Thursday, November 22, 2018

Guest Post: Garrett Lisi on Geometric Naturalness

Thanks to Sabine for inviting me to do a guest post. In her book, “Lost in Math,” I mentioned the criteria of “geometric naturalness” for judging theories of fundamental physics. Here I would like to give a personal definition of this and expand on it a bit.

Figure: Lowest million roots of E8+++ projected from eleven dimensions.
As physicists we unabashedly steal tools from mathematicians. And we don’t usually care where these tools come from or how they fit into the rest of mathematics, as long as they’re useful. But as we learn more mathematics, aesthetics from math influences our minds. The first place I strongly encountered this was in General Relativity. The tools of metric, covariant derivative, connection, and curvature of a manifold were irresistibly attractive structures that lured my impressionable soul into the mathematical world of differential geometry.

Studying differential geometry, I learned that the tensors I had been manipulating for physics calculations represent coordinate-invariant objects describing local maps between vector fields. That multi-dimensional integrals are really integrals over differential forms, with changing order of integration corresponding to 1-forms anti-commuting. That a spacetime metric is a local map from two vector fields to a scalar field, which can also be described using a frame – a vector-valued 1-form field mapping spacetime vectors into another vector space. And that the connection and curvature fields related to this frame completely describe the local geometry of a manifold.

With exposure to differential geometry and GR, and impressed by GR’s amazing success, I came to believe that our universe is fundamentally geometric. That ALL structures worth considering in fundamental theoretical physics are geometrically natural, involving maps between vector fields on a manifold, and coordinate-invariant operations on those maps. This belief was further strengthened by understanding Lie groups and gauge theory from a geometrically natural point of view.

Students usually first encounter Lie groups in an algebraic context, with a commutator bracket of two physical rotation generators giving a third. Then we learn that these algebraic generators correspond to tangent vector fields on a 3D Lie group manifold called SO(3). And the commutator bracket of two generators corresponds to the Lie derivative of one generator vector field with respect to the other – a natural geometric operation.

The principle of geometric naturalness in fundamental physics has its greatest success in gauge theory. A gauge theory with N-dimensional gauge Lie group, G, can be understood as an (N+4)-dimensional total space manifold, E, over 4-dimensional spacetime, M. Spacetime here is not independent, but incorporated in the total space. A gauge potential field in spacetime corresponds to a 1-form Ehresmann connection field over the total space, mapping tangent vectors into the Vertical tangent vector space of Lie group fibers, or equivalently into the vector space of the Lie algebra. The curvature of this connection corresponds to the geometry of the total space, and is physically the gauge field strength in spacetime. For physics, the action is the total space integral of the curvature and its Hodge dual.

The strong presence of geometric naturalness in fundamental physics is incontrovertible. But is it ALL geometrically natural? Things get more complicated with fermions. We can enlarge the total space of a gauge theory to include an associated bundle with fibers transforming appropriately as a fermion representation space under the gauge group. This appearance of a representation space is ad-hoc; although, with the Peter-Weyl theorem identifying representations with the excitation spectrum of the Lie group manifold, it is arguably geometric and not just algebraic. (The rich mathematics of representation theory has been greatly under-appreciated by physicists.) But the physically essential requirement that fermion fields be anti-commuting Grassmann numbers is not geometrically natural at all. Here the religion of geometric naturalness appears to have failed... So physicists strayed.

Proponents of supersymmetry wholeheartedly embraced anti-commuting numbers, working on theories in which every field, and thus every physical elementary particle, has a Grassmann-conjugate partner –  with physical particles having superpartners. But despite eager anticipation of their arrival, these superpartners are nowhere to be seen. The SUSY program has failed. Having embraced supersymmetry, string theorists are also in a bad place. And even without SUSY, string theory is geometric – involving embedding manifolds in other manifolds – but is not geometrically natural as defined above. And the string theory program has spectacularly failed to deliver on its promises.

What about the rebels? The Connes-Lott-Chamseddine program of non-commutative geometry takes the spectral representation space of fermions as fundamental, and so abandons geometric naturalness. Eric Weinstein's Geometric Unity program promotes the metric to an Ehresmann connection on a 14-dimensional total space, fully consistent with geometric naturalness, but again there are ad-hoc representation spaces and non-geometric Grassmann fields. The Loop Quantum Gravity program, ostensibly the direct descendent of GR, proceeds with the development of a fundamentally quantum description that is discrete – and thus not geometrically natural. Other rebels, including Klee Irwin's group, Wolfram, Wen, and many others, adopt fundamentally discrete structures from the start and ignore geometric naturalness entirely. So I guess that leaves me.

Enchanted with geometric naturalness, I spent a very long time trying to figure out a more natural description of fermions. In 2007 I was surprised and delighted to find that the representation space of one generation of Standard Model fermions, acted on by gravity and gauge fields, exists as part of the largest simple exceptional Lie group, E8. My critics were delighted that E8 also necessarily contains a generation of mirror fermions, which, like superparticles, are not observed.

My attempts to address this issue were unsatisfactory, but I made other progress. In 2015 I developed a model, Lie Group Cosmology, that showed how spacetime could emerge within a Lie group, with physical fermions appearing as geometrically natural, anti-coummuting orthogonal 1-forms, equivalent to Grassmann numbers in calculations. For the first time, there was a complete and geometrically natural description of fermions. But there was still the issue of mirror fermions, which Distler and Garibaldi had used in 2009 to successfully kill the theory, claiming that there was “no known mechanism by which it [any non-chiral theory] could reduce to a chiral theory.” But they were wrong.

Unbeknownst to me, Wilczek and Zee had solved this problem in 1979. (Oddly, first published in a conference proceedings edited by my graduate advisor.) I wish they'd told me! Anyway, mirror fermions can be confined (similarly to su(3) color confinement) by a so(5) inside a so(8) which, by triality, leaves exactly three generations of chiral fermions unconfined. Extending E8 to infinite-dimensional Lie groups, such as very-extended E8+++ (see Figure), this produces three generations and no mirrors. And, as I wrote in 2007, one needs to consider infinite-dimensional Lie groups anyway for a quantum description... something almost nobody talks about.

No current unified theory includes quantum mechanics fundamentally as part of its structure. But a truly unified theory must. And I believe the ultimate theory will be geometrically natural. Canonical quantum commutation relations are a Lie bracket, which can be part of a Lie group in a geometrically natural description. I fully expect this will lead to a beautiful quantum-unified theory – what I am currently working on.

I never expected to find beauty in theoretical physics. I stumbled into it, and into E8 in particular, when looking for a naturally geometric description of fermions. But beauty is inarguably there, and I do think it is a good guide for theory building. I also think it is good for researchers to have a variety of aesthetic tastes for what guides and motivates them. The high energy physics community has spent far too much time following the bandwagon of superstring theory, long after the music has stopped playing. It’s time for theorists to spread out into the vast realm of theoretical possibilities and explore different ideas.

Personally, I think the “naturalness” aesthetic of fundamental constants being near 1 is a red herring – the universe doesn’t seem to care about that. For my own guiding aesthetic of beauty, I have adopted geometric naturalness and a balance of complexity and simplicity, which I believe has served me well. If one is going to be lost, mathematics is a wonderful place to wander around. But not all those who wander are lost.


Henning Dekant said...

Beautifully written, and very convincing (at least to me).

Garrett said...

Thank you Henning. I suspect the main criticism here will be that I have set my own criterion for geometric naturalness, so that it is not so surprising I am the only one able to satisfy it. And that criticism is more-or-less fair.

deepak said...

Hmmm ... almost sends like a counterpoint to the thesis of Sabine's recent book - that "beauty" and "naturalness" are unfaithful guides for exploring the theoretical landscape.

In any case, it is surprising that Lisi feels that "fundamentally discrete" structures are not "geometrically natural". What criteria, other than a purely subjective viewpoint, could one use for favoring continuous over discrete structures as being more geometrically natural?

Moreover now it appears that to salvage his theory Lisi must extend E8 to E8+++ (or is it E8++++?). Isn't that then going down the rabbit hole the same way that string theory went in search of a compactification which would correspond to our universe?

While beauty and naturalness can provide signposts for which direction to look in another very important element is simplicity. That appears to be missing in Lisi's most recent formulation of EK+++ as compared to say LQG or Wen's spin nets. Though, I'm sure he would disagree!

Enrico said...


Supergravity in the 1970s attempted to unify general relativity and Standard Model. How does your E8+ group theory differ from supergravity? Does it include QCD, Higgs mechanism and electroweak symmetry breaking?

Tanner said...

Why am I not convinced? ��

Julian said...

Truth is Beauty !

Yehonatan Knoll said...

The undeniable beauty of differential geometry is the curse of modern physics.

