The lost thread

Posts from a deleted thread at PhysicsForums.com, discussing Sheppeard’s work.

Mar24-11, 02:26 AM

Does anyone else here read Marni Sheppeard’s blog, “Arcadian Pseudofunctor”? She is not exactly working in LQG, but I find what she’s doing by far the most stimulating set of ideas coming from that general direction, because it combines direct attention to particle properties like mass and the mixing matrices, with mathematically ambitious notions, like the derivation of these properties from motivic cohomology. I doubt that things work exactly the way she proposes, but some of the ingredients are good, and no-one else is trying to directly combine these things yet, so it’s a glimpse of things to come; and I spend a little time thinking about how to carry out the proposed synthesis in a more orthodox way.

The ingredients to her proposed synthesis that I would highlight are:

1) Bilson-Thompson’s mapping of three-strand braids to first-generation particles of the standard model

2) an interpretation of the quark and neutrino mixing matrices in terms of the Koide formula, also due to Carl Brannen

3) the proposal in twistor theory to represent mass using cohomology

4) her idea to obtain twistor diagrams from combinatorics of operads (entities from n-category theory)

5) the quest of Grothendieck and others for a universal cohomology

This all comes together with the idea that the universal cohomology, applied to Bilson-Thompson’s braids, should produce the mass quantum numbers as represented by generalized Koide formulae.

There is also a recurring theme of quantum information theory. The 3×3 mixing matrices are represented as products of special matrices which show up in the theory of “mutually unbiased bases”. She also takes an interest in the work of Michael Rios on the black hole / qubit correspondence – the idea seems to be that space-time will emerge from quantum information, but the right algebra(?) to give the real world hasn’t yet been explored. I can’t incorporate this part into an almost deductively motivated research program as I did with (1)-(5) above, but it’s also part of the conceptual mix.

At the moment she’s busy extending Bilson-Thompson’s correspondence. That is, Bilson-Thompson didn’t use all possible braids, and she’s trying to match up the unused braids (some of which can already be assigned a mass, under the Brannen-Sheppeard scheme) with very recent suggestions of CPT violation in neutrinos and quarks. (The working idea is that CPT is preserved, and that some of what are being interpreted as antiparticles are in fact “mirror particles”, spatial reflections of the braids in the original correspondence.)

From a particle physics perspective, I believe the biggest flaw in this program is that it can’t (yet? ever?) make contact with field theory. In this regard, the situation resembles LQG’s usual problems with the semi-classical limit. At present, Sheppeard’s program simply jams together braids and matrices; there is some logic to the mapping, having to do with permutation groups, but it’s an intuitive leap of the sort which usually doesn’t work in the end. But particles aren’t just a static set of quantum numbers; in action, they exhibit all the other phenomena of field theory like the running of masses and coupling constants with energy scale, loop corrections to tree-level perturbation theory, and so on. And – just as with LQG – it’s not clear that Sheppeard’s ideas actually admit an elaboration which can reproduce these properties. She did her thesis on the categorification of twistor scattering amplitudes, so it’s definitely an aspiration that everything should link up in the end, but that doesn’t mean that it will.

Nonetheless, I think what she’s doing ought to be getting a lot more attention, for two reasons.

First, all those people trying to get quantum gravity from quantum information, from spin foams, or from category theory, ought to be interested – especially since she’s trying to get the particle masses. No-one else is doing that!

The second reason is that she, along with Brannen, are tackling the origin of the Koide formula. This formula is extremely simple (it’s not a messy contrivance with powers of transcendental numbers and so forth) and it has been empirically validated to 1 part in 100,000. But it seems to exist in a blind spot as far as the particle physics mainstream is concerned, because of the belief that high energy = simple, low energy = messy. Koide’s relation concerns masses at low energies, and so apparently the reflex is to dismiss it. (There are some attempts, by Koide himself and by his colleague Sumino, to explain it using ordinary field theory, including a supersymmetric model, so it’s not impossible to do – it’s just unfashionable to think about.)

What Carl Brannen did was to represent Koide’s relation in a certain way, and then look at the neutrino mixing matrices from this perspective. Sheppeard later tackled the quark mixing matrix. The people who are trying to explain the mixing matrices using family symmetries should, in my opinion, be especially interested in just what is going on here. Certainly, I have found it hard to penetrate the logic at work, but that’s why I’m writing this little promotional screed. Despite all the theoretical idiosyncrasies and the known reasons why this shouldn’t work, at bottom we have an empirically validated relation, and a set of theoretical relationships, some version of which ought to be correct mathematically.

Personally, I am not so enthused about Bilson-Thompson’s braids – what interests me is the conjunction of twistor cohomology, motivic cohomology, and Koide mass matrices, preferably in the context of the new twistor/gauge/string relationship that has grown out of AdS/CFT and which is especially being promoted by Nima Arkani-Hamed. So I would be keen to have some statements, from people who know field theory and string theory much better than I do, about why there shouldn’t be simple relationships among low-energy masses – because there is evidently a loophole in the argument. But I know there are a few Bilson-Thompson fans here, and I think they should have a look at what Brannen and Sheppeard are doing from that angle. Together we might be able to dig a few layers deeper into this nexus of ideas.

