Garrett Lisi bid to join Std. Model w/ gravity

[marcus]

Garrett, occasional visitor here, posted this today:

http://arxiv.org/abs/gr-qc/0511120
Clifford bundle formulation of BF gravity generalized to the standard model
A. Garrett Lisi
24 pages
"The structure and dynamics of the standard model and gravity are described by a Clifford valued connection and its curvature."

congratulations.

 

[marcus]

several of us here took an interest in the Freidel/Starodubtsev paper
which IIRC we had a thread on, and which also John Baez was excited about

this Lisi paper builds on that, or more generally on the BF model

I don't know how successfully---need time to think and hope to hear other people's comments----but it makes sense as something to try. the BF model looked like a very promising way to treat gravity, so makes sense to see how the Std Mdl might be welded onto it.

for convenience, here is the Freidel/Staro paper
http://arxiv.org/abs/hep-th/0501191
Quantum gravity in terms of topological observables

here are some of Freidel other papers to see what he is doing by way of joining matter to gravity
http://arxiv.org/find/grp_physics/1/au:+Freidel/0/1/0/all/0/

Lisi's handle on the Standard Model comes from some 1999 work by Greg Trayling
http://arxiv.org/hep-th/9912231
A geometric approach to the standard model
Here is Garrett's conclusions paragraph giving a descriptive overview of what he has accomplished:
"This paper has progressed in small steps to construct a complete picture of gravity and the standard model from the bottom up using basic elements with as few mathematical abstractions as possible. It began and ended with the description of a Clifford algebra as a graded Lie algebra, which became the fiber over a four dimensional base manifold. The connection and curvature of this bundle, along with an appropriately restricted BF action, provided a complete description of General Relativity in terms of Lie algebra valued differential forms, without use of a metric. This “trick” is equivalent to the MacDowell-Mansouri method of getting GR from an so(5) valued connection. Hamiltonian dynamics were discussed, providing a possible connecting point with the canonical approach to quantum gravity. Further tools and mathematical elements were described just before they were needed. The matrix representation of Clifford algebras was developed, as well as how spinor fields fit in with these representations. The relevant BRST method produced spinor fields with gauge operators acting on the left and right. These pieces all came together, forming a complete picture of gravity and the standard model as a single BRST extended connection. If this final picture seems very simple, it has succeeded. As a coherent picture, this work does have weaknesses. Everything takes place purely on the level of “classical” fields – but with an eye towards their use in a QFT via the methods of quantum gravity, which must be applied in a truly complete model. The BRST approach to deriving fermions from gauge symmetries, although a straightforward application of standard techniques, may be hard to swallow. If this method is unpalatable, it is perfectly acceptable to begin instead with the picture of a fundamental fermionic field as a Clifford element with gauge fields acting from the left and right in an appropriate action. The model conjectured at the very end, based on the related u(4) GUT, is yet untested and should be treated with great skepticism until further investigated. In a somewhat ironic twist, after arguing in the beginning for the more natural description of the MM bivector so(5) model in terms of mixed grade Cl1,3 vectors and bivectors, this conjectured model is composed purely of bivector gauge fields. Although the model stands on its own as a straightforward Cl8 fiber bundle construction over four dimensional base, there are many other compatible geometric descriptions. One alternative is to interpret ⇁ ̃A as the connection for a Cartan geometry with Lie group G – with a Lie subgroup, H, formed by the generators of elements other than ⇁e, and the spacetime “base” formed by G/H. Another particularly appealing interpretation is the Kaluza-Klein construction, with four compact dimensions implied by the Higgs vector, φ = −φ ψΓ ψ, and a corresponding translation of the components of ⇁ ̃A into parts of a vielbein including this higher dimensional space. The model may also be extended to encompass more traditional unification schemes, such as using a ten dimensional Clifford algebra in a so(10) GUT. All of these geometric ideas should be developed further in the context of the model described here, as they may provide valuable insights. In conclusion, and in defense of its existence, this work has concentrated on producing a clear and coherent unified picture rather than introducing novel ideas in particular areas. The answer to the question of what here is really “new” is: “as little as possible.” Rather, several standard and non-standard pieces have been brought together to form a unified whole describing the conventional standard model and gravity as simply as possible."