Riemannian geometry, being a generalization of Gauss's work, is just what can be said of a manifold which is independent of a specific chart/atlas, used to describe the manifold. Riemann's metric, therefore, is positive definite as it makes no sense for points to have a negative mutual distance or for distinct points to have a vanishing distance. You can, as Einstein did, relax this condition and still use all the results of Riemann, but then you are not doing geometry; You are implementing general covariance. And although this may appear just a matter of terminology, there is a huge difference between the two.

Let me explain. Nowadays, we take coordinates for granted as part of any physical theory. But what are coordinates? The best image I could come up with, as an undergrad student, was some hypothetical `soup' of identical clocks, each carrying three (spatial) labels. But then I realized that a clock is some physical device whose description requires coordinates! You just can't avoid the circularity. The only way to think of coordinates is as some `scaffold' used in the construction of the `actual thing' which is a measurement - some (dimensionless) number. Being a mere scaffold, it must never appear in the measurement result and general covariance is but one convenient way of achieving that. No geometry and no gravitation.

The general covariance rather than the geometrical view, comes with many benefits.
The first (which I personally have not yet explored) is to look for formulations of physics not involving coordinates at all. The second is the unnatural (or even absurd) appearance of most geometry-motivated proposed theories (involving extra dimensions, non standard geometries etc.) when viewed in the light of general covariance. Finally (a path which I have extensively pursued), when you realize that general covariance is just a prescription for getting rid of coordinates in a final measurement, you start doing physics instead of pure mathematics, directly relating your mathematical structure to measurements. An excuse often heard from string-theorists regarding the lack of predictions is that the translation of their mathematical structure to the `language of labs' is a formidable task on its own. But this is just an advanced variant of what has been going on in high-energy physics for the past few decades: Except for S-matrices, the "spell of the gauge principle" has yielded almost nothing.

Koenraad Van Spaendonck said...

Lisi said :
"...a fundamentally quantum description that is discrete – and thus not geometrically natural"

The key is to join both into one concept : the discrete quantum and the geometrically natural language.

MartinB said...

Thanks for explaining (which I never really understood) why people are getting excited about E8.

JeanTate said...

How about experimental or observational tests? Even if in principle only, today (maybe feasible in a century or so’s time).

Bee wrote about one such, re quantum gravity, a while ago (it was an in principle one, which may be possible in a few decades’ time), and we know the fate of SUSY experimental tests. So how about E8?

belliott4488 said...

Goodness! Sabine H. has allowed a guest post that claims that aesthetics should guide physicists?? Shocking. ;-)

I've long believed that successful fundamental theories will necessarily have some kind of aesthetic beauty. I say this because they so often reveal new symmetries that unify constructs previously seen as disparate, and do so by applying new mathematical constructs that are naturally more elegant.

I wish you the best of luck, Garrett. I would love to see you succeed in this effort.

Michael John Sarnowski said...

If geometry is beauty, then the creation is stunning.

Lawrence Crowell said...

The E8 group is real, or 8 reals in pairs. By itself it is not a quantum unitary group. One needs a pair of them, which has a mermophism to SO(32). I think this holds for extended E8 groups.

Uncle Al said...

Forty years of theory failure are not wrong, merely non-empirical for 0.1 ppb beautifying baryogenesis' Sakharov conditions. You propose eternal math. I propose 24 hours of observation each in two experiments. Beautiful physical theory fears analytical chemistry ̶ the real world with all its interactive slop ̶ as fundamental falsification. LOOK

Euclid was Beautiful for 2000 years, until Bolyai, then Thurston.

Unknown said...

Dear Lisi,

Is not the Unruh effect incompatible with your principle of "geometric naturalness?" If a "geometrically natural" physical theory involves only "maps between vector fields on a manifold, and coordinate-invariant operations on those maps," then should one not conclude that the vacuum state of a QFT must be defined independently of a choice of a chart or of a reference frame on the underlying spacetime manifold? The vacuum state and other field operators should only depend on the intrinsically geometric properties of that spacetime, if we are to accept the "geometric naturalness" principle. In the case of the Unruh effect, observers belonging to a Rindler reference frame are supposed to experience a different vacuum from the observers of an inertial RF, even while the spacetime manifold is the flat Minkowski space.

Followig this viewpoint further, shouldn't a "geometrically natural" QFT be diffeomorphically invariant, in the sense that to each diffeomorphism transformation on the base manifold, one can associate an unitary transformation acting on the Hilbert state-space of the underlying QFT, in a way that generalizes the unitary transformations that realizes the Lorentz boots, e.g., on the Hilbert state-space of special relativistic QFTs?

Thank you,

clueless_grad_student said...

As a former student in physics and current student in math, I'm a bit confused by the use of the word "naturalness".
As far as I understand, usually when mathematicians say natural (which kind of has a precise meaning) they mean something that can be made sense of without you making a particular choice. This kind of means that the structure in question is "out there" and doesn't depend on you.
E.g. There is a natural map from a vector space V to V** (dual of its dual), but there is no natural map from V to V* (because to define such a map you need to choose a basis or a metric, and different choices give different maps).
Another way I've seen it used is that certain structures behave well with respect to functors between categories. (I don't know any higher category theory but maybe there's more to this here?)
None of Lie groups or Riemannian geometry or gauge theory is really "natural" in the sense above. You study them precisely because you care about specific objects with these specific properties (maybe because you want to study geometry or symmetry), not because these objects are in some sense "natural". (i.e. you study a specific lie group or manifold).
So I first learned about these structures and how they show up in physics, they were indeed very pretty (until you actually need to solve the equations at least..), but (based on my limited understanding of math and physics and everything else in general) I wouldn't go out of my way to call them special. There are lots of very interesting objects in mathematics (and in general), it just so happens these are the ones that are most aligned with how we've been thinking about physics. And the way we've been thinking about physics may not be the correct one, as we discover more physics we'd expect to use other bits of mathematics to build our models.
Also given how very few integrals in QFT exist (or I've been told) I always thought the expectation was that one day we'd discover some new mathematics to help solve our problems, when we first discover it we might say it's ugly and messy and difficult and unnatural, but after a few degenerations of struggle we finally give in and throw our hands in the air and say whatever, let's call this new structure "natural".

Please don't yell at me.

Garrett said...


Sabine's book is very well written and researched, and she presents good arguments, but yes, I am fundamentally in disagreement with its principle thesis. I do think beauty and naturalness (as described) is a good guide for exploring the theoretical landscape.

My view is of course subjective. And it is possible to arrive at discrete structures having started with the continuous. But I do feel it is the continuous that is fundamental.

As to which structure is fundamental, I do think E8+++ (E11) will be too small as well, but it's a good place to start. And there is a lot of exploring to do here. There are also DEN and AEN families and others to explore here... And these infinite-dimensional Lie groups are, technically, simple. ;)

Garrett said...


Supergravity assumes supersymmetry as fundamental. I do not. Past work with E8 does include QCD, Higgs, and EW symmetry breaking. It also includes mirror fermions. Current work with E8+++ (and others) solves the mirror problem and has three generations of fermions.

Garrett said...


Geometric Naturalness is just, like, my opinion, man.

Garrett said...

Yehonatan Knoll,

You make excellent points that I wholeheartedly disagree with.

Relaxing constraints on mathematical definitions is a wonderful road to progress, and has a long history of successes. And though it does change the character of the mathematics, it doesn't usually change it completely. Saying psuedo-Riemannian geometry is "not geometry" is just you being unreasonably rigid in your definition. Such rigidity is anathema to progress.

I do agree with you on coordinates being scaffolding though.

Garrett said...

Koenraad Van Spaendonck,

How to do the "join" though... Personally, I think best strategy is to start with a geometrically natural continuous structure, and produce the quantum.

Garrett said...

I will be happy first to reproduce everything in the Standard Model from first principles, before being confident about predicting new particles. But I do strongly believe in the primacy of the scientific method.

Sabine is one of a rare breed of humans willing to balance and be considerate of opposing points of view. And thanks. I agree with you on beauty. But, as Sabine points out in her book, beauty is such a subjective thing as to be possibly useless. Coming up with more precise definitions, such as "numerical naturalness" or "geometric naturalness" is trickier business, but potentially more useful.

Michael John Sarnowski,
It's in the eye of the beholder.

Lawrence Crowell,
I'm on to a different approach.

Garrett said...


What I found in Lie Group Cosmology is that de Sitter spacetime emerges naturally as a vacuum spacetime. And once there's such a vacuum spacetime, with well defined frame, it's possible to correctly consider acceleration with respect to that frame, and the Unruh effect. I don't have the full geometrically natural description of QFT worked out yet, but that is what I am working on. I will give a hint: annihilation and creation operators are part of an infinite-dimensional Lie group, and Fock space is its fundamental representation space.

qsa said...

I agree with your "Geometric Naturalness", but I think it should be even more simpler as numbers interpreted as points, then the the difference between them are lines, and that is all. So it would be something "like" Wolfram idea, what could be more natural than that.