Jul16-11, 12:46 AM

Sheppeard’s overall plan and philosophy has become somewhat clearer to me recently, especially thanks to this new preprint.

The big idea is to have a motive – a type of algebraic super-object – canonically determine a mapping from an exceptional Jordan measurement algebra onto a category of abstract geometric objects which will include braids. This is supposed to induce an extended version of Bilson-Thompson’s mapping (of triple braids onto Standard Model particles), which will include three generations, the particle masses and mixing matrices, and their interactions, including some beyond-standard-model elements. One should, I suppose, imagine the braids forming an algebra through side-by-side and end-to-end composition, just like subdiagrams of a Feynman diagram.

There’s also a cosmological aspect. Sheppeard appeals frequently to the varying speed of light (VSL) cosmology of Louise Riofrio, but the part that’s clearer to me – because it’s a rather vivid idea – is that all matter is Hawking radiation from a cosmological horizon. Her extension of Bilson-Thompson’s mapping includes a “mirror antineutrino” whose mass is determined by the mapping, and which matches the CMB temperature. This is interpreted as a sign of thermal equilibrium:

“Our background of photons is therefore in thermal equilibrium with the antineutrino that commonly partakes in the weak interaction with ordinary matter. If the photons are envisaged as coming from some cosmological horizon, then so are all other forms of matter. A quantum world is all Hawking radiation, and the cosmological constant is vanquished in favour of a new anti-realist vacuum fluid aether, modeled observationally by Riofrio’s varying speed of light critical density picture, which recovers Kepler’s law for orbital motion with respect to a cosmic time measure.”

Well, that last sentence eludes my comprehension as it goes on. It is also unclear to me whether this is supposed to be a steady-state cosmology (such as Erik Verlinde is now proposing?), or whether this particular equilibrium is only supposed to be a feature of the current epoch. But the reference to “anti-realism” should lead me to mention another feature of her synthesis, really a philosophical choice, namely a focus on epistemology rather than ontology. Of course this is common enough in quantum mechanics, but here it’s apparently taken to an extreme that should delight PF user Fra: Everything is quantum information, measurement algebras are central. The closest I can come to making sense of this outlook, is as if you regard yourself as a random sample from the class of all possible observers in Max Tegmark’s mathematical omniverse. Some properties of the world will be anthropically constrained, while others will genuinely be randomly determined. But this still doesn’t get the outlook right; I also sense that observers are supposed to be (partly) determining the world through their participation in it. Again, this is a familiar position from the history of quantum thought, e.g. one may say that the outcome of the observation is beyond control, but the observer chooses which observables to measure.

I mention these philosophical aspects for the sake of completeness – they clearly play a part in her thinking – but what I want to emphasize most is in the second paragraph: The idea of a motivically determined canonical mapping between a category of measurement algebras and a category of abstract geometric objects, especially a composition algebra of braids. She, Michael Rios (who works on Jordan algebras in M-theory), and Carl Brannen (who is extending the Koide formula for the charged lepton masses to all the particle masses and the mixing matrices) met on this very forum about five years ago, and from this was born the idea of Sheppeard’s motives mapping Rios’s algebras onto Bilson-Thompson’s braids and producing Brannen’s Koide mass matrices. It’s a very interesting and ingenious conception. Brannen and Sheppeard have had some of their work published, and Rios’s papers are on the arxiv, but the majority of this work is just on the web – on Carl Brannen’s website, or (in Sheppeard’s case) on Phil Gibbs’s alternative “vixra” archive.

So, why is this interesting? First, for its radicalism. Suppose you wanted to develop a physical theory which contained none of the ideas from the past forty years which, though experimentally not confirmed, make up the dominant paradigm? Here I mean the Higgs, grand unification, heavy superparticles, string theory, and the idea of cosmological “dark forces”. And suppose that you also wanted to do this in a mathematically sophisticated way, and in a way meant to explain the particle masses and mixing matrices? Well, here’s what such a theory might look like.

Second, I’m interested in the extent to which this program can be pursued within the standard paradigm. Certainly the part about motives and categories and exceptional algebras potentially applies directly to orthodox M-theory. It is entirely imaginable that many of the complex constructions currently employed are equivalent to much simpler combinatorial ones. The twistor revival already shows us what such radical simplifications can look like (and Sheppeard is a big fan of twistor variables, her thesis was about the application of category theory to twistor scattering). Doing away with dark energy and dark matter, in favor of cosmological quantum effects? Erik Verlinde is taking steps in that direction. No heavy superpartners? If we could realize the Rivero correspondence in a supersymmetric field theory, that would mean that we’ve already seen supersymmetry. (It would screw up the role that supersymmetry is supposed to play, in stabilizing the Higgs mass and providing dark matter, though.) Explain the particle masses and mixing matrices? GUTs do this a little, but mostly qualitatively. Predicting the actual particle masses seems to be left to the mid-term future, when the difficult moduli calculations in string theory become tractable. The only person I see developing field theory models meant to explain the Koide formula is Koide himself – and there really ought to be much more attention to his work, in which Yukawa terms are produced from the VEVs of new scalars, “yukawaons”. Surely these could be stringy moduli? Bilson-Thompson’s braids? Well, they are a little problematic from an orthodox perspective. Braided and knotted observables should exist – like Wilson loops – but I don’t see how you would get the particle species from those. Nonetheless, the recent discussion about (A)dS/LQG might eventually find some physical significance for the loop basis, in a way that could slot into the motivic approach.

