Final Thoughts

To sum up, let us revisit the Witten synopsis to see what GU has to say about it:

Figure 7: Edward Witten Synopsis.

Geometric Unity may be considered an alternative narrative that tweaks familiar concepts in various ways. As the author sees it, it is really a collection of interconnected ideas about shifting our various perspectives. Given the apparent stagnation in the major programs, GU has sought an alternate interpretation of either or both of the two incompatible models for fundamental physics of the Standard Model or General Relativity.

I think this represents a rather general perspective on the likely reasons for the impasse in fundamental physics encountered over the five decades since the early 1970s.

At almost every level, it appears to us as if the instantiations of the most important general ideas and insights hardened prematurely into assumptions that now block progress. In most cases, our shift in perspective is usually not a rejection of the current models at the level of ideas so much as a rejection of the pressure to communicate ideas concretely through instantiations.

In essence, we see an intellectual disagreement between the tiny group of physicists who have sought to discover physical law and the vast majority of theorists who attempted to work out its consequences.

What good is a beauty principle that works only in the hands of Einstein, Dirac, Yang, and a handful of others, while leading to failure and madness for others? Yet, in these matters, we have come to side with Dirac’s widely misunderstood perspective on the relationship between, instantiation, beauty, theory and experiment. In essence, a beautiful 13Note: we are speaking loosely here as if mass eigenstates and flavor eigenstates were one and the same.

theory is not its instantiation, but those who do not seek physical law cannot be forced to accept this critical issue.

To rephrase Witten’s paragraph then in light of Geometric Unity, it might be rewritten as follows:

“To Summarize the strongest claims of the strongest form of Geometric Unity, the basic assertions would be: i) Space-time X1,3 arises as a pseudo-Riemannian manifold from maps ג between 2 spaces Xn and Y n2+3n 2 (X) = Met(X) where Y is constructed from X at a topological level. ii) Over Y is a bundle C with a natural metric which is (semi-canonically) isomorphic to T Y , and one whose structure bundle carries a complex representation Spin(n 2 + 3n 2 , C) −→ U(2 n2+3n 4 , C) (12.23)

on Dirac spinors with structure bundle PH for H a real form of U, with no internal symmetry groups. There is an inhomogeneous extension G of the gauge group H of PH acting on the space of connections A(PH) where the stabilizer of any point A0 ∈ A gives rise to a non-trivial endomorphism τA0 : H −→ G. iii) Fermions on X4 are pullbacks ג ∗ (ν) and ג ∗ (ζ) of unadorned non-chiral Dirac spinors

ν ∈ Ω 0 (Y, /S) and 1-form valued spinors, ζ ∈ Ω 1 (Y, /S) on Y . In low

gravitational regimes, the equations governing the fractional spin fields decouple leading to emergent effective chirality that disguises the non-chiral fundamental theory, and leading to Witten’s representations R and R¯ which are not isomorphic exactly as according to the branching rules for Spinors.

The cosmological constant is actually the Vacuum Expectation Value (VEV) of a Field which plays the role of a fundamental mass, leading to the light Fermions being light in low gravity regimes. iv) If Super-symmetry is considered, it lives on the inhomogeneous gauge group and not the inhomogeneous Lorentz or Poincare group where gauge potentials take over from Galilean transformations and the affine space A plays the role of the Minkowski space M1,3

The lack of internal symmetries indicates why naive super-partners have not been seen as space-time SUSY may be implementing over the wrong group. v) Gravity ג lives on X while the fields of the standard model are native to Y leading to a reason for General Relativity to appear classical on X in contrast to the Quantum nature of the SM fields ω tied to Y . vi) Gravity is the engine of observation, so that where gravity is localized in different sections גa, גb, it pulls back different content while vii) Gravity on Y is replaced by a cohomological theory involving an obstruction δ

ω = Υ combining elements of Einstein-Grossman, Dirac and Chern-Simons theories, while there is a new 2nd order theory replacing Yang-Mills, Higgs and Klein-Gordon theories so that the cohomological theory δ

ω = Υ = 0 is a ‘Dirac square root’ of the second order theory. viii) The branching rules of ν leads to the appearance of one family of Fermions. ix) ζ branches as a second family due to gamma matrix multiplication on Y as T Y ⊗ /SY = /SY ⊕ /RY with a Rarita-Schwinger remainder. The Spin 3 2 portion of ζ breaks down under pull back to reveal a third ‘imposter generation’ that is merely effective, as it has different representation behavior in the full theory. x) The first order theory has a rich moduli of classical solutions and Υ = 0 carries an elliptic deformation complex in Euclidean signature once the redundant Euler-Lagrange equations are discarded.14”

We would like to end this speculative foray with a quote from the man whose question provided the impetus for this excursion. “The relativity principle in connection with the basic Maxwellian equations demands that the mass should be a direct measure of the energy contained in a body; light transfers mass. With radium there should be a noticeable diminution of mass.The idea is amusing and enticing; but whether the almighty is laughing at it and is leading me up the garden path — that i cannot know.” -Albert Einstein While we believe in the story of Geometric Unity, we find the above, now as then, to be sage words in all such endeavors.