While such system might be converted to group theoretic eventually, but I think group theoretic has outlived its usefulness in fundamental physics. But you know, humans are creatures of habits:(

But of course nothing takes away from your ingenuity.

Does your system predict the cosmological constant ?

Unknown said...

Dear Lisi,

I would like to add some further remarks:

1. Do you think that adopting a "geometrical naturalness" principle is sufficient to treat gravitation and the others interactions as equals?

The discovery of GR started in physics the long road of a research program whose main goal consisted of a reformulation of physical theories using only geometric objects. Examples are the failed attempts at a classical unified field by Weyl, Kaluza and Klein, Einstein, Eddington, Schrodinger and many others; Rainich's already unified theory based on the vacuum Einstein-Maxwell equations, or Wheeler's revival of the latter under the name of "geometrodynamics." In mathematics, however, a far more productive (at least in my view) research program started from the interest that mathematicians like, Cartan and Weitzenböck, took on the geometric structure of GR, culminating in the theory of connections on principal fiber bundles by Ehresmann, Koszul and Kobayashi. These two approaches, from physics and mathematics, seemed to have converged, with the work of Yang and Mills, in the modern theory of gauge fields.

As you said above, "The curvature [of this connection] corresponds to the geometry of the total space, and is physically the gauge field strength in spacetime. For physics, the action is the total space integral of the curvature and its Hodge dual."

However, I still believe that a fundamental distinction exists in the way that gravitation and Yang-Mills gauge fields are formulated even at the classical, geometrical level. First, if you consider GR as formulated by Einstein and Hilbert, using the metric-affine (Riemann/Levi-Civita) connection on the p-bundle of frames, the symmetry group of the fibers (or the gauge group, in physicist's language) is the non-compact GL(n,R), which is structurally quite distinct from Yang-Mills' SU(N). Moreover, the action functional of Einstein-Hilbert, being the integral of the curvature scalar density, is also structurally very different from the Yang-Mills action functional, the latter one involving the curvature and its Hodge dual.

An alternative formulation of GR, that is normally presented in the context of gauge theories, is the tetrad formalism (which seems to be the one that you use in your work on the E(8)-theory). In this case, one can formulate the gravitational potential as a 1-form connection taking its values in the Lie group associated to the group of affine translations T(4), as done by Trautman, Møller, Wallner and Thirring, and others. Now the Lagrangian density looks like the ones of Yang-Mills theories, but it still posses structural distinctions: the Lagrangian is only of first order in the Hodge dual of the associated curvature (L ~ e/\e/\*R); and the gauge group, being abelian and non-compact, is also quite different from the ones of the renormalizable Yang-Mills theories.

2. You have commented on some alternative theories such as non-commutative geometry program and LQG. Do you have anything to say regarding Penrose's twistor program? (I do not mean its perversion under string theory, but the original approach to quantum geometry.)

3. When you speak about "maps between vector fields on a manifold, and coordinate-invariant operations on those maps," this seems very much like the language employed by category theorists (functors between categories etc.) Do you see any advantage in using category theory in your work?


Josh said...

Great post, really looking forward to what you're going to come up with. Happy Thanksgiving Garrett!

WRL said...

It seems to me that you've contributing significantly to the state of being lost in math.
Given the fascination with E8 X E8, and the fact that QC/ED (..QFT) fits easily within a subgroup of a cross product of two wreath products... what is the relationship between E8 and the useful wreath product??
Or more to the point, why bother with all the extra metaphysics?

It is noteworthy here that spherical harmonics work ONLY in three space dimensions. So the 'extra' dimensions required can ONLY be dimensions of a finite representation geometry, yielding a consistent formulation - applicable across all physical scales.

..& please don't complain about proceedings papers. If it is published, it is published.. go ahead and peer review. Or not.

Enrico said...


The Rindler frame is an accelerating frame. By Einstein's equivalence principle, it is equivalent to a gravitational field. Hence, the local spacetime is curved and has intrinsic geometric properties. The observable universe is flat on cosmic scale and described by Minkowski spacetime but spacetime near a black hole has different geomertry.

Sabine Hossenfelder said...


The reason my book is called "Lost in Math" is that I point out arguments from beauty have gotten lost in math. By this I mean - as I explain in great detail - that physicists no longer recognize their subjective arguments for what they are; they have reformulated them as mathematical requirements and forgotten that they are really merely wishes.

Garrett at least states his assumptions clearly. I approve of this because that way we know what we are talking about. This is not to say that I support geometric naturalness, but that I endorse conceptual clarity.

Sabine Hossenfelder said...

ps: Should have added, of course the title does double-duty in referring to my own story.

Garrett said...


I agree with you here, but one of the points I'm trying to make with my post is that researchers should develop different criteria for what they consider "natural" and "beautiful," because when they employ such a criteria they are limiting their search space. But if everybody chooses the same limiting criteria, we might miss finding something new and fundamental that doesn't match it. So you should develop your own! At the same time, I am arguing for my criteria, geometric naturalness, based on mathematical structures I like and see in known fundamental physics, and I'm doing that mostly to get people more interested in my work. As for QFT... I believe I've found something wonderful that the community missed, and it matches well with the models I work with, so, I guess, stay tuned...

Garrett said...

Please, you are more than welcome to adopt and use your own criteria of naturalness. And I wish you luck with it, sincerely. As far as the cosmological constant, yes, de Sitter space with a positive cosmological constant emerges in Lie Group Cosmology.


1. Your understanding and opinions on how differential geometry made its way into physics via pseudo-Riemannian geometry and gravity, and then was given a different but equivalent description using fiber bundles, very much matches my own. And my own research path matched this. I started out trying to use Kaluza-Klein theory to also describe the gauge fields, as is well known to be possible, using the Einstein-Hilbert action. But I was never happy with fermions in that picture. I got MUCH more happy when I took the opposite approach of describing gravity as a sl(2,C)=spin(1,3) gauge field. Then I got even happier when I saw you could include the spin connection and frame as part of a spin(1,4) connection, a' la MacDowell-Mansouri. (If you haven't seen this formulation of gravity, as someone who appreciates the gauge theory approach, you'd probably like it.) And this unifies with the SO(10) gut in spin(11,3), which acts on a 64+ spinor of one generation of fermions. All of which (and more) is in E8. You are very correct that the gauge group is now non-compact, but the formulation goes through nicely. The MacDowell-Mansouri curvature ends up being F = R + e/\e, so the Yang-Mills action gives the desired e/\e/\*R term as well as an interesting R/\*R. This approach, which starts with Spin(1,4), is known as "Cartan gravity" and led to what I called Lie Group Cosmology.

2. Twistors are beautiful, geometrically natural structures, which I associate with the equivalence spin(2,4)=su(2,2). I did link to Roger Penrose's book, The Road to Reality, as a great place to learn about geometrically natural structures in physics. But I didn't include him in the list of rebel ToE researchers because he never seemed interested in building a unified theory.

3. For the most part I am categorically opposed. It's kinda cool, I just haven't seen it be usefully applied in physics -- to produce new results or easier calculations in anything related to anything measurable. I consider it an interesting form of mathematical abstraction, but I haven't drunk the cool-aid, and prefer to use older tools that I'm more familiar with and seem closer to known physics.


Garrett said...


I am, very much, a prime exemplar of mathematical lostness. This post was nothing if not a full expression of this disease. I have not, however, familiarized myself with the wreath product, so can't comment on it. The metaphysics... mostly serves to limit my search space for new theory. And it makes for interesting discussions.

Harmonics... My understanding (shoddy though it is) of the Peter-Weyl theorem is that it identifies representations of ANY compact Lie group with its harmonics. i.e. not just spherical harmonics, but Laplace eigenstates in general on the relevant Lie group manifold. There is more recent work -- by Harish-Chandra I think? -- on this identification for non-compact Lie groups. And an active research community of mathematicians working on related things for infinite-dimensional Lie groups and representations.

Wasn't complaining about proceedings. Was amused my grad advisor, Henry Abarbanel, was editor -- must have been when he was in high energy, before switching to dynamical chaos.

Garrett said...

Not sure what you're asking about here?

Unknown said...

Congratulations Mr Garrett Lisi!
Beautiful article in our beautiful world.

distler said...

"Unbeknownst to me, Wilczek and Zee had solved this problem in 1979."

Alas, you have also misunderstood Wilczek and Zee.

They start with a chiral spinor of Spin(18) and, after gauging a Spin(6) subgroup (which confines), they end up with chiral spinors of Spin(10). This (unlike your attempted constructions) is not in conflict with basic physical principles.

So, no, Distler and Garibaldi were not wrong.

The chiral spinor of Spin(18) is a complex representation (not equivalent to its conjugate). I am shocked that you have still not understood this after more than a decade.

Enrico said...

248 and infinite-dimensional E8 groups are unobservable. Why is it more credible than 4-dimensional non-unified general relativity and Standard Model?