All in all, although I still have a lot of problems with the overall synthesis, I’ve found it extremely fruitful to try to understand this nexus of ideas, and especially to use it as a counterpoint to orthodoxy.

Aug19-11, 11:16 PM

I have been wanting for a while to make another post on this thread, but it’s a little difficult, because this is not a case where I feel free to simply discuss the merits of a person’s ideas and leave it at that. Marni Sheppeard has formerly had positions at Perimeter Institute and at Oxford, but right now she isn’t just another academic happily posting mathematical speculations on the arxiv; she’s holed up in some mountain town in rural New Zealand, living on a shoestring budget while trying to preserve her intellectual life, and on her blog she regularly appeals for assistance in getting back into a research institution. Furthermore, she attributes her situation in large part to academic sexism, while pouring scorn on the numerous standard but unvalidated concepts (superpartners, Higgs boson, “dark forces”) which make up conventional thinking about what lies beyond the standard model. The situation would be simple if I agreed with all or even most of her theoretical work – I could then “promote” it without a second thought; for that matter, it would also be simple if she could just be dismissed as a crackpot. But the actual situation appears to be that she is ahead of her time in some respects, but is going down the wrong path in others.

The hardest part for me is her status as an exiled mathematical physicist, wanting to leap directly back into the elite world of theoretical physics research. One way to do this would be to have an unassailable body of independently obtained results in pure mathematics. Another way to do this would be to have an unassailable grasp of the high technicalities of advanced quantum field theory. So far as I can make out, her actual situation is that she has an excellent command of category theory, and that she wants to apply this to the twistor revolution; but she doesn’t have papers which stand alone as works of pure mathematics (e.g. by containing significant new theorems), nor does she have papers which apply the technical machinery of QFT in order to make new calculations (e.g. in developing the empirical implications of a new model). What she has, are a few first steps towards categorifying twistor scattering amplitudes, an ambitious program to recast the whole of physics in the language of motives, and a few simple quantitative results deriving from a concrete physical picture that I, at least, don’t believe (more on this below).

This would not be such a problem if she already had a secure position somewhere. Any seasoned reader of the arxiv would be aware that there are many papers there which are highly speculative; and Marni Sheppeard’s speculations are far more interesting than most. But it’s this combination of having a conceptually radical agenda, an unwanted status of academic exile, and very few publications that aren’t part of the big new physical picture, which makes “rescue” difficult for her. Furthermore, even in the era of arxiv and conference video, it’s difficult for people outside the archipelago of well-funded elite institutions to catch up with the always-developing technical expertise that exists within them. So far as I can see, there are three ways into such institutions. Either you have money to pay your way in, or you apprentice yourself to someone who is already tenured, or you force yourself upon their attention through sheer brilliance. On account of Marni’s financial situation and heterodox agenda, the third option seems like her best chance, and yet it’s by the far most difficult, even if you are highly intelligent. No matter how smart you are, if you’re an outsider, you are almost certainly lacking expert knowledge which is essential for making discoveries of central, rather than peripheral, importance. Experts are hit-and-miss with their ideas too, but the expertise that comes from professional acculturation helps them retain some credibility even when their own ideas all fail.

Hundreds of people read this forum, and the posts all go into the search engines, so I can only hope that this ambiguous defense of Marni’s status, as a person who ought to be obtaining support from somewhere, will reach someone who knows someone who knows someone who can and will help.

Aug19-11, 11:38 PM

Now, what about the physical concepts themselves? I think the best places to start are her thesis from 2007 and her latest paper. Also, although her last paper from the arxiv was withdrawn, you can still see it, and it may help provide some context for her later efforts. (And it shows that she was talking about Goncharov years before the twistor world discovered his relevance.) The thesis is where the first steps towards categorifying twistor amplitudes are taken, and the recent paper outlines the motivic program, which would recover physics from a mapping which takes categorical structures to exceptional algebras.

I believe, more or less, that some form of this program ought to work. Arkani-Hamed, in his recent talks, has described a perspective in which gauge theory and string theory are two redundant expressions of a third theoretical framework which employs neither the language of fields or of strings, and which we glimpse in the twistor Grassmannian polytopes from which the planar amplitudes in N=4 Yang-Mills are being recovered. This high-flown formalism of categories and motives has an excellent chance of being the language of the third theory.