 

[marcus]

just as additional context
here is Not Even Wrong blog from earlier this year where John Baez comments and mentions Garrett in connection with the work by Greg Trayling that this paper draws on

http://www.math.columbia.edu/~woit/wordpress/?p=173

 

[garrett]

Hey Marcus, thanks for mentioning my work. Everything went down just as you said -- I was playing around with the MacDowell-Mansouri idea recently revived by Freidel et al, and I was amazed to find all the other pieces coming together and fitting more or less perfectly. Thus the paper.

 

[marcus]

Hey thanks for visiting us. We should make a list of recent ventures at putting gravity together with Std Mdl. maybe they have underground tunnels connecting them

I can think of yours and of the preon-spinnetwork approach. Can you suggest others?

 

[garrett]

Well, the best attempts, IMO, are all old. The Kaluza-Klein idea is probably the best overall. But it has problems with chiral spinors, and more problems with "towers" of states. And the KK idea had a rebirth in the 80's with supergravity, and again with string theory. But I think the attempts to get the standard model out of string theory are pretty bad -- way too many unjustified assumptions.

Trayling's model, and mine the addition of gravity to his, is really just the Kaluza-Klein idea wearing Clifford robes.

In the "final" theory, if there is one, I think everything is going to have to come together at once. It's a lot to tackle, which is why most contemporary physicists don't want to touch it. There's also the taboo subject of trying to "derive" quantum mechanics from something. This whole field is also chalk full with crackpots, so it's damn near academic suicide to enter it. It also doesn't help that you probably can't, or shouldn't, be publishing a paper a month when you're working on the big questions.

 

[selfAdjoint]

We should make a list of recent ventures at putting gravity together with Std Mdl

http://www.arxiv.org/abs/gr-qc/0511089" is not quite that, but it is an interesting venture in combining matter into GR; sources of matter arise as changes of differentiable structure!

Differential Structures - the Geometrization of Quantum Mechanics

Torsten Asselmeyer-Maluga, Helge Rose'

Abstract:
The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics. In Einstein's theory gravitation is expressed by geometry of space-time, and the solutions of the field equation are invariant w.r.t. a certain equivalence class of reference frames. This class can be characterized by the differential structure of space-time. We will show that matter is the transition between reference frames that belong to different differential structures, that the set of transitions of the differential structure is given by a Temperley-Lieb algebra which is extensible to a $C^{*}$-algebra comprising the field operator algebra of quantum mechanics and that the state space of quantum mechanics is the linear space of the differential structures. Furthermore we are able to explain the appearance of the complex numbers in quantum theory. The strong relation to Loop Quantum Gravity is discussed in conclusion.

 

[garrett]

Hey SA, that does look interesting. I'll give it a read. Here's the arxiv link for it:
http://arxiv.org/abs/gr-qc/0511089
I also liked this paper by Ashtekar and Schilling:
Geometrical Formulation of Quantum Mechanics
http://arxiv.org/abs/gr-qc/9706069
Though it was more conservative.
I do think quantum field theory is going to have to get a geometric description eventually. It's hard though. And there are a lot of wild and crazy ideas out there too. One of the wackier approaches is to try to do Bohmian quantum mechanics, with an objective reality and pilot waves, applied to field theory -- usually capitalizing on some scalar field.

 

[marcus]

http://arxiv.org/abs/gr-qc/0511124
Spin Gauge Theory of Gravity in Clifford Space
Matej Pavsic
9 pages, talk presented at the QG05 conference, 12-16 September 2005, Cala Gonone, Italy
"A theory in which 16-dimensional curved Clifford space (C-space) provides a realization of Kaluza-Klein theory is investigated. No extra dimensions of spacetime are needed: "extra dimensions" are in C-space. We explore the spin gauge theory in C-space and show that the generalized spin connection contains the usual 4-dimensional gravity and Yang-Mills fields of the U(1)xSU(2)xSU(3) gauge group. The representation space for the latter group is provided by 16-component generalized spinors composed of four usual 4-component spinors, defined geometrically as the members of four independent minimal left ideals of Clifford algebra."

garrett, is this someone you might want to get in touch with?

