Finally, Garrett's model with 3 generations

[garrett]

A "deformation" of a Lie group is meant in the same way that our lumpy spacetime is a deformation of de Sitter spacetime.

 

[E8(8)]

Can you give a precise sensible mathematical definition of what you mean by "deformation of E_8"?

 

[Jimster41]

A "deformation" of a Lie group is meant in the same way that our lumpy spacetime is a deformation of de Sitter spacetime.

I'm just a layperson trying to get a sense of what what the view is here. Your paper has sent me into a fugue of wikipedioia.
is there any intuitive way to understand (enough to appreciate the view) the difference between the number of dimensions and the number of roots, of an algebraic structure like the e8?

Your statement above seems relevant to my confusion here, I have been trying to tie the E8 and representation theory to the experience of looking at objects in the world. I get the idea that the algebra describes the constrained behavior of a set of objects - potentially those objects are the basis of the SM, the fundamental structures of matter in space time.

If that is in some sense getting the meaning of it, Do you have any thoughts on what process is causing deformation? What is causing the evolution of species, as it were, such that we don't just have the raw basis, but instead... The world with all its richness, in time. Is there a sense in which the algebra must be "run"?

 

[garrett]

Yes, E8(8), a deforming Lie group is described by a variation of a Lie group manifold's Maurer-Cartan connection away from zero curvature.

 

[E8(8)]

Can you give a a precise definition of what you mean by "deformation of E_8"? What you just said does not seem to have any sense. Can you at least point out to a mathematical reference where such concept is defined and explored?

 

[garrett]

I'm just a layperson trying to get a sense of what what the view is here. Your paper has sent me into a fugue of wikipedioia.
is there any intuitive way to understand (enough to appreciate the view) the difference between the number of dimensions and the number of roots, of an algebraic structure like the e8?

Hello Jimster41,
Sure, for an N-dimensional Lie algebra (corresponding to an N-dimensional Lie group), there can be a maximum of R mutually linearly independent commuting generators, which span what is called a Cartan subalgebra. With respect to these mutually commuting generators, acting via the Lie bracket on the rest of the Lie algebra, there will be N-R eigenvectors, called root vectors, with eigenvalues called roots.

Your statement above seems relevant to my confusion here, I have been trying to tie the E8 and representation theory to the experience of looking at objects in the world. I get the idea that the algebra describes the constrained behavior of a set of objects - potentially those objects are the basis of the SM, the fundamental structures of matter in space time.

Yes, in physics, the roots, and specifically the eigenvalues with respect to the mutually commuting generators, correspond to elementary particle charges.

If that is in some sense getting the meaning of it, Do you have any thoughts on what process is causing deformation? What is causing the evolution of species, as it were, such that we don't just have the raw basis, but instead... The world with all its richness, in time. Is there a sense in which the algebra must be "run"?

This is a harder question. The best answer I know of comes from quantum mechanics, which basically says nature tries everything.

 

[garrett]

Can you give a a precise definition of what you mean by "deformation of E_8"? What you just said does not seem to have any sense. Can you at least point out to a mathematical reference where such concept is defined and explored?

Yes, that is the point of the entirety of the paper. What part is confusing to you? Do you not understand that the Maurer-Cartan form is a Lie algebra valued 1-form, with vanishing curvature, defined on the Lie group manifold? And if you do understand that, is it not sensible to consider variations of this 1-form? If you clarify the problem you're having with this, I may be able to help by providing more information (that's in the paper), but I can't help if your objection is vague.

 

[E8(8)]

My objection is not vague. What do you mean by "variations of the Maurer-Cartan form"? How is that related to deforming the underlying Lie group? Can you give a mathematical reference were such concept is defined and explored?

 

[Jimster41]

Sorry if I this covered in the paper, does the deforming Spin(1,4) Lie group containing the rigid Spin(1,3) sub algebra have to be DeSitter to work, to allow GR to be represented as the Cartan Connection between them (I hope that's not butchering it too much), or can it also/instead be AdS?