Now let’s turn to the practical application of these ideas. Carl Brannen, who would be known to many of the readers here, has a preon theory and a slightly unusual quantum formalism, both of which were inspired by the Koide relation connecting the masses of the charged leptons. Carl himself, like Marni, is one of those quasi-alternative physicists who is good enough to have a few publications, but also different enough to be struggling with the system. In any case, he and Marni have, for several years now, being trying to obtain the particle masses and mixing matrices by combinations of certain special classes of matrices (e.g. “cumulants”). One version of this worked for the Koide relation, they want to extend this to all the other mass and mixing parameters, they have a rationale for this (obscure to me) in quantum information theory, and Marni also wants to obtain these mappings in her categorical framework.

I would be incredibly surprised if this part of the program works out. The use of the cumulant basis to obtain a parametrization for these 3×3 matrices, and the information-theoretic rationale for this, sounds very dubious to me. An alternative way of proceeding would be start with a field-theoretic explanation of the Koide relation (such as Sumino’s), embed it in string theory (either supersymmetric or nonsupersymmetric), and then look for the twistorial or motivic reformulation of that string background. I can’t prove that the cumulant parametrization is meaningless or doomed, and it may have some value as a way of starting with the numbers obtained from experiment and working towards an explanation, but I think this simple idea will eventually have to be junked. At the very least, these matrices need to be put back in their field-theoretic context, and embedded in a renormalization group flow.

Aug21-11, 12:38 AM

Originally Posted by mitchell porter
“cumulants”

I meant circulants.

Oct4-11, 12:14 AM

Many attempts to explain OPERA’s neutrinos also involve dark matter, dark energy, and the gravitational sector. For example, the first serious riposte to the Cohen-Glashow argument – that a superluminal neutrino ought to lose energy via weak-force-mediated bremsstrahlung of electron-positron pairs – has just come from Aref’eva and Volovich, who propose that this can be avoided if there are right-handed “dark” neutrinos which are singlets for the standard model gauge group. (I don’t actually understand the difference between their “dark neutrino” and the usual “sterile neutrino”.) Dvali and Vikman think a second, massive graviton, coupled to the Earth, is the only way for OPERA’s results to be consistent with SN1987A. And “OPERA and a Neutrino Dark Energy Model” speaks for itself.

So it’s worth noting that Sheppeard was working on the neutrino sector before OPERA’s announcement, and tied it to the gravitational sector in an arcane but intriguing way. For reference, here are her recent papers, here is the original statement of her categorical theory of gravity, and her thesis on getting gauge theory from category theory. The ultimate conception is somewhat akin to noncommutative geometry but even more abstract. Imagine obtaining the particle masses by applying a motivic version of Witten’s twistor formulae to Bilson-Thompson’s braids. I was always skeptical about the possibility of identifying particles with these braids, but recently I began to see ways in which braids might show up in M-theory (through timelike braiding of point excitations on an M2-brane, or via knotting of open M2-branes stretched between M5-branes), so it’s even conceivable that some form of M-theory (perhaps seven-dimensional “topological M-theory”) admits of such a description. Maybe even some version of M-theory phenomenology would look like that, if twistorial variables were employed!

Oct19-11, 06:50 AM

It’s time for another installment in my long-running attempt to understand Marni Sheppeard’s work and relate it to other trends in theoretical physics. At the level of particle physics, it’s a version of Bilson-Thompson’s helon model, resembling a categorified LQG, in which algebraic inputs inspired by M-theory are meant to explain Carl Brannen’s extensions of the Koide relation to cover all the particle masses and mixing matrices.

Bilson-Thompson’s helons are topological preons which can be twisted and braided, in order to produce one generation of Standard Model fermions, and all the gauge bosons. Or rather, there is a proposed mapping between braids of helons, and these particles – see Bilson-Thompson’s paper – but it is rather far from reproducing all the quantum-field-theoretic properties that a particle should have.

Last week there was an M-theory paper by Cecotti et al which I think makes it much more plausible that something like Bilson-Thompson’s correspondence could make field-theoretic sense. I wrote about it here and here. Cecotti et al’s braids describe changes in the triangulation of some compact dimensions, and there are particle states corresponding to topological classes of M2-brane states trapped in the transition zone. I find it remarkably conceivable that some version of these topological/geometric transitions which propagates in space might underlie a Bilson-Thompson correspondence. In particular, I’ve noticed that you can braid branes. (Think of the braiding of points in a 2-dimensional space, and then imagine taking the product of that embedded braid with another space which is the worldvolume of the brane.) Is it conceivable that the known particles correspond to M2-branes trapped in propagating braidings of M5-branes?

The paper by Cecotti et al constructs field theories with N=2 supersymmetry, and it’s generally thought that this is irrelevant for phenomenology, since the real world is chiral, and extended supersymmetry doesn’t permit ordinary chiral mass terms. However, it’s not so simple! An N=2 extension of the standard model was studied ten years ago. The fermion masses come from supersymmetry breaking terms. Phenomenologically, such models are supposed to be ruled out by bounds on “oblique electroweak corrections”. I should say that for N=2 supersymmetry, you have three new particle species for each known particle. There’s the visible particle, its superpartner, and then there are mirror partners for each of those. For each standard model generation, there’s an “anti-generation” which transforms under the complex conjugate representation of the standard model gauge group. It’s the existence of three heavy anti-generations which is supposed to be ruled out by the bounds on oblique corrections.