You wrote in your paper:
“Lie Group Cosmology also suggests that our universe is fundamentally de Sitter—infinite in spatial and both temporal directions. Physically, this means our universe existed forever, before the big bang, and has been forever cooling and accelerating in its expansion, and will continue this expansion, approaching perfect emptiness and symmetry.”

One end of a timeline is the beginning of time. The other end is the end of time. Since the timeline is infinitely long in both directions, any point between the two ends is infinitely far from the beginning of time. You will not get a big bang. You will get absolute zero cold and empty space.

Yehonatan Knoll said...


You seem to have missed my point: In both geometry and physics, coordinates serve as scaffolding, leading to similarly looking mathematics. This superficial, anecdotal similarity has led generations of physicists to identify physics with geometry. I, too, was absolutely mesmerized by the beauty of geometry. But I quickly realize the danger of "getting lost in math", not finding my way back to the lab (and in my opinion, even predicting yet another resonance doesn't qualify as finding the way back; This is too low of a standards for a physical theory).

A. Mikovic said...

When one is looking for a mathematical structure that can unify SM and GR, there are two criteria: (1) simplicity, (2) possibility to explain the current puzzles like the number of particle generations, particle masses, dark matter, dark energy etc.

The approach of G. Lisi is natural in the sense that it uses the well-known mathematical structures like smooth manifolds and Lie groups, while the simplicity is attained by using a unifying Lie group. However, the criterion (2) is more difficult to satisfy, and for this one needs to quantize the theory, which
also means to introduce a new mathematical structure, since quantizing GR is not a unique procedure. Note that there is already a minimalist quantum gravity proposal called ASG (asimptotically safe gravity), where it is assumed that one can use the QFT formalism for GR because there are indications that GR is renormalisable in a non-perturbative regime. However, even if that claim is correct, the ASG approach does not say much about (2). This can be improved by applying ASG to E8 unified theory. However, the criterion (2) is still difficult to satisfy, and furthermore many physicist would also like to see some new physics coming out of a new

Mozibur said...

A great article and very informative too. I don't know whether this is relevant at all but I read somewhere - I think it was an expository article on Grassmanns work - that Grassmann, when he was theorising the Grassmann numbers, discovered that the product had to be either commutative or anti-commutative on geometric grounds. I can't recall what exactly exactly this geometric grounds were, but I think it was due to a transformation law that a vector would follow.

Unknown said...

Beauty is the promise of happiness.

randall morris said...

A more fundamental way to address naturalness is to map bispinors into tensors fields and rewrite the Dirac equation as a tensor equation. This has been done by F. Reifler and published. The tensor form of the Dirac Equation turns out to be a constrained Yang Mills equation. It was shown that this system can be quantized to produce fermions with the correct commutation properties, even though the tensors do not have the 720 degree property of bispinors, which is not directly observable in any case. From this viewpoint, bispinors appear to be coordinates which linearize a constrained Yang-Mills equation into the Dirac equation. It was further shown that the standard model can be reproduced with this system in a "natural" geometric interpretation at the first quantized level, which also includes gravity, though this unification does not solve the more fundamental problem of multiple GR spacetimes that must be attached to each particle. Making sense of QM seems to require a universal time or set of time frames which are not local.

JeanTate said...

@Garrett Thanks.

How long do think it will be before you can start testing against experiment and observation? And before you start looking for new particles, will you be able to reduce the number of ad hoc/arbitrary constants in the SM (~25) at least somewhat? How about the g-2 anomalies? And the neutrino, Majorana or Dirac?

Reimond said...


Your geometric naturalness is indeed much more well-founded than just naturalness, which as you say is a red herring.

I only know the basic stuff like SO(4) is locally isomorph to SO(3)⊗SO(3) and SO(3) is double covered by SU(2), SO(3,1) is locally isomorph to SU(2)⊗SU(2), SO(3,1) is the analytical continuation of SO(4), …. I was already happy to understand spinor representation and I never reached some elaborate level like you did. Amazing that all of the standard model, gravity and even a relation like Λ=3/4Φ² is included in E8. Greatest respect for your work and congratulation.
Actually, I wondered how to bring non-compact, real SO(3,1) and compact, complex U(1), SU(2), SU(3) together in one group without possible topological issues later on. And it is amazing that you can also include fermions without their mirrors in E8+++. This is probably a stupid question, but is the gravitational connection torsion-free? And obviously you get around the Coleman-Mandula theorem somehow - is it because de Sitter spacetime is a coset manifold SO(4,1)/SO(3,1)?

“But is it ALL geometrically natural?”
In [1] you write “… action for the theory are chosen by hand to match the standard model - this needs a mathematical justification.” You use the modified BF action for gravity and all fits remarkable well.
But you seem, also in [1], not yet decided which kind of dynamics to choose, since you write “In any case, the dynamics depends on the action, and the action depends on the curvature of the connection.”
So, I guess as in [2] you have the Relational Quantum Mechanics interpretation in mind as dynamics which makes the world change. A huge path integral over the whole action. But would not this be a quantized spacetime? Maybe I am wrong here.
BTW, since you also mentioned A. Zee and I just saw what you did in [2] do you have a clue why Euclidean quantum field theory in (D+1)-dimensional spacetime, with cyclic boundary condition becomes quantum statistical mechanics in D-dimensional space?
(see here on page 288 eq. (5)). Maybe a connection with Rovelli´s thermal time?

“Spacetime here is not independent, but incorporated in the total space”
Would not a non-quantized spacetime be much more geometric?
Like Penrose objective reduction (OR), since you mentioned “Road to Reality”, I see more the tension between (non-quantized) spacetime and quantized matter as a potential dynamics.
Tiny process steps everywhere, all the “time”, bringing linear QM and the non-linear GR together, patching the spacetime like a sheaf cohomology with every tiny backreaction. Unitary evolution in between reductions and E8 in every tiny step acting in the tangent space. This would resemble what a minimal surface does.
In this comment at the end the quote by George Ellis and Sabine should indicate the tension between exclusively unitary evolution and the notion of probabilities.

In Penrose OR particles are getting more and more entangled until the mass/energy exceeds a certain threshold and an observer independent reduction is triggered. (Even to explain why in the Unruh effect each different accelerated observer sees its own temperature is easy. If the entangled particles partly belong to an observer, then this of course becomes apparently observer dependent, so I guess you would not need a preferred, a “well defined frame” here.)


Reimond said...

Considering Elie Cartan’s formulation of the differential geometry of Riemannian manifolds. (Ignoring exterior products /\ and Hodges *) Given the spin connection 1-form ω with de + ωe = 0 and transforming the frame e under e → Λ(x)e with Λ(x)∈ SO(3,1) results in:
R = dω + ω²; ω → ΛωΛ^-1 + ΛdΛ^-1;
Yang-Mills later did the same for the gauge connection A. Transforming the spinor ψ under ψ → U(x)ψ with U(x) ∈ U(1), SU(2), SU(3) results in:
F = dA + A²; A → UAU^-1 + UdU^-1;
An analogy between the classical and QM perfect like “The Perfect Wave”, but it breaks, since the classical Einstein-Hilbert action S_EH ∿ R and the QM Yang-Mills action S_YM ∿ F². Further the Einstein field eq. is derived by varying δS_EH = 0, but for QM probabilities one needs the path integral ∫e^(iS).
Because of this mismatch in the actions physicists gave up on this. I find this surrender premature, for two reasons:
1.) there is another striking analogy between the metric g, being the square of the frame -g=e² and the probability ∿ |ψ|².
(-g=e² works also with Dirac´s γ matrices as {eγ, eγ}=2g, which you also use in [1] (page 27) Tr(eγγe)=4 )
2.) When we compare these two actions the classical and the QM one by dividing S_EH by ℏ we get 1/(16πGℏ) having set c=1. This relates to the Bekenstein-Hawking entropy area/(4 l_Planck²).

My point is maybe we should take this apparent conflict more seriously and use it for the dynamics.

“complexity and simplicity“
Let me finish with what Sabine says in her book: “If you start at a larger scale, the scale of, say, galaxies, and go down, it doesn’t get simpler - it first gets more complicated, as with life crawling around on planets and all that. It’s only past the level of biochemistry that it starts getting simpler again.”
Do you think that the dynamics, the evolution of our world will be a deterministic, an exclusively unitary one? Do you believe that the next wave you will surf is already set by the initial condition in the very beginning of “time”?

[1] “An Exceptionally Simple Theory of Everything
[2] “Quantum mechanics from a universal action reservoir

ben6993 said...

I think that your E8 ideas are excellent.

1. If the elementary particles are at vertices of the E8 structure pictured in the inset to your article. It seems naively to have a symmetrical structure. The electric, colour and weak isospin charges average out to zero across the whole structure? Is it not-so-beautiful/symmetrical that the masses of particles do not average to zero in such a symmetric structure? (I say this because I have a naive computer simulation using negative masses to mimic some dark mass and dark energy effects.)

