Chirality Is Merely Effective

12.9 Chirality Is Merely Effective and Results From Decoupling a Fundamentally Non-Chiral Theory

Consider a stylized system of equations for a world Y with metric g, having scalar curvature R(y), and endowed with a non-chiral Dirac operator operating on full Dirac Spinors,

… (12.12)

12The author finds supersymmetry unnecessarily confusing as an as-if symmetry and is uncomfortable saying much more about it.

which are nonetheless decomposed into chiral Weyl component-spinors. Solving both of these equations together yields a system of coupled equations: /∂AψL(y) = R(y) 4 ψR(y) (12.13)

/∂AψR(y) = R(y) 4 ψL(y) (12.14)

leading to a stylized massive Dirac Equation with mass m = R(y)4

for any fixed background metric for which the scalar curvature R(y) is approximately constant in a region under study. However, in any region where the scalar curvature was zero or sufficiently close to zero,

R(y) ≈ 0 (12.15)

the differential equations would decouple as they are only linked by the scalar curvature term of order zero.

/∂AψL(x) ≈ 0 (12.16) /∂AψR(y) ≈ 0 (12.17)

This however, is not the end of the story when the tangent bundle has further structure. In the neighborhood of an embedding such as we have in:

… (12.19)

from our previous discussion.

However at the level of the chiral Weyl halves of the total Dirac Spinor we have two decompositions:

…

Luminous Light Standard Model Family Matter

…

(12.20)

Dark Decoupled Looking Glass Matter

requiring a different view of chirality as both Left and Right handed spinors emerge from the branching rules of both Weyl halves confusing the picture. Left handed spinors on Y do not remain exclusively Left handed on X. It may be asked what the relevance of the above stylized toy example is to the model under discussion. Quite simply, for every field on Y in the Observerse, there is both a naive spin and a true spin. The naive spin of a differential form valued in another bundle is taken to be the spin of the form field if the tensored bundle were taken formally to be purely auxiliary. Thus, for example, an advalued one form would carry naive spin 1 whether or not the ad bundle was derived from the structure bundle of the base space on which it lives.

Thus, for example, our bundle Ω1 (Y, ad) of ad-valued 1-forms has naive spin one, but this disguises the fact that it also contains an invariant subspace that derives from Λ1 ⊗ Λ 1 ⊂ Λ 1 ⊗ Λ ∗

This space of naive spin 1 would appear to be truly spinless from the point of view of Y . Thus, in some sense, the field playing the role of the fundamental mass for the generalized Dirac equations is actually part of the gauge potential. This sets up a three way linkage: Cosmological ‘Constant’ Λ ↔ Spinless Gauge Field ↔ Fermion Mass (12.21) and it is in such ways that GU seeks to attack non-anthropic fine tuning problems by having the same fields do multiple service.

In essence here, a fundamentally non-chiral world of Dirac Spinors in this simplified example would appear chiral in regions of low scalar gravity. From beings made of such chiral matter, they would naturally view the universe as being mildly chiral much the way each of the two hands in Escher’s drawing is separately approximately symmetric about its middle digit.

But raised high, the symmetry breaks down as digits two and four are only approximately symmetric in most people, and one and five are undeniably different. Yet it is not only the two middle fingers which are beautiful and symmetric about themselves, because the proper symmetry is left pinky to right pinky, left thumb to right thumb etc. and not left pinky to left thumb, right pinky to right thumb which is not broken as a symmetry, but simply accidental as well as being false.

12.10 Three Generations Should be Replaced by 2+1 model of two True Generations and one Effective Imposter Generation

At the time of this writing, the author is not convinced that we have three true generations of matter which differ only by mass. We instead posited here that the so-called third generation of matter is instead part of pure Rarita-Schwinger Spin−3 2 matter on Y and its Spin− 1 2 appearance on X is the result of branching rules under pull back from Y where it is native:

… {z } Imposter Third Generation … (12.22)

Thus, part of the field ζ ∈ Ω 1 (Y, /SR) is an ordinary second generation spinor in Ω0 (Y, /SL ) via the Dirac gamma matrix contraction while the complement /RR(T Y ) corresponding to the sum of the highest weights contains the imposter third generation which is only revealed under decomposition as in the above. Thus, it is not a true generation as it has a different representation structure than the other two beyond its obvious mass difference.13