However, phenomenology is complicated and usually requires many simultaneous assumptions. I am especially interested in a recent challenge to supersymmetric orthodoxy which challenges conventional wisdom about the need for two Higgs doublets in the SSM. If you go to section III of that paper, you find a model which in a sense has no Higgs. You have three standard model generations, no Higgs fields, and then one anti-generation (as described above). The sneutrino of that anti-generation plays the role of up-type Higgs, and the down-type masses come from “wrong Higgs” couplings. What if we extend this to three anti-generations? Based on a superficial examination of the literature, this should make it harder for the N=2 SSM to meet the electroweak bounds, because it relied on the second Higgs to cancel some of the corrections coming from the heavy mirror fermions. However, couldn’t the sneutrinos from the extra two anti-generations do this? I don’t think this is a matter that can be decided easily.

So, there seems to be a startling possibility that an extended version of Cecotti et al’s construction – embedded in the full 11 dimensions of M theory, where all the details of supersymmetry breaking and vacuum energy are determined – could provide a phenomenologically valid realization of Bilson-Thompson’s correspondence, in the form of an N=2 supersymmetric standard model, broken to N=0. I haven’t shown that this is a possibility in any quantitative sense, of course – just that a line of inquiry is open.

Returning to Marni Sheppeard, the part of her theoretical synthesis that always gave me the most trouble was her cosmology. She claims as her inspiration the varying-speed-of-light cosmology of Louise Riofrio, but that in itself has not been the problem; I just couldn’t figure out the basic concepts she was using – concept of time, concept of cosmological evolution, etc. Well, eventually I found the broad outline of it. It’s an expanding universe. It goes through epochs, just as standard cosmology does, though not necessarily the same epochs appearing in the conventional timeline.

More striking are the assertions that (1) the matter of the visible universe has its origin in Hawking-like radiation from the cosmological horizon (2) “mirror neutrinos” from the horizon are responsible for particle mass, by way of an electroweak-like interaction. Sheppeard’s concept of a mirror neutrino is specific to the Bilson-Thompson framework; the mirror neutrinos are supposed to correspond to certain electrically neutral braids that were unused in the original framework. Also, in his original paper Bilson-Thompson observed that his model provided no counterpart to the Higgs boson or to the graviton. So Sheppeard hypothesizes that these new neutrino-like braids are somehow playing that role.

Anyway, now we are in the brave new world of possibly tachyonic neutrinos, thanks to OPERA – but recall, that MINOS made a similar claim a few years before that, and there were even earlier suspicions of tachyonic neutrinos arising from beta decay experiments. And though Brannen’s Koide-inspired formulae for neutrino masses gives positive values, Sheppeard has joined the caravan of theorists taking OPERA’s measurement at face value, and trying to explain it, e.g. in terms of the mirror neutrinos.

What I want to mention here is the remarkable degree to which even this element of her synthesis can be connected to ideas found elsewhere in the literature. Here I won’t focus on the braiding aspect, but rather the idea of tachyonic dark energy in a holographic cosmology. There are a number of papers along those lines (1 2 3). But the really intriguing development has been to discover a Matrix theory description of black hole infall in terms of tachyonic strings! The strings in question connect branes inside the black hole with branes outside the black hole, and they become tachyonic when the brane falling in crosses the event horizon.

These are tachyons in the usual field-theoretic sense – where the imaginary mass indicates a vacuum instability, such as occurs in spontaneous symmetry breaking (the “unbroken Higgs” is a tachyon in this sense) – not in the faster-than-light sense. But let’s put aside the FTL aspect for now. What happens is that the appearance of the tachyons signals thermalization. The modes associated with the brane (and strings attached to it) that has fallen into the black hole enter into the same high-entropy thermal equilibrium already pervading the black hole. In terms of matrix theory, this is a state in which off-diagonal modes, strings between different branes, have an energy similar to diagonal modes.

So first of all, it would be very interesting to understand whether tachyonic models of dark energy can be understood in a similar way, but the tachyonic strings lead out of our Hubble volume, beyond the cosmological horizon. Can the accelerating expansion literally be explained in terms of the tension of horizon-crossing strings?! I see hints of this idea in recent very speculative work by Tom Banks and Erik Verlinde, and it may even have a more rigorous expression in some of the work on string theory in de Sitter space that I haven’t read.

Ordinarily, one assumes that a tachyonic field is just unstable. It is a standard result, possibly dating back to this paper by Bert Schroer, that even for imaginary values of mass, a Klein-Gordon scalar still propagates at less than c. The argument is also presented in this sci.physics FAQ on tachyons dating from 1993, but along with a loophole: superluminal propagation might be exhibited, if you were dealing with tachyonic states which were delocalized from the beginning. Arguments like Schroer’s assume that tachyons are to be localized like ordinary particles.

