The Associahedra and Permutohedra Yet Again

## Three papers and an essay

## Marnihedra versus arkanihedra

The publication of an article on “the amplituhedron” in *Quanta Magazine*, a journal put out by the Simons Foundation, apparently marks the entry of twistor polytopes into the media pantheon of theoretical physics concepts. The article is being discussed in popular Internet forums, and the amplituhedron already has its own Wikipedia page too.

So spare a thought for Marni Sheppeard, who truly suffered for many years, waiting on tables while she did a PhD on twistor polytopes in a country at the very edge of the global physics map, who blogged openly about her ideas and was excoriated as a crackpot for her efforts, who starved herself in order to keep working on physics – and who now has to live with the spectacle of this media celebration, in which already well-paid and well-connected physicists dwelling in the elite institutions of the northern hemisphere are feted for their discoveries, while she remains as unknown and vulnerable as ever, and required to keep enduring whatever grim circumstances she is currently dwelling in.

There are many examples of people who were ahead of their time; who only got belated credit or no credit at all for their work; who anticipated a big discovery, either in a broad way or a highly specific way. The phenomenon comes in many forms. In this instance, here are some of the questions I would like to see answered: What is the relationship between Sheppeard’s body of work, and the new twistor physics now in the headlines? Is there anything that the twistor researchers with salaries and academic appointments can learn from the work that Sheppeard has managed to produce? And, is it still possible to get her into a position where she can work properly, make further contributions, and communicate with her peers?

I still do not know how to turn on comments for older posts, but hopefully the comment section for this one is open and working.

## A note on cosmology

For a long time, I wanted to do a post here on Louise Riofrio’s cosmology, because it was part of Sheppeard’s synthesis, but I felt I couldn’t do it justice. It’s a varying speed-of-light cosmology, in which there is no dark energy or accelerating expansion; light is just slowing down, because of the mass of the universe, according to the formula “GM=tc^{3}“.

But I could never quite grasp the logic whereby the density fractions were being predicted, and that was the big empirical claim to success. I also thought I saw a similarity to the Milne universe, but again, I never quite got it clear in my own head.

Much later I came to understand that the model was empirically problematic, because dark matter and dark energy density fractions evolve with time, so any numerology based on present-day observations is just picking out one moment in the history of the universe. It remains possible that the present-day values are special, e.g. perhaps they are the asymptotic values. One should also bear in mind that for Sheppeard, Riofrio’s handful of equations are a classical ansatz to be derived from a theory of quantum gravity.

The Planck satellite has in any case given us new values for the density fractions, that are no longer close to Riofrio’s numbers, so it’s an open question as to whether Riofrio’s ideas are still part of Sheppeard’s thinking. However, Sheppeard continues to champion another item of astrophysical numerology, which I believe was co-discovered by her collaborator Graham Dungworth – that the temperature of the cosmic microwave background equals the rest mass of a certain “mirror neutrino”, as predicted by a version of Brannen’s formula for Koide mass triplets.

This would be even more “problematic” than Riofrio’s numbers. The CMB temperature varies with the expansion, but that’s not fatal to the idea; quintessence models show that it’s possible to have mirror neutrino masses evolving with the size of the universe. The real difficulty lies in explaining the connection between CMB photons and mirror neutrinos at the level of particle physics and cosmic history.

One normally supposes that the CMB photons are simply photons which decoupled from the primordial plasma, once it had thinned out enough for neutral atoms to form, and which, ever since that time, have been travelling without scattering again. Neutrinos, mirror or otherwise, feature nowhere in this story, and yet they ought to, if the posited relationship is supposed to be real. Either the mirror neutrino ought to play a role in photon decoupling, or a completely different origin must be ascribed to the CMB photons, one with room for the mirror neutrinos to play a role. (Or the connection is spurious, which is what I think; but the ideas are still interesting to play with.)

## Synopsis of the new paper

This is a challenging document to assimilate because it mixes unfamiliar mathematics with unfamiliar physics. But here is a rough outline.

Chapters 2 through 5 are spent defining many simple combinatorial objects in the not-so-simple language of category theory. There are numbers, sets, trees, braids, polytopes, and matrices, among other objects.

In chapter 6, we make contact with current applications of twistors to quantum field theory. This is already an area in which combinatorial objects and simple symbolic calculi are playing a role, and Sheppeard wants to interpret the subject using concepts and results introduced in the preceding chapters.