2. An even more symmetrical structure would be an N dimensional sphere. But that would be too featureless to be useful. So naturalness of forms requires asymmetry or broken symmetry. In the same way that fractals have interesting and beautiful features which are not blandly or simplistically symmetric.

3. The general feeling is, I believe, that the last turtle in the pile is the elementary particle of the Standard Model. And these fit your E8+ vertices. But if preons did exist, and they were fewer in number than the elementary particles, could they live in a simpler structure than E8.
Or would the structure have to be more complicated? In my view there would have to be the same number of dimensions used for both SM and preons. But maybe have a simpler structure for preons since: does a brick need to have in it all the information needed to design and build a house.

4. Artists likewise input the design information to make a picture but the colours of paint are the building blocks of the picture. To what extent does a mathematical Group need to include the artist as well as the paint in its Group relationships.

5. A Picasso painting of his blue period would be asymmetric of colour. Artists sometimes paint in two colours: one colour plus its complementary colour. A symmetrical painting in this style would be net grey i.e. equivalent to average of zero colour. So an interesting painting could be asymmetric (net blue) but a different painting could be net grey with interesting zones of blue and anti-blue (orange) within it. And that is ambiguous as a beauty criterion. Also the general public used to be said to be fifty years behind professional artists in taste. So the lay person is likely to lag way behind professional physicists in what is held to be a beautiful Group structure?

6. I prefer the fundamentality of continuous over discrete. In the SM, some elementary particles seem to annihilate into energy and then be created from energy. That seems to me to imply that there is discreteness within the SM-level of discreteness which implies an arbitrary granularity of discreteness if not quasi-infinite divisibility.

Austin Fearnley

Garrett said...


What I understand is that you are either confused by what is going on, or are obfuscating with your choice of language. Finding how mirror fermions are confined away is EXACTLY what Wilczek and Zee did. Real, pseudoreal, and complex representations aren't the issue here. The existence of mirror fermions (what you, unlike everybody else, chose to call an "anti-generation") is the issue, and, contradicting your claims to the contrary, there is a way for mirror fermions to exist and be confined (using a spin(5), not a spin(6) as you wrote), producing three generations of chiral fermions as we observe.

Garrett said...


These unified models are not more "credible," than GR+SM. The idea is that they are more unified than GR+SM, and they have more particles that might someday be observed, and might be able to solve theoretical gaps in GR+SM.

If the universe has been expanding eternally, that leaves an infinite amount of time for particle creation as it expands, leading to and past the era of the big bang we see evidence for.

Garrett said...

Yehonatan Knoll,
I understand your position, I simply disagree with it strongly. I don't think the similarity of geometric and physics tools is superficial, and I do think fundamental physics is geometry. If it's not, nature has played a dirty trick on us.

Garrett said...

A. Mikovic,
I agree with you 100%, and I think I have recently found a unified and geometrically natural quantization approach which I am in the process of writing up as a paper.

Thank you. Perhaps the article you are recalling was describing the link between spin and statistics (anti-commutation).

That can work the other way around too.

Garrett said...

randall morris,
There is an enormous landscape of theoretical approaches to these structures. This is why it is helpful for each of us to have our own guiding aesthetics. The approach of going from tensors to spinors, and vice-versa, can be fruitful, but it seems to complicate things, as you have well described in the example you laid out. And I am choosing to hope that nature has chosen a simpler, more direct structure.

I don't know; but the first task of any fundamental theory, before predicting something new, is to match what we already know, and we know a lot.

Unknown said...

Terrific, Sabine! I just a moment ago read it all out loud to myself; it takes 11 minutes to emote! It is beautifully constructed: even if one knows nothing of the details (me) and only is generally familiar with the terms and with the concepts, it gives a thrill of exploring a wonderful mental jungle, now veering left, now veering right, and then back again; trying, trying, to find the garden of Eden, somewhere, in that jungle. It is sheer poetry!

Lee McCullochJames said...

If you commit to the full-on geometrization explanation you probably should formulate your CFT according to Ehresmann with Lagrangian as a function on fields (sections of a smooth fiber bundle) and finitely many derivatives of the fields -finite jet bundle. You then hit a load of formalism which I am not clever or patient enough to see it reveal any more than Cartan's original multi vector form exposition.

What physics insights have been revealed by the jet bundle re-interpretation of higher order (co) tangent spaces? I note the contact form can be used to nicely reveal Einsteins vaccuum equations as integrability conditions for Rarita Schwinger. Does the insight that the first variation giving the Euler-Lagrange equations of motion, while second variation deliver linearised solutions come from an appreciation of such jet bundle hierarchies?

Anyway, how useful is the jet program if it does not work for QFTs?

aleazk said...

To be honest, this pretty much sounds like being "lost in math" to me... as if physics were utterly secondary, we instead look at the math and magically guess from it the ultimate theory about nature based on some aesthetic criteria. This epoch of theoretical physics (and this type of naturalness, in particular) seems very similar to Plato or Kepler, both of them were fascinated by the beauty of the platonic solids and their place in geometry and thought that this was an indicator of something deep in nature, the former postulated that they corresponded to the four classical elements (earth, air, water, and fire), and the latter postulated that they corresponded to the orbits of the planets. They were utterly wrong, of course. Einstein's way of thinking had nothing to do with this, and he arrived at his geometrical theory of gravity not by mythical arguments, but through down to earth physical argumentations (the equivalence principle and the story we all know). He went astray when he started to sacralize geometry in his quest for a unified theory.

Garrett said...


Thanks for the kind words. For the gauge groups, U(1), SU(2), SU(3), these appear in physics as real Lie groups, not their complex versions. They do act on spinors in complex representations, but, for example, there are 8 gluons, not 16.

In what I have done so far, the spin connection does get torsion, sourced by fermions.

For the Coleman-Mandula theorem, there is no spacetime and thus no S-matrix until AFTER symmetry breaking, when gravitational so(3,1) and gauge fields separate, at which point the restrictions of the theorem are satisfied. (It was surprising to me ten years ago when so many people got hung up on this. But there's a lot of inertia in particle physics.)

I have spent a lot of time going back and forth and thinking about what the correct action is. I eventually settled on ∫F*F, for everything -- possibly erring on the side of simplicity.

Rovelli's thermal time hypothesis is certainly intriguing. I don't yet fully understand QFT geometrically, but I'm actively working on a new approach on that. It's very... different. If it turns out to be correct, it's going to be revolutionary. And simple.

I agree that there's something funny about the mismatch between the Einstein-Hilbert action and the Yang-Mills action. But you really can get the former (plus a bit) from the latter. Maybe see my Lie Group Cosmology paper on how.

Your final question... I don't currently think time had a beginning. But my current opinion is that yes, I seem to be experiencing the unitary evolution of a vector in infinite-dimensional Hilbert space.

Thanks for the interesting topics.

Garrett said...


First off, thanks.

1) Yes, it starts out perfectly symmetric. Then this symmetry breaks, with some states getting vevs, making it lopsided.

2) Yes.

3) I am not violently opposed to preons... But there seem to be a lot of clues that we are approaching a complete set of truly fundamental particles, that aren't made of anything else but themselves. This is an exciting time in theoretical physics! Maybe we only have a couple decades, max, before the machines figure it out, and probably don't, or can't, tell us.

4) Oh, Lie groups are amazingly pretty and have a lot of intricate structure, even being made of elementary parts. Just go look at the "Hopf fibration" to see what I mean. And that's the simplest nontrivial one.

5) Well, the exceptional groups have often been considered the most beautiful Lie groups. And I thought E8 was the most beautiful Lie group, until I stopped being such a mathematical pansy and dove into infinite-dimensional Lie groups...

6) Maybe, as a surfer and paraglider and fan of fluids and differential geometry, I'm biased, but I do think reality is fundamentally continuous. Specific solutions within this continuous reality though, especially quantum solutions, are discrete.

Chris Sonnack said...

Very interesting post; thanks to both Sabine and Garrett Lisi!

Any thoughts on how this might be similar to, or different from, Cole Furey's work with octonions? In particular, how "geometric" is that approach? (I'm not at all qualified to judge!)

Nick W said...

Read "Lost in Math", loved it. From that perspective, here is a non-mathematical critique:

As scientists observe smaller things, a pattern emerges that the things get easier to describe, and there is less variation possible. For example: Description and Variation of Human >> D&V of cells >> molecules >> atoms >> subatomic particle zoo >> something new.

If I were to try and extrapolate this pattern, I would guess that the "something new" has less variations and takes less information to describe than the parts that make up the atoms.

This system does not seem less complicated than quantum physics. I used to have fun graphing multidimensional equations too. This shape is pretty awesome, like a super duper complicated periodic table of elements. If humans can come up with this, I am sure we can figure out a theory which can predict galaxy rotation curves based on their distance and brightness, and why we have yet to detect "dark matter"!

distler said...