Appendix: Other Elements of Shiab Constructions

Continuing on from our earlier discussion of Shiab operator construction, the author simply wanted to note some of the gadgetry that has come up in the construction of these operators in past years. Most of this is obvious, but the fact that there are two products on the Unitary group Lie algebras given by matrix commutators, and anti-commutators multiplied by i, is an example of something that can be easily forgotten. The author may have forgotten other tools in the Shiab workshop over the years as well.

Wedge

The wedge product passes to bundle valued forms from the usual DeRham complex.

Hodge Star

As we have assumed our manifold to be oriented from the beginning, every time a metric g on Y is chosen it induces a non-vanishing volume form dvol 14The so-called Seiberg-Witten equations were first found this way around 1987 as the simplest toy model to proxy this moduli problem. 65 compatible with the metric and orientation. This in turn induces a Hodge Star operator ∗ : Ωi (B) −→ Ω d−i (B) (12.24)

which passes to forms valued in arbitrary bundles B over Y .

Contraction Various forms of contraction can be defined either with co-variant against contravariant tensors in the obvious way or via the wedge and star operations between forms as in: φ ∨ µ = ∗(φ ∧ ∗µ) (12.25) Adjoint Bundle Bracket

As with any Lie Group, U(64, 64) carries a Lie Bracket structure. Given that it lives embedded within the Clifford Algebra ClC(7, 7) = C(128), it can be constructed from the matrix algebra product in the usual fashion: [a, b] = a · b − b · a (12.26)

Symmetric Product

Unlike most Lie Algebras, there is a second symmetric product on u(n) gotten from taking: {a, b} = i(a · b + b · a) (12.27)

Volume Form

The analog of the Hodge Star operator is multiplication with the Clifford Volume form λ.

Appendix: Thoughts on Method

A few words are in order about what the author sees as unbridgeable differences with the mainstream of the community of professional physicists.

Experiment and The Scientific Method

I understand the scientific method differently from many others and particularly from many within the world of String Theory. In essence there are general ideas and multiple instantiations of those ideas. The author believes that many who put their faith in the scientific method do not understand the danger of being pressured to discard ideas because one of their instantiations was invalidated by experiment.

This is, in essence, the very point Dirac raised in his 1963 Scientific American Article where he warned that beauty rather than the scientific method should be used as a guide to progress:

“It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one’s work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further developments of the theory.”

It is the misinterpretation of this very clear point that the author finds chilling. Dirac was clearly not saying that if a theory is beautiful, it need not agree with experiment, and yet he is frequently lampooned as such. Why is this?

I believe there is a principle, by which the scientific communities push most members for hyper explicit claims so as to learn the general idea and to wed the author to a prediction that can be easily falsified.

Should the author succumb to associating her or his more general idea with a particular instantiation that fails to be confirmed, that idea is now ‘up for grabs’. Further, established players can speak more generally allowing different members different privileges.

I am:

• proud to be able to offer algebraic predictions as to the ‘internal quantum numbers’ of new particles but would need the help of Quantum Field Theorists to see whether these can be sharpened further to include energy scales.
• not equipped to undertake that effort alone but considers the predictions already offered to be considerably more explicit than many of the current contenders for a theory of everything on a relative basis.

My experience is that in calling such quantum numbers ‘predictions’ is that those farthest away from making such predictions are paradoxically the most likely to complain viciously about the lack of an energy threshold so as to deflect criticism from their own theory’s failure to be able to make such claims. Isolation

I think that almost no professional mathematicians and physicists have any concept what it is like to be isolated from the community for 20 years or more at a time. Geometry and field theory are languages that in this author’s experience, decay exceedingly rapidly when there is no one with which to speak them, and it is nearly impossible to find it actively maintained anywhere outside of the profession.

It has been over 25 years since I was in a professional environment where anyone else was conversant in the topics discussed here.

My apologies are offered for any inconvenience caused, but the author’s ability to converse with the professional community, but, in full candor, the ability to communicate was likely to get even further degraded via additional years of isolation.

Appendix: Locations Within GU

We collect here for convenience the usual ingredients that constitute fundamental physics and give their intended address within the framework of Geometric Unity.

Usual Name GU Location Higgs Field ג