Summary

The approach of Geometric Unity as a candidate physical theory of our world is to work with a variety of different bundles in both finite and infinite dimensions which are all generated from a single space X4.

The structure of the relationships may be summarized here:

We recall that there is a theory of sections above the infinite dimensional constructions on the right hand side involved with superspaces and so-called ‘induced representations’, but at this point cannot remember even the standard theory and so have not entered into it here and may do so in further work if there is sufficient interest and ability to recall.

12.1 Equations

In Geometric Unity, we believe that the Einstein, Dirac, Yang-Mills and KleinGordon equations for the metric, Fermions, internal forces and Higgs sector respectively are not to be unified directly. Instead, the Einstein and Dirac equations are to be replaced by the reduced Euler Lagrange equations

Π(dI 1 ω) = (δω) 2 = Υω = 0 (12.2)

for a first-order Lagrangian after removal of redundancy through projections Π Then the Yang-Mills-Maxwell equations and Klein-Gordon equation for the Higgs follow from a second related Lagrangian

Π(dI 2 ω) = D ∗ ωΥω = 0 (12.3)

whose Euler Lagrange equation are automatically satisfied if the 1st order theory is satisfied.

12.2 Space-time is not Fundamental and is to be Recovered from Observerse.

There has always been something troubling about the concept of Space-Time as the substrate for a dynamic world. In a certain sense, space-time is born as a frozen and lifeless corpse where the past is immutable and the quantum mechanically unknowable future hovers above it probabilistically waiting to be frozen in the trailing wake of four-dimensional amber which is our geometric past.

To have a hope of contributing insight, GU must, it must recover this established structure as an approximation within the theory. But at its deepest level, it seeks to break free of the tyranny of the Einsteinian prison built on the bedrock of a single space with a common past.

There is something very special about the arrow of time mathematically.

Only in dimension n = 1 is R n always well ordered. For every dimension n > 1 there is no such concept without additional structure chosen (e.g. indifference curves and surfaces foliating the space of baskets in consumer choice theory).

In our case we have moved to a world X4 in which we believe all signatures are in some sense ‘physically’ real, with X1,3 and X3,1 being the only two to be provably anthropic, and the others being disconnected and unreachable by the condition of non-degeneracy.

Yet in Geometric Unity, hovering above the world we see, there is always a second structure Y 14 looming with multiple spatial and temporal dimensions beyond our own.

This capacious augmentation of non-metric X4 as proto-spacetime allows us to wonder about the nature of time without a clear arrow being interpreted on a different space where the arrow is enforced by anthropics.

However, we have found it quite challenging to think through the tension between two such worlds related by a bridge ג which must measure in order to observe. Thus, the idea of measurement and observation are forced to be intrinsically tied and the concept of multi-dimensional arrowless time above is shielded from us living as if in Plato’s cave below.

12.3 Metric and Other Field Content are Native to Different Spaces.

If the metric on X4 and the observed Bosonic and Fermionic fields are native to the same space, then there is likely a need to put both of them in the same quantum system. However, if they originate intrinsically from different spaces, then the possibilities for harmonizing them without putting them into the exact same framework increase. It may fairly be pointed out that we have a metric in the derived space Y that will have to be put in a common framework with the other fields, but even there we have a new twist. In this work we have not been considering unrestricted metrics on Y.

In fact, almost all of the ‘metric’ information is built into the construction of Y , so that our subset of ‘metrics’ under consideration is really equivalent to the space of connections that split the long repeating exact sequence we have discussed between T Y and T ∗Y .

This is not accidental but desired, as one of the goals of GU as connections, unlike metrics, have an adequate quantization theory as exhibited by QED, QCD and other theories. Hence the Zorro construction puts the only true metric field ג on a separate space from the main quantized structures, but uses a connection to derive the highly restricted metrics on Y .

12.4 The Modified Yang-Mills Equation Analog has a Dirac

Square Root in a Mutant Einstein-Chern-Simons like Equation Without the quadratic potential term in the earlier example of a GU Bosonic Lagrangian, we are left with an expression of the form:

…

If we were to attempt to compare it to other Lagrangians, it would be seen as having some aspects of both the Einstein-Hilbert and Chern-Simons Lagrangians. The Einsteinian character comes from the fact that it produces a linear expression in the curvature tensor making use of Riemannian Projection via }· ω. The Chern-Simons-Palatini like properties come from the fact that it is a Lagrangian that takes connections and ad-valued 1-forms as its natural parameter space.

A comparison of the two expressions may be helpful to motivate some readers more familiar with one than the other:

… (12.4)

Where ω = (ε, $) ∈ G = H n N and the connection 1-forms

A = ∇A − ∇0 $ = ∇$ − ∇g (12.5)

are measured relative to the trivial connection in the usual Chern-Simons theory, while Geometric Unity is inclined to use the spin Levi-Civita connection. The displaced torsion on the other hand

Tω = ∇$ − ∇gℵ · ε = $ − ε −1 (d∇g ε) (12.6)

is measured relative to the gauge transformed Levi-Civita spin-connection ∇Bω =

∇g · ε. The operator }· ω depends on the gauge transformation and, like the Einstein-Ricci projection, always kills off the Weyl curvature. Unlike the EinsteinRicci projection map, however, it does so in a gauge covariant fashion.

In the Chern-Simons case, the ad-valued 1-form A is differentiated by the exterior derivative coupled to the trivial connection. Within Geometric Unity, it is differentiated by the exterior derivative coupled to ∇Bω , the Levi-Civita spin connection gauge transformed by ε.