So this makes me wonder… if you didn’t redefine the vacuum to a stable one, but instead tried to describe the interactions of a tachyonic field in terms of delocalized tachyonic states, could you see transient superluminality in a tachyonic background? Causality violation might be avoided by the probabilistic nature of the effects, as discussed here.

This is piling speculation upon speculation, but I need to add one or two more ingredients. First, tachyons are normally scalars; how do you get a neutrino to be a tachyon? I don’t have a fixed idea here, but a sneutrino is a scalar, and a pair of neutrinos can form a scalar condensate (and both of these have been used to explain the Higgs). It’s already been suggested (in order to evade the Coleman-Glashow argument) that OPERA’s superluminal effect might come from oscillation between a subluminal neutrino that feels the weak force, and a superluminal sterile neutrino that doesn’t; could superluminality similarly arise from “superluminal spreading” due to a very-low-mass neutrino condensate, or some sort of oscillation involving the sneutrino? (No, I have no idea how the latter would work.)

On a completely different note – I mentioned that Tom Banks has some matrix models of quantum gravity in de Sitter space which by my reading involve effects across the horizon. He has also proposed a “cosmological supersymmetry breaking” in which supersymmetry is broken by virtual gravitinos from the de Sitter horizon. Recall that in the N=2 supersymmetric standard model, fermion masses have to come from supersymmetry breaking. Now let me observe that it has also been proposed that a sterile right-handed neutrino could come from a bulk gravitino! Amazingly, this all seems to dovetail with Sheppeard’s otherwise eccentric-sounding model, in which mass is due to mirror neutrinos coming from a cosmological horizon.

Nov22-11, 08:41 AM

I want to report a significant advance in my understanding of what Sheppeard is up to.

An issue that has troubled me, but which only came into sufficient focus to be solved, after I had grasped the basic moving parts of Sheppeard’s synthesis, is, how does she get the field theory limit? Her area is quantum gravity, but when and how do we get QFT back? I had to ask, even though the answer is there in her thesis: categorification of twistor polytopes. Having grasped this, I’m now totally convinced this is a major research program, and that however it turns out, we will learn something significant as a result.

Twistor polytopes are what the group around Nima Arkani-Hamed work on, day and night, and category theory has been a player in quantum gravity for many years. I’m not aware of anyone else combining the two topics yet, though cohomology has been part of twistor theory from the beginning and it is a categorical subject. In any case, this gives her theoretical framework unimpeachable credibility from a formal perspective.

But the insanely inspired step she took, which has caused her all sorts of career grief in the short term, but which in the long term will get her into the history books, not just as a mathematical technician, but as someone who had a big new physical idea, was to think of combining this formalism, first with Bilson-Thompson’s braid correspondence, and then with Brannen’s density operator formalism for the Koide formula, and Riofrio’s VSL cosmology. The significance of drawing on Brannen’s and Riofrio’s ideas is that they are paying attention to numerical relationships that scarcely feature in current theorizing. I have yet to see any other cosmology papers trying to explain the 3:1 ratio between dark energy and dark matter (let alone the similarity of the sum to 3/pi), And astonishingly few people have tried to explain the Koide formula, perhaps because it’s hard to do so within the framework of ordinary field theory – simple mass relations ought to be obscured by quantum correction at low energy; Koide has to suppose that complex family symmetries exist to protect his relation. Brannen, on the other hand, simply built a whole new physics around his representation of the Koide formula.

Now much or all of Brannen and Riofrio’s ideas may be wrong, but it is a highly significant step just to start paying attention to the numbers, and to try to embed them in a functioning theoretical framework. That’s why I consider it certain that Sheppeard’s program leads somewhere. Presumably it leads to falsification in some forms! But it is a path that has not been taken; and there must be work for dozens of people there.

I said Sheppeard suffered for her boldness. I don’t know the details, but from her blog I know that after these connections were first made, around 2005 (and some of them took place here in this very forum), she worked as a waitress while she got her PhD, and she’s now living in scenic destitution in some New Zealand mountain town. She has to post her papers at vixra.org, arxiv.org’s unmoderated twin. Obviously she’s very unhappy with the state of her career – the motivic cohomology of polytopes in twistor space is perhaps the most advanced perspective on QFT we have at the present time, and she was pushing towards that years before the crowd arrived – and I believe that the uncanny similarities with an old phenomenological approach to N=8 supergravity (from the year before the superstring revolution), which I’ve posted about here, only strengthen the case that there is a lot to investigate.

Ultimately she needs to escape from her exile and find some sort of academic post. She also needs citations, which are the oxygen of modern scientific life. I think the main barrier to the uptake of her ideas, especially after 2005, must have been incomprehensibility and incredulity: people just haven’t understood them, and when they did understand, they didn’t believe. Although I think that by now I have to count as at least half Sheppeardian in my own thinking, I still find passages in her work to disagree with. In her thesis, she says in passing some odd-sounding things about truth and time – but which physicist, working in quantum gravity, doesn’t say odd metaphysical things? The important thing is to have grasped the fundamental conceptual integrity of her research program. It does make sense, and it deserves a chance to play out and make its contribution to human knowledge.