 

[garrett]

My objection is not vague. What do you mean by "variations of the Maurer-Cartan form"? How is that related to deforming the underlying Lie group? Can you give a mathematical reference were such concept is defined and explored?

Yes, the main idea and mathematical formulation of Cartan geometry has been explored in many previous works, perhaps most extensively in Sharpe's book on the subject. What is provided in LGC is a generalization and application to physics.

 

[garrett]

Sorry if I this covered in the paper, does the deforming Spin(1,4) Lie group containing the rigid Spin(1,3) sub algebra have to be DeSitter to work, to allow GR to be represented as the Cartan Connection between them (I hope that's not butchering it too much), or can it also/instead be AdS?

The symmetric space resulting from modding Spin(1,4) by Spin(1,3) is de Sitter spacetime -- in this case, no deformations are necessary, and de Sitter spacetime can be considered the "rest state," described by the Maurer-Cartan form on the Lie group. One only needs to consider deformations in order to describe our lumpy spacetime, with matter in it, described by a Cartan connection. If you wanted to describe Anti-de Sitter spacetime, you'd have to start with Spin(2,3).

 

[E8(8)]

Yes, the main idea and mathematical formulation of Cartan geometry has been explored in many previous works, perhaps most extensively in Sharpe's book on the subject. What is provided in LGC is a generalization and application to physics.

I am obviously not asking about the mathematical formulation of Cartan geometry. I know that is a solid theory of mathematics, and I know Sharpe's book. You are obviously evading the question. I will repeat it:

What do you mean by "deforming a Lie group"? What do you mean by "variations of the Maurer-Cartan form"? How is that related to deforming the underlying Lie group? Can you give a mathematical reference were such concept is defined and explored?

Can you AT LEAST give one mathematical reference where this is properly defined , because in your paper it is not? And by this I evidently don't mean a general book a bout Cartan geometry, I mean a reference where the deformations that you use are defined and studied. Can you at least do that? It is not that hard if you know what you are talking about. Because right now it looks like you don't know what "deforming a Lie group" is and that your paper is nonsense.

 

[garrett]

I provided a solid reference, and a succinct description. Perhaps you don't know Sharpe's book, or anything for that matter, as well as you think you do. This is from Sharpe:

A Cartan geometry on $M$ consists of a pair $(P, \omega)$, where $P$ is a principal bundle $H \to P \to M$ and $\omega$, the Cartan connection, is a differential form on $P$. The bundle generalizes the bundle $H \to G \to G/H$ associated to the Klein setting, and the form $\omega$ generalizes the Maurer-Cartan form $\omega_G$ on the Lie group $G$. In fact, the curvature of the Cartan geometry, defined as $\Omega = d \omega + \frac{1}{2} [\omega, \omega]$, is the complete local obstruction to $P$ being a Lie group.

The manifold $P$ may be regarded as some sort of “lumpy Lie group” that is homogeneous in the $H$ direction. Moreover, $\omega$ may be regarded as a “lumpy” version of the Maurer-Cartan form. The Cartan connection, $\omega$, restricts to the Maurer-Cartan form on the fibers and hence satisfies the structural equation in the fiber directions; but when $Ω \ne 0$ we lose the “rigidity” that would otherwise have been provided by the structural equation in the base directions and that would have as a consequence that, locally, $P$ would be a Lie group with $\omega$ its Maurer-Cartan form. Thus, the curvature measures this loss of rigidity.

I prefer the descriptive "deforming" to "lumpy," but the concept is the same. Perhaps you should compose a note to Sharpe to inform him that his work is nonsense.

 

[garrett]

The LGC idea begins with Cartan geometry, as a deformation (or excitation) of a Lie group $G'$, and then generalizes to what can happen when $G'$ is a subgroup of a larger group, $G$. And, yes, it's simple, and, yes, that's all one needs to build a ToE.

 

[E8(8)]

The LGC idea begins with Cartan geometry, as a deformation (or excitation) of a Lie group $G'$, and then generalizes to what can happen when $G'$ is a subgroup of a larger group, $G$. And, yes, it's simple, and, yes, that's all one needs to build a ToE.

