex looks quite different under the above tangent space decomposition:

for Spin(1, 3)×Spin(6, 4). The idea being explored here is that the full operator depicted decouples effectively into two separate Dirac like operators, when there is no vacuum expectation value pulling the various sub-fields of $ to values significantly above zero. Thus we assert that a non-chiral total theory splits at the emergent level into two separate chiral theories and that the one above the dashed line corresponds to matter in our world with the other sectors not labeled by F to the left and above the line are currently dark to us. 11.3 Explict Values: Predicting the Rest of Rabi’s Order. With all that said above, we can now predict what the internal quantum numbers will likely be if GU is correct as per the following:

Names Multiplicity Dimension Structure Notation Name(s)

after reductions of the structure groups. A more violent regime would be expected to reveal differences that are more profound than mere mass discrepancies.

Another surprise would be a new cousin spin- 3

the logic of the known matters is reversed in the sense that it is right handed matter and left handed anti-matter that feel the effects of Weak-Isospin. Names Multiplicity Dimension Structure Notation Name(s)

Number Multiplicity Dimension Structure Electric Charge Name(s)

11.4 Bosonic Decompositions

As we have argued previously, the observed standard model appears to be consistent with a reduction of structure group of the full Dirac Spinors on Y in 3 stages

1. First: to a splitting of T

2. Second to a Maximal Compact Subgroup of structure group of the Normal bundle Nג from Spin(6, 4) to Spin(6) × Spin(4) ∼= SU(4) × SU(2) × SU(2) in accordance with the Pati-Salam theory.

3. Lastly to a complex structure on the Normal bundle from Spin(6)×Spin(4) to U(3) × U(2) where a final reductive factor may be removed or not by privileging a complex volume form.

As far as understanding the picture of the force carrying gauge potentials, we are really looking to understand what parts of Ω1 (Y, ad) carries meaning for us already within the standard model. To that end we can see that at the level of vector spaces if not as algebras, the ad−bundle is equivalent to the exterior bundle on Y . Both the branching of the 1-forms Ω1 (Y ) and the ad-bundle are straightforward under pull back of ג

Space-Time Cosmological Constant and Dirac Mass

Fiber Cosmological Constant and Dirac Mass