EDIT since time hasnt run out and I can still edit, I will reply to your next post here!
Garrett, glad to here you have ALREADY been in touch with Pavsic, whose work although different has some similar directions. the way it looks to me as (fairly naive) outsider is that it is good to have potential allies who can help each other get a hearing for their work. I was interested by this QG '05 conference held just this past September in SARDINIA. several people whose names I recognized were there, and also Pavsic whom I didnt know of until now.

 

[garrett]

Hey Marcus,
Yes, I've read Matej Pavsic's papers and corresponded with him. And our work shares many common lines. But I came to the conclusion after playing with it a bit that the standard model just doesn't naturally fit in Cl_{1,3}. This isn't to say I don't think his work is good, and he never says it does fit -- he just says "it might." But, most of the stuff I've laid out in my paper are independent of the dimension and signature of what Clifford algebra you want to work in. It's a fairly compact introduction to model building with Clifford valued forms. I just picked Trayling's model as the one that looks by far the best in the end, especially when joined with gravity.

 

[Mike2]

http://www.arxiv.org/abs/gr-qc/0511089" is not quite that, but it is an interesting venture in combining matter into GR; sources of matter arise as changes of differentiable structure!
Differential Structures - the Geometrization of Quantum Mechanics
Torsten Asselmeyer-Maluga, Helge Rose'
Abstract:
The usual quantization of a classical space-time field does not touch the non-geometrical character of quantum mechanics. We believe that the deep problems of unification of general relativity and quantum mechanics are rooted in this poor understanding of the geometrical character of quantum mechanics.

How is he defining a "topology change" as opposed to the a change in geometry? Thanks.

 

[selfAdjoint]

How is he defining a "topology change" as opposed to the a change in geometry? Thanks.

A geometry change in this paper is a curvature change. The authors show that unlike a diffeomorphism itself, a change of differential structure introduces an extra term in the connection, and hence a change in curvature of the manifold. They identify this with sources of matter.

A topologocical change would be something like the appearance of a new hole or handle on the manifold. The authors show that in spite of using radical topological surgery, cutting out tori and sewing them back in twisted or knotted, they do NOT introduce topology change.

 

[garrett]

SA,
The differential structures idea is very interesting. Unfortunately, I only have a poor physicist's understanding of the topic. I know a differential structure is an equivalence class of atlases, and an atlas is a collection of maps from a manifold to R^4 and transition functions. But I don't get how to enumerate the various differential structures, or why there are an infinite number of them for four dimensional manifolds, and one or a finite number for many others. Although I see it is directly related to the number of smooth structures over n dimensional spheres. But I don't understand that well either. Do you know of any introductory description of this stuff, preferably online?

This would integrate very well with the paper I wrote if it works like they seem to be saying it does. My paper lays everything out, the standard model and gravity, as a single connection over a four dimensional base manifold. The differential structures idea could give a natural explanation for the quantization of such a connection.

 

[Helge Rosé]

SA,
The differential structures idea is very interesting. Unfortunately, I only have a poor physicist's understanding of the topic. I know a differential structure is an equivalence class of atlases, and an atlas is a collection of maps from a manifold to R^4 and transition functions.

Hi garrett, thanks for your interest in our paper.
In the physical point of view an atlas is set of reference frames which are needed to describe measurements at different space-time regions. With Einstein all reference frames are physically equal if the charts can be transformed by diffeomorphisms - the charts are compatible.
In 1,2,3 dimensions all charts (reference frames) are compatible. In 4 dimensions you can find one set S1 of charts which are compatible with all other charts in this set. But you can also find a further set S2, were all charts compatible in S2 but with no chart in S1. S1 and S2 are two different representants of two atlases. S1 and S2 belong to two different differential structures.

But I don't get how to enumerate the various differential structures, or why there are an infinite number of them for four dimensional manifolds, and one or a finite number for many others. Although I see it is directly related to the number of smooth structures over n dimensional spheres. But I don't understand that well either. Do you know of any introductory description of this stuff, preferably online?

In mathematics the differential structures are called exotic smooth strctures. It can be shown that for a manifold (e.g. Dim=7 like Milnor) atlases exist which are not compatible (transfromable by diffeomorphisms). It is also known that for a compact 4-manifold the number of non-compatible altlases are countable infinite. But the structure of the set of differential structures was unknown. We have shown that the set of the changes of a differential structure is a Temperley-Lieb algebra and the set of differential structures is a Hilbert-space (Dim H = inf). This is a mathematical fact like: "the number of integers is countable infinite". Torsten is writing a book about https://www.amazon.com/gp/product/981024195X/?tag=pfamazon01-20
At the mean time you may looking for the mathematical papers about "Exotic Structures", but this is hard to cover.