You are not generalizing anything. You are just using Cartan geometry and you are not even doing that right. The fact that you are calling "deforming Lie group" a Cartan geometry, which is something very classical and very well known, is simply absurd, ridiculous. The abstract of your paper: "Our universe is a deforming Lie group.", can be thus rewritten as:

"Our universe is a manifold."

Because a Cartan geometry has as underlying space a manifold. So great, Garret, you really have obtained a TOE by modelling our universe using a manifold. I think no one thought of that one before! ;)

 

[strangerep]

The abstract of [Garrett's] paper: "Our universe is a deforming Lie group.", can be thus rewritten as:

"Our universe is a manifold."

Because a Cartan geometry has as underlying space a manifold.

This is an oversimplified criticism, hence unhelpful. Of course a Cartan geometry has an underlying manifold, but it has more structure. One equips the tangent bundle with an (abstract) Ehresmann connection to distinguish horizontal/vertical directions in a coordinate-independent way. Then one specializes the Ehresmann connection to be a Cartan connection with extra, more concrete, properties) to bring the framework closer to physics. The fact that a Cartan connection can be regarded in terms of Lie group deformation theory is not "absurd". (Whether this point of view is in fact useful for real world physics remains to be seen, but let us keep the criticisms on an informed and constructive level.)

 

[strangerep]

For other readers who were wondering what "Lie Group deformation" means, it's essentially a procedure of (continuously) modifying the structure constants of the Lie algebra. For infinitesimal modifications, one may then analyze whether the so-deformed structure constants may be converted back to the originals by linearly mapping the generators among themselves. If this is possible, the Lie group is said to be "Lie-stable", meaning that one does not obtain a fundamentally different group by such deformation, but merely a renaming of generators. An example of a non-stable algebra is the Galilei algebra, whereas the Poincare algebra is stable (and hence more likely to be a more accurate description of physics).

(I'm still reading Garrett's paper -- maybe I'll try to clarify further when I understand it a little better... )

@garrett: I've also seen (elsewhere) the term "rigid" used to mean "Lie-stable" in the above sense. I notice you use the term "rigid" for subgroups. Do you mean "rigid" in the above sense, or with some other meaning? (Apologies in advance if I haven't yet read far enough into your paper to answer this for myself.)
[Edit: I just read the extract from Sharpe on your p18. I see that he uses "rigidity" with a different meaning, i.e., "non-deformed"(?). And curvature expresses the extent to which the Lie subgroups on neighbouring fibres are exactly the same, hence indirectly the obstruction to global (horizontal) integrability.]

[Edit: Question: since the maximal subgroup $H$ on a fibre is the same for all points of the base manifold, is the Ehresmann--Cartan connection therefore essentially equivalent to the "linear mapping of generators among themselves" that I mentioned above? ]

[Edit: Even though "lumpy Lie group" is Sharpe's phrase, I think it's a choice that screams out "misinterpret me!". ]

[Edit: @garrett: Would you please number ALL of your equations? Incomplete equation numbering makes it annoyingly cumbersome to discuss details...]

 

[strangerep]

@garrett: OK, another question: I'm up to p20 and getting a bit confused by the various "curvatures". You say (top of p20) that "the curvature of the de Sitter connection vanishes", which superficially contradicts the fact that curvature of ordinary de Sitter spacetime does not vanish. I'm guessing this is because there's more than one "curvature" involved here? I.e., the curvature $F(x)$ of the Cartan connection over $M$ given by your eq(8.7) which involves both the Riemann curvature $R$ and the torsion $T$?

Is that right? (If so, it wouldn't hurt to be more explicit. Does $F(x)$ have a name? E.g., "Cartan curvature"??)

Edit: next question: on p24 you use the adjective "wavy" to describe $G'/H$. This term is not defined, afaict. I'm guessing you mean that deformations are applied to $G'/H$, but not to $H$ nor to $G/G'$ ?