"Real, pseudoreal, and complex representations aren't the issue here."

They are fundamentally the issue here.

You have spent more than a decade attempting to start with fermions in a real representation of the gauge group, wave a magic wand, and end up with fermions in a complex representation.

Despite both the wand and the incantation changing multiple times over the years, you have never succeeded. That's because you've failed to understand the nature of what you are attempting to do.

"using a spin(5), not a spin(6) "

SU(4) is Spin(6), not Spin(5), but this is a minor point of reading comprehension, when the major point continues to elude you.

Garrett said...

Lee McCullochJames,

I have studied the jet bundle approach, both for QFT and for Hamiltonian/symplectic geometry, and my feeling on it is pretty similar to that on category theory: I can see the power there, and in its way it is certainly correct, but it seems overkill for physics that can be described with more pedestrian structures. And, as you said, it hasn't provided new insights. The Ehresmann connection formalism though... I have fully embraced, and I think it finds its ultimate application in Cartan geometry and its generalizations.

Garrett said...

Yes, I have been "Lost in Math" for quite a while now. I think Sabine had me make this guest post because I am a prime example of what she was writing about -- except for being someone doing it on their own, so my mistakes can't be blamed on herd mentality. I am guilty of everything you are describing. And yet, I still hope to succeed.

Sabine Hossenfelder said...

Tbh I can't recall offering Garrett to write a guest post. But it sounds like the thing I would have done to apologize for not going into any detail explaining what he means by geometric naturalness in my book, so he probably remembers this correctly.

Seth Thatcher said...

Sabine, Garrett is a true gem. Thanks for letting him post. I found the chapter on him in your book to be the most interesting. Though you both disagree, you go about it in such an agreeable fashion.

Sabine Hossenfelder said...

^^ I don't think "agreeable" is quite the right word ;)

Garrett said...

Chris Sonnack,
Geometric naturalness is agnostic on octonions. They are, largely, orthogonal religions. However, the particular structures I've been using are all about octonions. The structure of the exceptional groups comes directly from octonionic multiplication. And when one extends, and uses so(8) triality to confine mirrors, that can all be done using octonions. It is in the exceptional Lie groups that the algebra of the octonions is realized as natural geometry.

Garrett said...

Your ability to persist in being both condescending and wrong simultaneously is truly impressive.

Yehonatan Knoll said...


Geometry vs. general-covariance (GC) is not just a matter of perspective. For example, in the GC approach, the `metric' tensor has no a priori metrical role (in the physical metrical sense). By `relieving' the metric tensor from its metrical duty, one can do away with the contrived notions of dark-matter, dark-energy and inflation. Moreover, the conceptual difficulties with quantum gravity likewise disappear - space and time no longer have any meaning other than that related to the readings of physical devices (clocks etc.)

Igor Mol said...
This comment has been removed by the author.
ben6993 said...

Garret said, in reply to ben6993:

"4) Oh, Lie groups are amazingly pretty and have a lot of intricate structure, even being made of elementary parts. Just go look at the "Hopf fibration" to see what I mean. And that's the simplest nontrivial one."

Hi, Thank you for replying. I did look (naively) at Hopf fibrations and wrote elsewhere about them (non-mathematically) in the context of preons. I find them to be a likely pattern of the structure/motion of RGB branes within the preons.

IMO there is an important symmetry broken in going from preons (symmetry present) to Standard Model elementary particles (symmetry broken). And that is a direct relationship between electric charge and colour charge. That relationship cannot be seen without investigating preons. It also means that a colour brane [e.g. red] is really a multicolour brane with any number and combination of colour and anticolour branes present, but giving a net red total. So I see string-like objects, but very complicated ones, more like twisted ropes than strings, as present within Standard Model particles.

And for usefulness of beauty as a guide, it seems to me that only the bland simplistic symmetry can be a guide, but it is a useless guide. Lobsided broken symmetry can arise in so many ways that it is hard to see it as a guide. For example we can have N-legged creatures, all beautiful. My wife even likes spiders.

Which would be more beautiful/useful, the broken symmetry in the Standard Model which separates electric charge from colour charge or my hypothetical direct relationship between electric and colour charge at preon level which implies an unbroken (bland?) symmetry at that level?

Austin Fearnley

JimV said...

"Maybe, as a surfer and paraglider and fan of fluids and differential geometry, I'm biased, but I do think reality is fundamentally continuous."

And yet, the water and air and all known fluids are not continuous, but made of discrete molecules, without the mathematical paradoxes (e.g., Cantor's Hotel, Zeno's) that are dispelled by discreteness. Calculus in general and differential geometry are wonderful approximations of the finely-discrete structures we see in nature; that's what I appreciate them for.

Great post, thanks very much for it. You are very fortunate to able to be able to do such work. It's almost enough to restore some of my lost faith in humanity.

Yehonatan Knoll said...
This comment has been removed by the author.
Yehonatan Knoll said...

Dear Igor Mol,

GR is a classical theory. So suppose we had a classical theory-of-everything, allowing us to write explicitly the energy-momentum (e-m) tensor sourcing Einstein's equation. You can easily convince yourself that, in this case, the result of ANY measurement (whether of time, length, luminosity...) must - by definition - be computed from the e-m tensor alone. Now, it's true that, in freely falling `labs', the invariant interval would also be nearly proportional to local time/length measurements, but this is just a corollary of the mathematical structure of GR and, at any rate, you would still be missing the proportionality constant (the importance of this point to cosmology cannot be over emphasized).

One may argue that the above picture is based on the false premise that our world is fundamentally classical. I think I have good arguments to the contrary and, at any rate, I have no idea what it would mean for our world to be `fundamentally quantum'...

Incidentally, Einstein later regretted the metrical significance he attributed to coordinates in his work on SR, but could not do much about it; He was unable to write a consistent e-m tensor representing matter, referring to the r.h.s. of his equation as "made of cheap wood".

Garrett said...

I'm somewhat familiar with braid models for preons and particle physics. (But for me, this is where my preference for geometric naturalness diminished my enthusiasm.) At one point Sundance Bilson-Thompson and I worked through the construction of standard model particles using braid numbering, and saw that the numbering corresponded to weights of SU(4), and thus the preon braid model was a way of transcoding the Pati-Salam GUT into braids, complete with the correct emergence of electric charge. I consider this pretty cool, but, fundamentally, my own preference takes me back to Lie groups.

Garrett said...

I know I am lucky. Sorry about my... indiscretion.

Garrett said...

Yehonatan Knoll and Igor Mol,

I think a fundamentally quantum description that is naturally geometric must ultimately exist. It is a very ambitious thing to shoot for though, and few would dare. I think 't Hooft has recently tried something along these lines, but not with a lot of success.

But, on the subject, I agree with Igor on chronometry being fundamental in GR. And, linking this with the quantum, particle physics provides quantum clocks via decay times, such as for the muon.

We need to resolve the dichotomy between marble and wood, and have quantum marble for both sides.

Unknown said...

Hi Garrett,

Have you read the latest publication by Meissner and Nicolai?

"Standard Model Fermions and Infinite-Dimensional R-Symmetries", Krysztof A. Meissner, Hermann Nicolai, 2018

With regards to recent deveolopments in N=8 supergravity, are you able to provide your perspective on potential parallels/similarities and strengths/weaknesses of the above work compared to E8 models?

Thank you for your time and insights.

Garrett said...

Yes, I saw this paper when it came out a few months ago. It is a case of convergent evolution. Building upon SUSY, and strings, and supergravity, and M-theory... the hep community got to E10. From my side, I built on differential geometry, and Clifford algebra, and Lie algebra, and E8... and also got to E10 and higher, weirder infinite-dimensional Lie groups. And now that they and I have arrived... To understand this object, these authors are relying on their toolkit they used to get there, which I don't care for, and they are finding it mostly puzzling but interesting. And I am using the tools I used to get here, and finding it pretty fantastic. So... we live in interesting times, and will see how it goes.

Plato Hagel said...

Hi Garrett,

So are your saying that the E8 structure is a quasi-crystal structure that has yet to be identified??

Lawrence Crowell said...

Are there references to E8+++ and what this means?

ben6993 said...

Garret said, in reply to ben6993:
"I'm somewhat familiar with braid models for preons and particle physics. (But for me, this is where my preference for geometric naturalness diminished my enthusiasm.) ... [...edited...] ... fundamentally, my own preference takes me back to Lie groups."

I, myself, do not like braiding. I do not see that the ordering of one braid with respect to another as always important. To me what is important [for a particle property] is the end product not the braiding [which is only the path to the end].