This would integrate very well with the paper I wrote if it works like they seem to be saying it does. My paper lays everything out, the standard model and gravity, as a single connection over a four dimensional base manifold. The differential structures idea could give a natural explanation for the quantization of such a connection.

This is very interesting, please sent me a copy: rose@first.fhg.de

 

[Mike2]

Torsten is writing a book about https://www.amazon.com/gp/product/981024195X/?tag=pfamazon01-20

Do you have more info on this book, such as a table of contents?

Secondly, what are they talking about when they mention singularities? Is this supposed to be a topological entity, or is this a scalar function defined on a manifold whose value goes to infinity at various points on the manifold? Thanks.

 

[Helge Rosé]

Do you have more info on this book, such as a table of contents?
Secondly, what are they talking about when they mention singularities? Is this supposed to be a topological entity, or is this a scalar function defined on a manifold whose value goes to infinity at various points on the manifold? Thanks.

For the table ask Torsten (torsten@first.fhg.de).
In our paper the term singular has the usual mathematical meaning in the context of maps:
The map f: M -> N is http://mathworld.wolfram.com/SingularPoint.html" at points were its differential
df: T_x M -> T_f(x) N
is vanishing. We do not mean poles etc. and it is not a topological entity (M,N are homeomorph). The map f is smooth but because of the singular points f^-1 does not exists (Like: a curve with a self intersection). f is a map between 4-manifolds, the set of singular points is 3-dimensional. We mean this singular 3d-set of map f when we use the term singularity - it is a 3d-region in the 4d space-time, like a particle. This singular set determines a 1-form -> the connection change between M, N -> differential structure change. Thus a 3d-set determines the transition of DS of a 4-MF.

 

[marcus]

For the table ask Torsten (torsten@first.fhg.de).
In our paper the term singular has the usual mathematical meaning in the context of maps:
The map f: M -> N is http://mathworld.wolfram.com/SingularPoint.html" at points were its differential
df: T_x M -> T_f(x) N
is vanishing. We do not mean poles etc. and it is not a topological entity (M,N are homeomorph). The map f is smooth but because of the singular points f^-1 does not exists (Like: a curve with a self intersection). f is a map between 4-manifolds, the set of singular points is 3-dimensional. We mean this singular 3d-set of map f when we use the term singularity - it is a 3d-region in the 4d space-time, like a particle. This singular set determines a 1-form -> the connection change between M, N -> differential structure change. Thus a 3d-set determines the transition of DS of a 4-MF.

thanks for the extra explanation. it is appreciated. sometimes just saying the same thing over (repeating what you already said in the paper with maybe a few more words) can be very helpful

it looks like we need to refer to Brans papers
I will try to find arxiv numbers

http://arxiv.org/abs/gr-qc/9604048
Exotic Smoothness on Spacetime

http://arxiv.org/abs/gr-qc/9405010
Exotic Smoothness and Physics*

http://arxiv.org/abs/gr-qc/9404003
Localized Exotic Smoothness*

http://arxiv.org/abs/gr-qc/9212003
Exotic Differentiable Structures and General Relativity*

*these are cited in the recent paper

we also need to be able to check out some Sladowski papers
I am getting sleepy, and also I am worried now that this approach could be crazy. It looks too good. maybe I had better go to bed and try to understand some more tomorrow. thanks for your help so far Helge

 

[Mike2]

For the table ask Torsten (torsten@first.fhg.de).
In our paper the term singular has the usual mathematical meaning in the context of maps:
The map f: M -> N is http://mathworld.wolfram.com/SingularPoint.html" at points were its differential
df: T_x M -> T_f(x) N
is vanishing. We do not mean poles etc. and it is not a topological entity (M,N are homeomorph). The map f is smooth but because of the singular points f^-1 does not exists (Like: a curve with a self intersection). f is a map between 4-manifolds, the set of singular points is 3-dimensional. We mean this singular 3d-set of map f when we use the term singularity - it is a 3d-region in the 4d space-time, like a particle. This singular set determines a 1-form -> the connection change between M, N -> differential structure change. Thus a 3d-set determines the transition of DS of a 4-MF.