 

[E8(8)]

This is an oversimplified criticism, hence unhelpful. Of course a Cartan geometry has an underlying manifold, but it has more structure. One equips the tangent bundle with an (abstract) Ehresmann connection to distinguish horizontal/vertical directions in a coordinate-independent way. Then one specializes the Ehresmann connection to be a Cartan connection with extra, more concrete, properties) to bring the framework closer to physics. The fact that a Cartan connection can be regarded in terms of Lie group deformation theory is not "absurd". (Whether this point of view is in fact useful for real world physics remains to be seen, but let us keep the criticisms on an informed and constructive level.)

First of all, it is not a criticism, it is a fact. The fact that Garret refers to a Cartan geometry as a "deforming Lie group" is misleading to the point of being absurd. Writing an abstract saying "The universe is a deforming Lie group" is a joke, plain and simple. It is like if someone writes a paper and instead of "manifold" writes "deforming flat space" or something like that. He should have written an abstract saying something like:

"We consider a classical cosmological model based on a Cartan geometry with group..."

which is what he is really doing (although he is not doing it right because he does not know the mathematics involved). Of course, if he writes that abstract, no one would even open that paper because that is like the oldest idea around, and it is classical. Garret wanted to create some fake fuzz and that is why he wrote that other absurd abstract. I guess that didn't work either: you can fool people once (with his Simple Theory of Everything), but it is much more difficult to do it twice.

I know that a Cartan geometry has more structure than a manifold, in particular it is a fiber bundle, but that does not change the fat that the underlying space is a manifold and that the structure is completely classical. Saying that he has obtained "TOE" just because he is considering a model which is fiber bundle with a connection is again, a hilarious joke. That is one of the oldest ideas of classical geometry and that is the classical description of gauge theories, all pretty well known. There is no "quantum" in that description, everything is classical.

You are wasting your time reading that paper; it is very poorly written, it is inconsistent and for sure, it is not a TOE.

 

[strangerep]

[Garrett] should have written an abstract saying something like:

"We consider a classical cosmological model based on a Cartan geometry with group..."

which is what he is really doing

I've now finished a first pass through the paper. I agree that the above is "what he is really doing". Although there are some vague allusions to something beyond that, they are embryonic at best and therefore strike me as far too speculative.

(although he is not doing it right because he does not know the mathematics involved).

Please give some specific references in his paper where he makes mathematical mistakes.

I know that a Cartan geometry has more structure than a manifold, in particular it is a fiber bundle, but that does not change the fat that the underlying space is a manifold and that the structure is completely classical.

Yes. Although he mentions "superposition" of 3 "spacetime-like" regions inside the large group, I did not see a proper development of that idea, certainly no unitary rep structures, or whatever.

Saying that he has obtained "TOE" just because he is considering a model which is fiber bundle with a connection is again, a hilarious joke.

I agree that it is not a TOE, but perhaps my sense of humor is not as harsh as yours.

You are wasting your time reading that paper;

Well, I do not consider it a waste of my time -- it forced me to clarify some aspects of my own understanding of Cartan geometry.

it is very poorly written,

I did indeed perceive that (especially) the later sections where he describes his theory (after the general framework) were rather rambling. OTOH, I have read other poorly written papers that nevertheless contained a seed of an idea, or a different angle on an old idea, that stimulated my own thoughts. So, again, I don't consider reading Garrett's paper to have been a waste of my time. Nevertheless, YMMV.

it is inconsistent

Again, please give some specific references in his paper where he commits mathematical inconsistencies.

and for sure, it is not a TOE.

I agree that in its present classical form, it is not a TOE.

I was also a bit disturbed early in the paper by how Garrett describes SU(10) GUT as a "success". I would have thought the opposite was true.

 

[garrett]

(Sorry for the delay, it's been a busy few days for me. Also, my responses here are not necessarily in correct temporal order.)

I was also a bit disturbed early in the paper by how Garrett describes SU(10) GUT as a "success". I would have thought the opposite was true.