14 Responses to “The lost thread”

  1. Wow, I’m amazed you saved this, Mitchell. Good one.

  2. Thanks Mitchell.

    I have tried to understand her work in many years too, despite my outsider position. It is interesting, but the cosmology? She ought to spend more time to look on the frame, when she declares that her cosmology begans from scratch. Who can even think of citing her in that situation?

    I would also want to see clearer the relations leptons – fermions. Now they seem intermixed. Baryonygenesis?

    Her fairy fields as quantum information and abstract fields are nothing else than the virtual fields behind the Higgs mechanism. Why does she deny that? Also the eventual supersymmetry would need some light?

    There should maybe be more thinking from this bottom-up view, as an effort to understand the microstates, but the endless possibilities in string-theory talk against it. How is the criticality, the finiteness, solved in her theory? Exactly what form of strings does she use? F-theory? She talks of a tripartistic world, and that should be seen in her strings (interactions)? The fact that she use many triangular forms does not explain her strings? In that case they are hadrons?

    The situation now is very frustrating, and I understand also for her. I have tried to have her talk about these things, but she gets silent. Also her quarrel with Matti Pitkänen (his refusal to cite her, in spite of her being helping him with the categories) ended with her silence, instead of declaring her standpoints. That makes me suspicious. He talk of her science being numerology. That I think is going too far, but explains maybe her bad success.

  3. Does anyone else here read Marni Sheppeard’s blog, “Arcadian Pseudofunctor”?

    Yes, I do read it everyday!

  4. Hello everyone… Ulla, if you make a post on your blog, we can discuss your questions there. Some make sense, others don’t, and though I can’t speak for Marni, I am happy to give my opinions – but at your blog, not here. For discussions on this blog, I want to assume that the reader has a background in physics and mathematics.

  5. Thanks, Mitchell, I will, but next week maybe. I give a link here, if it is ok?

    I understand the math can be difficult to explain in a simple way, but she can give it a try? My questions have irritated her, I know. And I am just an amateur.

  6. An interesting summary of Marni’s research framework, Mitchell. Indeed, much of the mathematics has yet to be developed to fully capture the physical picture that is envisioned. The “M-theory” Marni mentions in her blog is more general than the d=11 theory that unites the various string theories and maximal supergravity. The ‘M’ is closer to “motive” or some deeper mysterious theory that also unifies mathematics.

    It was very convenient, in retrospect, to find Riemann’s non-Euclidean differential geometry (from the 1850’s) useful in Einstein’s general relativity. However, it’s much too optimistic to expect the same classical manifold to be useful in a theory of quantum gravity. This is why there is frequent mention of “emergent space” and noncommutative geometry on Marni’s blog. Alain Connes, the father of noncommutative geometry, put forth a noncommutative geometric GUT model, which had problems in its details, but overall may have pointed the way towards a proper mathematical path to quantum gravity – one where spacetime dissolves into mere morphisms between objects.

    As is well known to all modern string theorists, string theory is no longer a theory of strings. It is a theory of branes, and branes in turn, have positions given by matrices. For a system of N identical branes, their positions must be described by NxN position matrices. If the position matrices commute, they can be simultaneously diagonalized and their eigenvalues describe positions in a classical sense. The physics of the N branes is described by a gauge theory, with gauge group SU(N). For example, if one makes a black hole out of N branes, one would have an SU(N) gauge theory describing the black hole.

    From a mathematical perspective, brane positions are generally described by elements of C*-algebras. Actually, if the position matrices are Hermitian (as is usually the case), brane positions take values in Jordan algebras, which are the self-adjoint parts of C*-algebras. For example, take N=3, where one has three coincident branes described by elements of the Jordan algebra J(3,C) – the (nonassociative) algebra of 3×3 complex Hermitian matrices. The space of rank one elements of J(3,C) is the complex projective plane CP^2. The isometry group of CP^2 is SU(3), the automorphism group of the Jordan algebra J(3,C). J(3,C) sits inside M(3,C), the C*-algebra of 3×3 complex matrices and in noncommutative geometry M(3,C), just like any finite-dimensional C*-algebra, can be considered as a (noncommutative) algebra of functions on a finite set of points, spec(M(3,C)). In our N=3 example, this means our three branes have a description in terms of functions over a 3-point space, i.e., as noncommutative morphisms over three objects. In categorical terms, the three branes can be described by a 3-object category X where the collection of morphisms Hom(X)~M(3,C).

    There are more general C*-algebras, whose self-adjoint parts are, for example, the exceptional Jordan algebra J(3,O), the nonassociative algebra of 3×3 Hermitian matrices over the octonions. Now, turning the problem on its head, can we start with the math and try to envision a generalized brane system described by J(3,O)? We know the rank one elements of J(3,O) form the octonionic projective plane, OP^2, the space recently used by Sati as a fiber in extending M-theory (arXiv:0909.4737v2 [hep-th]. Moreover, the isometry group of OP^2 is the exceptional group F4, which is also the automorphism group of J(3,O). Naively, one could envision a system of three “octonionic branes” described by an F4 gauge theory. However, does such a system even make sense in string/M-theory? Well, it just might.