In chapter 7, some of this apparatus is employed to introduce an extended version of Bilson-Thompson’s correspondence (of braids and Standard Model particles) and an explanation of Koide formulae in terms of circulant matrices associated with the braids.

So to summarize, chapter 6 is about the frontier of current theory and chapter 7 is about phenomenology, and in both cases there is an attempt to make the categorical combinatorics of chapters 2 through 5 relevant.

Chapters 8 and 9 return to pure math – knots, polytopes, and categorical diagrams.

Chapter 10 is about black holes – the “black hole / qubit correspondence” – and chapter 11 is about cosmology – the dark sector and cosmic time.

## Twistors and motives

We should soon be hearing again, any day|month|year now, from Nima Arkani-Hamed’s group, about the new motivic twistorial basis of quantum field theory that they and others are unearthing. While we wait, the reader may wish to puzzle over Sheppeard’s thoughts on the matter, as expressed in her thesis and preprint.

## Braids, circulants, and Jordan algebras

Sheppeard brings together a number of correspondences and formal schemes in her attempt to recreate the standard model. Here I will touch on Sundance Bilson-Thompson’s braids, Carl Brannen’s circulant matrices, and Michael Rios’s Jordan algebras of observables for quantum gravity. In a nutshell, the fermions of a single generation are represented by braids, the circulants encode information about the generations, and the Jordan algebras associate the braids with the circulants.

In Bilson-Thompson’s braid correspondence, the fermions of a single standard model generation are represented by braidings of three “ribbons”. A ribbon may itself contain a twist; the electric charge of a braid containing *m* twists and *n* antitwists is (*m*–*n*)/3. The correspondence suggests a topological realization of the rishon model; but Bilson-Thompson’s correspondence still lacks a counterpart to the Harari-Seiberg hypercolor lagrangian. That is, no-one has put together a well-defined dynamical theory in which the braids interact as they should.

Carl Brannen is working on an algebraic preon model which seeks to explain the Koide mass formula. It turns out that masses satisfying the formula are eigenvalues of a 3×3 circulant matrix. Brannen and Sheppeard have worked on expressing all the standard model mass matrices in terms of circulants.

Finally, Michael Rios has written a series of papers in which he aspires to have M-theory emerge from a matrix model based on the exceptional Jordan algebra. Here he defines the algebra of observables in terms resembling loop quantum gravity; here he claims the result resembles an octonionic twistor string. Loop quantum gravity is the framework within which Bilson-Thompson’s correspondence is usually pursued, and it seems that Sheppeard hopes to motivate an association between braids and circulants via something like Rios’s constructions.

This effort has been going on since 2005 or so, and features in several of Sheppeard’s recent papers.

## viXra:1201.0097 – “A Ribbon Dark Sector and Koide Triplets”

Sheppeard’s latest paper is very ambitious. It is a paper of many firsts. It is the first paper anyone has written that mentions Alejandro Rivero’s “new Koide tuple”, discovered in November 2011. It is the first paper which addresses and tries to explain Louise Riofrio’s observation that the dark energy fraction of the energy density of the universe is about three times the dark matter fraction. It appears to be the first paper extending Sundance Bilson-Thompson’s braid scheme for the standard model to the dark sector.

The paper undoubtedly looks strange by ordinary standards. There is no hint of a dynamical framework. Mathematically, it only contains simple algebraic formulas. There are leaps of logic and many peculiar statements. Combined with the general unfamiliarity of Koide tuples, Riofrio cosmology, and even Bilson-Thompson braids (which, despite having been the subject of a *New Scientist* cover story, are a fringe topic within theoretical physics), I think that most physicists, even if they somehow found themselves reading the paper, would quickly give up and put it aside.

I’ve had a few weeks to think about it, and for me, the jury is still out. The basic empirical facts – the various Koide relations, now extending to all the standard model fermions; Riofrio’s observations – are certainly worth thinking about. There is a 3:1 ratio appearing in both domains (Koide relations, dark sector fractions), and it is not beyond imagining that in both cases it is a manifestation of the 3:1 ratio of colored quark states versus colorless lepton states. It is even conceivable that the specific algebraic relations that Sheppeard proposes (qutrit path counts, sums of 2π/27 phases) are part of the explanation. But it is also very conceivable that they are not.

## Introduction

This blog will serve to record occasional thoughts and observations about the work of the physicist Marni Sheppeard.