I was referring to the success of the Spin(10) GUT in several different aspects; specifically, in the appearance of correct Standard Model hypercharges when fermions are assigned to the 16 rep space, the roughly correct weak mixing angle, and in roughly convergent coupling constant strengths. Of course, there are deficiencies, such as the non-observance of proton decay, so the degree to which the Spin(10) GUT can be considered a success is certainly a matter of opinion.

For other readers who were wondering what "Lie Group deformation" means, it's essentially a procedure of (continuously) modifying the structure constants of the Lie algebra.

In general, I think a Lie group deformation can be thought of as, essentially, a modification of a Lie group's Maurer-Cartan form. This is different than a modification of the structure constants.

I've also seen (elsewhere) the term "rigid" used to mean "Lie-stable" in the above sense. I notice you use the term "rigid" for subgroups. Do you mean "rigid" in the above sense, or with some other meaning?

I mean rigid in the same sense as Sharpe, in that the relevant curvature of the connection vanishes, and the geometric structure of the corresponding subgroup or subspace is preserved.

Question: since the maximal subgroup $H$ on a fibre is the same for all points of the base manifold, is the Ehresmann--Cartan connection therefore essentially equivalent to the "linear mapping of generators among themselves" that I mentioned above?

There are a few twists here. First, when thinking about the Ehresmann-Cartan connection, the relevant "base manifold," over which it is described, is the deforming Lie group Manifold. What has happened is that the Maurer-Cartan form over the Lie group manifold, valued in the Lie algebra, has varied in a nontrivial way to become the Ehresmann-Cartan connection, which is still valued in the same Lie algebra but now describes the geometry of a deforming Lie group manifold and not the Lie group manifold. As a map, the Ehresmann-Cartan connection maps vectors on the deforming Lie group manifold to Lie algebra elements.

Would you please number ALL of your equations? Incomplete equation numbering makes it annoyingly cumbersome to discuss details...

Apologies. I'm in the bad habit of only numbering equations I refer to later, which makes it harder for others to refer to the unnumbered equations.

 

[garrett]

@garrett: OK, another question: I'm up to p20 and getting a bit confused by the various "curvatures".

Best way to keep track of them is to think of them with respect to their corresponding connection. (Of which there are many.)

You say (top of p20) that "the curvature of the de Sitter connection vanishes", which superficially contradicts the fact that curvature of ordinary de Sitter spacetime does not vanish. I'm guessing this is because there's more than one "curvature" involved here? I.e., the curvature $F(x)$ of the Cartan connection over $M$ given by your eq(8.7) which involves both the Riemann curvature $R$ and the torsion $T$?

Yes, the curvature of the de Sitter connection includes the torsion, and the Riemann curvature, AND minus the frame squared. Because the torsion vanishes and the Riemann curvature equals the frame squared for de Sitter spacetime, the curvature of the de Sitter connection vanishes while the Riemann curvature does not.

Edit: next question: on p24 you use the adjective "wavy" to describe $G'/H$. This term is not defined, afaict. I'm guessing you mean that deformations are applied to $G'/H$, but not to $H$ nor to $G/G'$ ?

Correct.

 

[marcus]

Garrett, could you summarize in simple terms how your model includes the cosmological constant Lambda?
Someone was asking in another thread:
https://www.physicsforums.com/threads/e8-theory-according-to-the-dark-matter-and-energy.828982/

when I look in the first part of your new paper, on page 15, I see:
==quote==
Physically, a de Sitter universe corresponds to empty space with expansion driven by a cosmological constant, Λ =

3α². As a spacetime of constant curvature, this vacuum solution satisfies Einstein’s equations

of General Relativity, and it approximates our physical universe, with large scale observations

currently indicating Λ ≃ 2.036 × 10⁻³⁵ 1/s² and an expansion parameter of α ≃ 2.605 × 10⁻¹⁸ 1/ s

Of course, the existence of matter disturbs spacetime away from this de Sitter vacuum state.

==endquote==

It seems to me that you have a parameter alpha in your model which corresponds more or less to the longterm asymptotic expansion rate H_∞ that I'm used to.