I have a rubic cube analogy for braiding versus group theory if you care to hear it. Take a 3x3 Rubic cube in its bland state of six pure colour faces. Rotate it various ways to give a cube with asymmetric colours of faces. These could represent six particles. What is important IMO for the properties of a particle is the counts of cells of each colour on a face. Group theory IMO is akin to having an algorithm for the rotations to take the cube from from bland to asymmetric. Braid theory is equivalent to having a braid attached to each cell and keeping track of the braids during the rotations. IMO I can get to the same end point in more than one way. Going down blind alleys and returning. There could be a minimum braid I suppose, and a minimum-step algorithm for rotations. Both ways of tracking the path are informative but for me the essential feature of the particle properties is the end product or lob-sided face and not the path to get there.

Further, consider a bland red face as a brane, and each of the (nine) red cells to contain [as with a fractal] all the information in that brane. That is why the path is extra information, as all the information in the branes is IMO stored fully within every particle. This is also why I asked in a previous post how important, and where it should be stored, is the information that the artist puts into making a picture.

Finally, if the nature of the red brane/dimension is very important, will knowing the braiding or group theory throw light on what the brane is? Will knowing a knitting pattern help us to find out what wool is?

Garrett said...

Plato Hagel,

Lawrence Crowell,
Working on the paper now. You can also research E11 and see more about it and what others have done with it. My approach is quite different though.

Your analogy is more than an analogy. In Lie algebras, you can use raising and lowering operators (keeping track of your rubik turnings) to specify Lie algebra elements.

Enrico said...


“If the universe has been expanding eternally, that leaves an infinite amount of time for particle creation as it expands, leading to and past the era of the big bang we see evidence for.”

Do you suppose the conservation of mass-energy does not hold in your infinite universe? So it has different laws of physics like the multiverse? Expansion leading to the big bang? So the past era is in the singularity. GR breaks down at the singularity. Does your theory apply to the singularity?

The Rindler frame is accelerating but if you measure relative velocity with respect to another frame with equal acceleration, they would appear to be at rest. You can change the coordinate system to make an accelerating frame appear to be inertial. For analogy, a logarithmic function is non-linear but if you change the coordinates to logarithmic scale, you get a straight graph. This is math not physics.

ben6993 said...

Garret said, in reply to ben6993:
"Your analogy is more than an analogy. In Lie algebras, you can use raising and lowering operators (keeping track of your rubik turnings) to specify Lie algebra elements."

I am very grateful to you for replying. A final point, if I may. I suggested particles as the faces rather than the vertices of the rubic cube/group structure so as to allow sub-division of particles (into the nine cells on a face). By choosing particles as vertices of the rubic cube, or of the group structure as in your model, is that pre-judging the particle to be indivisible? Or would your model allow further sub-division of vertices e.g. by adding more dimensions?

Austin Fearnley

Yehonatan Knoll said...

Garrett said: "We need to resolve the dichotomy between marble and wood, and have quantum marble for both sides."

I have absolutely no idea what physical picture you have in mind. Do you? I mean, after another decade of heroic mathematical struggle, when you finally put your hands on that quantum marble, will you be able to say anything more than predict yet another resonance, or some dent in a scattering cross section? (Note that, in this regard, you are in good company with two generations of high-energy physicists).

My solution to Einstein's (classical) marble=wood problem is simply to write a consistent classical energy-momentum (e-m) tensor. Quantum mechanics emerges simply as a statistical description of that e-m tensor (I have shown this in the case of flat spacetime but the generalization requires no conceptual leap). Indeed, the unitarity of QM highlights its statistical designation.

mls said...

Garret, ben6993

You might find the paper "The (16,6,2) Designs" by Assmus and Salwach to be of modest interest. There are three non-isomorphic designs with those parameters. One of those designs has a subgroup that fixes a block and corresponds with the symmetry group of the cube. Another of those designs also has a relationship with cube faces. In this case, the subgroup fixing a block has a six-point orbit (not a block) that is identified with cube faces.

There is a paper by Kibler in the bibliography which details designs with these parameters for 12 of the 14 groups of order 16. The group of Pauli matrices is one such group. Since it is not the dihedral group, it has one of these designs associated with it, although it may not be isomorphic with the designs mentioned above.

The quaternions are represented in the groups with the second design mentioned above. The other design is associated with the group Z_2 x Z_8. One cannot relate this directly to quaternions. However, the near-fields wich generate the miniquaternion geometries are derived from a Galois field whose multiplication is a cyclic group of order 8.

Not physics. Just some interesting finite geometry relating to cube faces if you are not aware of it.

Louis Tagliaferro said...

If you work hard enough, and are smart enough, you will find science that supports your beliefs. Science is about seeing with complete objectivity (easier said than done for everyone), and not seeing what you think is there. I remember you talking fondly about Lie groups many years ago, it doesn't surprise me you've found reasons to think your on the right track.

Humany objectivity is an oxymoron, it's the same for all of us, me included. Recognizing how much we are unknowingly influenced by are beliefs, self-interest, ego, etc. helps a lot in actually being more objective. That trait in researchers would help the pace of scientific advancement significantly.

David Schroeder said...

Garrett, I watched the wonderfully done TED talk in 2008, this morning, on your E8 theory on my sister's computer, where unfortunately I can't respond. I'm on my notebook computer and just tried to access that talk, but it was no longer available, or possibly was the wrong one. Anyway, in it you mention two types of gravitational charge, and I was curious what was meant by that. Towards the end of the talk you personally speak with a man coming from the right side of the stage. That short conversational exchange was enlightening as I sort of understood what constitutes a particle in the E8 model. I plan to look at more of your of your TED, and other youtube talks, especially more recent ones to better understand your theory. Also looked at your 2010 Scientific American article on E8, which was excellent, when I was back home in New Hampshire. I'm going to try to find it again on my sister's computer, as it was an ideal introduction of the concepts behind E8 for laypeople.

Garrett said...

The classical vacuum I see in Lie Group Cosmology is de Sitter spacetime, which expands forever but doesn't have a singularity. The big bang is just what it looks like back when you can't see earlier. And, correct me if I'm wrong, but I don't think mass-energy conservation holds in an expanding spacetime, because it's not time symmetric.

I wasn't talking about rubik cubes specifically. But about the relationship between raising and lowering operations and twisting moves. And then also about the relation to braids and particle charges and preons. (This has been a wide ranging discussion!) Personally, I think our known elementary particles really are fundamental. But that's an opinion mostly on aesthetics.

Yehonatan Knoll,
Yes, if I can successfully predict a new particle resonance using a unified model I will be pretty damn happy. Hmm, I think 't Hooft is trying to do something similar, getting QM behavior from a classical model. And I certainly wish you luck with your effort!

Thanks -- the mathematics of symmetry is endlessly fascinating.

Louis Tagliaferro,
I am pretty aware of my biases. And I try to balance them and behave reasonably while also supporting the optimism and enthusiasm necessary for making progress in an area that is absurdly difficult and frustrating. If I am wrong about Lie groups being foundational, at least I will only have wasted my one intellectual life tilting at windmills, and perhaps a little bit of others' time. But I'm not wrong. ;)

David Schroeder,
The two types of gravitational charge are, essentially, spin and helicity. (+-½ and L/R). After the TED talk, what I SHOULD have told Chris Andersen, but hadn't figured out at the time, was that our universe was actually INSIDE the E8 Lie group, not the other way around. Thanks for the kind words.

Reimond said...


Just a short question: Does Wigner´s little group play any role in your reasoning?
I ask since Igor a while ago said: “… in a way that generalizes the unitary transformations that realizes the Lorentz boots …”
Just started to read your “Lie Group Cosmology” – I first had to catch up.

Lawrence Crowell said...

So this means E8+++ = E11, or is similar. I am not an expert at these, but E9 has an additional su(2) Dynkin node, which as I recall means an extension of E8 with a ladder of states assigned to each of the weights of E8. E11 has a more complex form, I would suppose an A3 Lie valued extension.

Are there forms of the Jordan 3x3 matrix?

As pointed out this is a bit in opposition to Bee’s thesis on lost in math.

Louis Tagliaferro said...

Garrett, It's refreshing to read you think about the influence bias may have on your work, I laughed at your quip, "But I'm not wrong. ;)"

I would advise you that I believe no matter how much effort you put into being objective it is impossible for humans to be completely objective. Also, the deeper you are embedded in an idea the harder it becomes to extract yourself from it regardless if it is right or wrong.

-Best of luck.

Enrico said...

A = My theory does not have a big bang singularity.
B = My theory proves big bang singularity does not exist.
B does not follow from A. I suppose you mean A

Time asymmetry is an artefact of how we slice spacetime. Photons have their own internal clock (frequency = f) but we measure time with Earth reference frame clock. When a photon redshift due to spacetime expansion, its wavelength (y) increases. The inverse of frequency is period (t), a measure of time, and this also increases by the same proportion such that:
t = 1/f
y/t = y/(1/f) = f y = c = speed of light

No matter what the redshift is, the product of length (y) and time (t) is always c. The constancy of speed of light proves spacetime expansion is symmetric. If you find a redshifted photon that’s faster or slower than light, that will prove time asymmetry.