    J(3,O) sits inside a C*-algebra called the exceptional Jordan C*-algebra J(3,CxO), the algebra of 3×3 Hermitian matrices over the bioctonions. This algebra includes the split-exceptional Jordan J(3,Os), the algebra of 3×3 Hermitian matrices over the split-octonions. J(3,Os) and its subalgebra J(2,Os) are the charge spaces for black holes in D=5,6 N=8 supergravity, i.e., toroidally compactified M-theory on T^6 and T^5. Duff et al. have also used these algebras in the black hole/qubit correspondence (arXiv:1101.3559v2 [hep-th]. With time, it also became clear that one should define generalized qubits and qutrits for any composition algebra (not just over R,C), to strengthen the correspondence. This allows one to associate 1/2 BPS black holes in D=5,6 N=8 supergravity with qutrits and qubits over the split-octonions, for example.

    Many more mysteries remain, but it helps to be able to study quantum gravity through many different lenses, as Marni does. Marni’s approach is much like Grothendieck’s approach to mathematics. All the various parts could be written up into a “multiplodocus”, as Grothendieck and Serre refer to such long papers. The various chapters would include topics from twistor amplitudes, spectral triples and bubble operads to Shimura varieties, automorphic black holes and L-functions for motives. After covering these prerequisites (which would fill volumes), one could hopefully proclaim, “Let’s start doing quantum gravity!”

  7. Thanks for the excellent algebraic summary, kneemo. Of course, unlike many, I never set out to develop ‘my own’ theory, because (i) I was brought up to be an obedient and modest young lady, and (ii) I have a strong enough maths background to know that many great minds are needed to solve the essential puzzles of our time (as is always the case, if one knows one’s history). Newton did not work in a vacuum.

    In other words, the forum censorship cannot justify its action on the ‘personal theory’ grounds. What they really mean is that they don’t allow any discussion outside Formula Loopy Stuff, Formula String Stuff, Formula QI stuff or Formula NCG stuff. All I am doing is carefully choosing elements from all these approaches, according to what I believe are deeper categorical measurement principles. The literature is now heading in this direction anyway, as leaders recognise the failure of mainstream ideas. What concerns me most about the corruption here is the effect it has on the snail’s pace of progress. We have lost many opportunities already to make sound predictions before experiments, which is everything in science.

    As for kneemo’s textbook idea … that is one textbook, but I would rather write the Physics version, which is exceedingly elementary. It would start with measurement principles, qubit and qutrit secondary polytopes, then further categorical structure, then some algebra and motivic functions, and then slowly work its way up to motivic maths.

  8. My original plan for this blog was (1) I would save the lost thread (2) no-one would be reading the blog (3) I would put up a long post on some aspect of Marni’s work every few weeks (4) eventually it would show up in the search engines. I suppose that still is the plan.

    But with all this discussion, I thought I might after all respond to some of Ulla’s comment, since it’s about some down-to-earth issues, which could provide a counterpoint to the high-flown math. So, just four points:

    1. Leptons – such as electrons and neutrinos – are fermions. All the “matter” particles are fermions, that exclude each other from the same place, and the “force” particles are bosons, that can condense together at the same point. This is standard particle physics.

    2. In the Higgs model, the mass of a particle is essentially just how much the Higgs field slows it down. Mass has a very different origin in the original approach to twistors.

    3. I don’t see any strings in Marni’s work. When she talks about her work as “M-theory”, she only means that some of the same algebra is involved.

    4. I don’t know what you mean by her “triangular forms”, or why that is connected to strings.

    People who know better are welcome to comment on points 2-4…

  9. Point 2, yes. Even in the ’70s and ’80s, twistor theorists were thinking about how to extend the twistor H^1 sheaf cohomology to higher cohomology, in order to deal with massive field equations. Most simply, the Klein Gordon equation (see eg. Hughston and Hurd) can be equated with an H^2 setup involving a pair of massless solutions. This is physically inadequate as a replacement for the Higgs, but already very suggestive. But the right kind of cohomology theory for gravity has always been lacking. Only now it is clear that the emergence of classical geometry is closely linked to Grothendieck’s Motives, a universal cohomology theory, in a purely category theoretic sense. Moreover, modern twistor scattering theory proves (beyond any reasonable doubt) that the insights into non locality in twistor geometry are more physical than the usual Lagrangian business.

  10. One example of the highly interesting ‘triangular forms’. This is complex.i can understand that it comes from different matrices, which then can be interpreted as ‘strings’ formed of three forms of numbers (helices?, knots?). Kneemo says there are no strings, just braids (3-surfaces?). But how does braids and manifolds differ? Both are surfaces?

  11. Ulla, your attempts to discuss mathematics are not welcome here.

  12. Thanks everyone for providing future readers with that glimpse of how it was, in the physics blogosphere of 2011. Comments will now be closed, until I have a plan for this blog, beyond simply preserving the record.

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