Then on page 36, in the summary towards the end you again refer to this expansion parameter alpha:
==quote==
...and φ₀⁴ = α/2, a Higgs vev, with expansion parameter α. ...
==endquote==

Maybe it would be possible to respond to the question in that other thread in a satisfactory manner.

 

[marcus]

Actually what I get for alpha using google calculator is:
1 / (17.3 billion years) =
1.8317205 × 10^-18 hertz

So 3α² comes out 10.0656 × 10^-36 1/s²

or around 1.00656 × 10^-35 1/s²

Your figure for Lambda may be off by a factor of 2, on page 15

 

[garrett]

Marcus, you caught me being lazy. I must have been so enraptured with theoretical manipulations that when I hastily looked up the value of the cosmological constant I pulled it from this paper:
http://cds.cern.ch/record/485959/files/0102033.pdf
But that value is very outdated, and now off by a factor of 2. Thanks for catching that. I'll update it in a revision next week.

 

[martinbn]

Can someone give a brief description of the model, assuming that the reader is OK with the maths. Just the general idea.

 

[DaniV]

This video helps me understanding theorticlly without matematics this theory, it's very easy underdtanding the basis of the theory although video is not explaining the whole theory processes and its conclusions

 

[garrett]

Hello martinbn,
The general idea of the model is to begin with a high-dimensional Lie group, which has a natural connection and metric, then embed de Sitter spacetime in this Lie group and allow the Lie group to deform, described by variations of the connection, guided by a Yang-Mills-type action extended to fermions. By choosing a nice Lie group, and allowing deformations that preserve some of the Lie group's structure, one can obtain all of the particle-fields and dynamics for the Standard Model and gravity. The model proposed in this paper is a prequel, laying a geometric foundation for previous work on E8 theory, but can be readily adapted for use with other Lie groups.

 

[Alberto Garcia]

Hi Garrett, and what is the problem to calculate the bare mass of an elementary particle (fermion)?

In the paper you wrote:
'...we must presume that each generation of fermions is only accessible from one of three triality-related regions of E8; only then do each of these regional fermion generations correctly match known fermion properties. Each generation will transform under a different triality-related regional spin connection...'
'...all three generations of chiral fermions transform correctly, with no mirror fermions. Also, because the vacuum Higgs field may be different in each region, each generation of fermions may have a different bare mass...'
'...the triality-related boson generators in the three regions act on the corresponding fermions in agreement with their familiar Standard Model spins and charges...'

 

[garrett]

Well, this is the million dollar question. Somehow, according to this new theory, our spacetime must be a superposition of these three triality-related spacetimes, with fermions having masses related to the parameters of how these spacetimes are embedded and related. This would need to produce the (bare) masses and CKM/PMNS mixings we observe. The superposition currently posited in LGC is too simplistic to produce this -- I think the final description, if there is one within LGC, will be more... complex.

 

[Alberto Garcia]

To complete the theory, undoubtedly, a monumental amount of work has to be done , but maybe, you are in the right track, and your efforts could lead to the beginning of a new science revolution, which are stages required by history of science for the progress of knowledge, as T. Kuhn showed us.

 

[garrett]

Thank you, Alberto, that's very flattering. But it may also be that this new idea is wrong. I think it has a lot going for it, but I'm biased, and one always has to remember that, in science, human wishes and dreams, no matter how wonderful, can never change nature's mind. So if a theory is wrong, no matter how nice and popular it gets, it will keep being wrong.

 

[MTd2]

Garrett, I don't know if this exists, but, a Fourier transform is a relation that sends variables between a collection of variables where each one has a complex value. That is, it is a map that relates 2 2d spaces, where the observable is a 1D quanitty.

Your triality could mean mapping 4 3d spaces where the "meta- observable" is a 2D quantity, a complex plane. So, it's going to functions in 4 quaternions, and descending to a 2D space which is wavefunction of the the triadic object.

Depending on how you define the triality, it can yield a fermion on a boson when taking the "meta observable".