Sabine Hossenfelder said...


Your comment is wrong on so many levels I don't even know where to start. To begin with, photons don't have "their own internal clock". Also, guess what, light always moves at the speed of light. Having said this, I have no time and no patience to correct all those mistakes and since I don't want to contribute to the spread of misinformation, I will not approve further comments on the topic.


Could you please omit submitting your personal theories of everything to this thread? This is not the place to discuss them.

Anonymous coward said...

Fun read. Have you mentored any students? Or since maybe you've been out of the academic mainstream, have you thought about what would be your rough equivalent of a student?

Garrett said...

I have mostly concentrated on unification of off-shell symmetries, including spacetime, before reduction to subgroups via symmetry breaking and further dynamical reduction to solutions of the EOM -- but Wigner's Little Group as it applies to such solutions are certainly on my mind.

Lawrence Crowell,
Yes, this is all correct. Infinite dimensional, generalized Lie algebras have, so far, been wonderful math to be lost in.

Louis Tagliaferro,
Agree completely. In order to be motivated enough to make progress, one needs to be invested in the ideas; but, at the same time, be aware of this for the purpose of making rational assessments and decisions.

Anonymous coward,
I am not an accredited university, but I have had students visit, often during summer months. We tend to have a steady stream of visitors coming through the Pacific Science Institute, who interact with others to varying degrees, so it's a good and productive atmosphere. Also lots of time for running around and enjoying the island. And if there's interest I'm happy to talk about the specific math I'm lost in, give introductory talks, or guide students in projects. It has been more casual than academia but, IMO, more fun and productive.

Lawrence Crowell said...

@ Garrett: I have not studies these extended E8 groups. So before I should do this I might want to ask a few questions. The E8+ group or E9 is an extension of the E8 with the addition of a ● in the Dynkin diagram. This might then be compared to extending the E8 with an A1 ~ SU(2). This is a form of Kac-Moody algebra, as is my understanding. I am not exactly sure how that is done.

The standard Kac-Moody algebra is form from the Virasoro algebra of operators L_n with

[L_n, L_m] = {m-n}L_{m+n} + 1/12(n^3 – n)δ_{mn}.

The Kac-Moody operators a^μ_n form these with

L_n = sum_m a^μ_{n-m}a^ν_m g_{μν}

I am a bit unclear whether with E8+ and beyond there is some connection to the Virasoro algebra.

Now here is a sort of idea I might have about this, which may or may not be wrong. I can think of the SU(2) ~ SO(3) added to form E8+ as describing angular momentum states, such as seen in atomic physics. Then for a high Rydberg atom this symmetry describes a large ℓ quantum systems with a large number of azimuthal quantum number m. In the limit ℓ → ∞ this approximates a harmonic oscillator system. In this way it might appear that am E8 = E8+/SU(2) with e8+ = e8⊕su(2) so the roots and weights of e8 are extended into this angular momentum system that in the infinite limit is a set of harmonic oscillator states. I would then assume if this is correct that E8++ and E8+++ are extended by the more complex roots of A_2 ~ SU(4) and A_3 ~ SU(6).

ben6993 said...

Garret said, in reply to ben6993:
"I wasn't talking about rubik cubes specifically. But about the relationship between raising and lowering operations and twisting moves. And then also about the relation to braids and particle charges and preons. (This has been a wide ranging discussion!) Personally, I think our known elementary particles really are fundamental. But that's an opinion mostly on aesthetics."

Thank you again for replying.
Two more points if I may. One about static v dynamic and one about energy in your model.
In the recent FQXi essay contest, there was some support for static, crystalline models being somehow more aesthetic than others. The pictures of your model look, at face appearance, like beautiful static crystals. But if your structure has raising and lowering operators as elements of the group structure is that a dynamic element by proxy? That is, one (presumably) needs to apply those operations as well as describe the structure of those operations. And a follow on point is whether anything at all is dynamic in the model? Especially, rotation never really happens in the laboratory, but only in an abstract consideration of exchange of properties of particles?

Even if an SM elementary particle is indivisible, its qualities are multiple not singular eg charge, spin and weak isospin. So in a way I can think of abstract rotations of these particle properties as one moves around the group structure.

The raising and lowering operators change states of particles which I understand can be either changes in particle fundamental properties [as in rotations around the model structure] /or/ of energy levels.

My 'energy' question is can you take a pair of electrons and annihilate them completely by using repeated lowering operations? [and in so doing create a photon pair]. But one needs to remove the electron multiple properties as well as its energy. Are there separate operators for raising/lowering energy levels and for rotations around the structure, because energy is completely different to other particle properties? I have very little understanding about energy but assume it is completely different from structural rotations.

Austin Fearnley

Garrett said...

Lawrence Crowell,

Infinite-dimensional Lie groups are fascinating! Yes, your description of E9 is correct. The root system is an infinite "stack" of E8 roots. In general, one generates the Cartan matrix from the Dynkin diagram, and then generates the root system (or a truncation of it) from the Cartan matrix, and also can build the Lie algebra generators and structure using the Chevalley-Serre relations. The "presentation" of the Lie algebra can also be related to other presentations -- a different set of generators for the same Lie algebra. Another path to E8+ from E8 is as an "extension" of the E8 Lie algebra. The Virasoro algebra is the central extension of the Witt algebra. Your interpretation in terms of angular momentum states seems more-or-less correct, but I don't know if it will end up being fruitful. I do think physicists should play in this area more!


My models are based on natural geometric structures (as I defined above), not on discrete structures such as crystalline lattices. However, the root diagrams used to describe Lie groups do resemble crystalline lattices, and some end up confusing the map for the territory I think.

For dynamics, one needs to allow the Lie group to "deform," as I defined that in a recent paper, Lie Group Cosmology. Deformations in time, or not, relate to energy and its conservation. For particle annihilation, the quantum numbers (roots) of a particle and anti-particle cancel out, but the energy, momentum, and angular momentum are converted to those of a pair of photons.

Lawrence Crowell said...


Since you appear to say I have somewhat of the right idea this then leads to my next question. The infinite stack of E8 roots in my sense might then possibly be seen as due to an infinite sequence of prolongations, such as Beckland transformations. Is this possible?

Unknown said...

@ Garrett, I have a little erratum. It is B{\a"}cklund transformations and not Beckland.

Lee McCullochJames said...

The original modified BF theory for Gravity was Plebanski's complex chiral Lagrangian. The Weyl curvature was the Lagrange multiplier that ensured the complex bivector came from the co-frame in the field equations. Problem was that "off shell" extraneous conditions needed to be imposed on the complex bivector to ensure reality of metric.

Also in being complex using both Hodge and Lie-Hodge duals acting as Idempotents on the PFB gauge algebra, presupposed metric structure on the Bundle base space. As such, the sense that geometric structures arise naturally out of the Variational principle were rather lost. The field equations not being the only equations in town, Ward-like identities need to hone too much gauge over explanation.

A Clifford valued BF suffers does it not from both such undesirables?

WRL said...

Odd that you said that, bee. That is very much the same thing Murray Gell-Mann said when I suggested that they explore partitioned strings (so as to model preon algebra at root level) _before_ building the Super-conducting Supercollider - in the 80s!
Yet even today there is no appropriate venue. I've even had to split my talks into two general areas in order to present a comprehensive theory, partly because the Physics Ordering schema doesn't include such a category.

There is NO Theory of Everything anyway... take a look at the variety of FQXi papers attempting to respond to the question "What is Fundamental?". It seems no-one can properly define the limits of "Everything" fundamentally required in such a theory anyway.
No wonder so many theorists are lost: "Without a Vision, the people perish"??

WRL said...

"I wasn't talking about rubik cubes specifically. But about the relationship between raising and lowering operations and twisting moves. And then also about the relation to braids and particle charges and preons. (This has been a wide ranging discussion!) Personally, I think our known elementary particles really are fundamental. But that's an opinion mostly on aesthetics."
I was talking about the Restricted Rubik's cube algebra when I gave the very well-attended talk "QCD via Rubik's Cube" at the APS Centennial in 1999. I'd love to have a collaborator, but it seems NO venue (except the back rooms of academia?) is appropriate to establish collaboration.

Note I use the equivalence {R,D,B,L,U,F}Std = {R,Y,B,G,P,O}Eur = {R,G,B,C,M,Y}QCD where R,B are always shoulder-to-shoulder, and thusly colorize the cube according to the original 8-fold-way charge-color symmetry diagram.
Second publication at: - all the figures, including the assignment of Red, Green, Blue to TVV, VTV, VVT ordering - which Bilson-Thompson was awarded a Ph.D. for 'originally' using... WERE COPYRIGHTED in 1990 and published many time since.