I appreciate the help, of course.

So as I understand it, you've made a "connection" between 3D particles and 4D geometry, right? So that if we have means of determining the QFT of the vacuum, then we can connect this via differential structures to the background metric, is this right? It sounds like this effort might be getting close to figuring out the quantum geometry of the background (read quantum gravity) from QFT. Or is this saying that the same ZPE of QFT exists for any amount of curved space (the operator algebra is the same in all curvatures)? Thanks

 

[Helge Rosé]

I appreciate the help, of course.
So as I understand it, you've made a "connection" between 3D particles and 4D geometry, right?

Yes, the singular 3d-support of the 1-form determines the change of connection of the 4-MF and thus the differential structure (DS). But note the DS is not the only structure which influences the geometry of the 4-MF . You have also the freedom to choose the metric and by this you can modify the geometry but the DS is fixed.

So that if we have means of determining the QFT of the vacuum, then we can connect this via differential structures to the background metric, is this right?

if you mean the metric is fixed by the DS - no! I do not know how strong metric and DS are independent, but there is no one-to-one connection.
And note in our model is no background metric: Also for a fixed DS the metric is dynamical - determined by Einsteins eq.

It sounds like this effort might be getting close to figuring out the quantum geometry of the background (read quantum gravity) from QFT. Or is this saying that the same ZPE of QFT exists for any amount of curved space (the operator algebra is the same in all curvatures)? Thanks

As I can see, this can not answered yet. In any case the quantum geomety will be dynamical and we need to better understand the dynamics, i.e. the field eq. of DS - but we don't know.

 

[marcus]

Helge, thanks for the explanation you have given so far!
Do you have any suggestions of papers to read as preparation for your paper with Torsten?

So far I just see the citations to papers by Brans and by Sladowski (including one that Torsten co-authored with Brans)
are these the best to read or are there also others you might suggest?

I know this is a very hard question to answer since we here are totally miscellaneous----some of us are self-taught, others are, or used to be, engineers and mathematicians. So it is a total mystery what information would help us to understand. But go ahead and take a chance. If you have anything to recommend, please do.

this whole subject is incredibly interesting.

 

[garrett]

Hi Helge, I like your idea. But I'm still just learning about this and I want to ask you some questions. I think it's probably best to move this discussion over to its own thread though, so I'll post there.

 

[Helge Rosé]

Thats right! sorry. Some times the threads live their own live

 

[Mike2]

Yes, the singular 3d-support of the 1-form determines the change of connection of the 4-MF and thus the differential structure (DS). But note the DS is not the only structure which influences the geometry of the 4-MF . You have also the freedom to choose the metric and by this you can modify the geometry but the DS is fixed.

if you mean the metric is fixed by the DS - no! I do not know how strong metric and DS are independent, but there is no one-to-one connection.
And note in our model is no background metric: Also for a fixed DS the metric is dynamical - determined by Einsteins eq.

Isn't there a requirement that the number of particles/singularites must effect the metric on the 4-MF so that the more mass there is, the more the spacetime metric curves? Isn't this what you have? Or have you only connected the QFT algebra with a 4-manifold?

 

[Spin_Network]

Isn't there a requirement that the number of particles/singularites must effect the metric on the 4-MF so that the more mass there is, the more the spacetime metric curves? Isn't this what you have? Or have you only connected the QFT algebra with a 4-manifold?

Mike2, we discussed this some time ago in the M-Theory thread?

Lets ask this again:When is a singularity, NOT a singularity?

Look here:http://arxiv.org/abs/gr-qc/0511131

The answer is in the question

Singularities can be imbedded as far as observers are concerned:http://arxiv.org/abs/gr-qc/0511135

again there are some older papers that are pretty neat, but here is another interesting recent article:http://arxiv.org/abs/gr-qc/0511139

The thread paper of Helge Rose et-al I have not had time to absorb,yet!

M-theory thread:https://www.physicsforums.com/showthread.php?t=77653

actually here is a major paper that answers a lot of the relevant questions:http://arxiv.org/abs/gr-qc/0508045

 

[Spin_Network]

The first paper linked above shows the topological structure/detail, of interest.

 

[Helge Rosé]

Isn't there a requirement that the number of particles/singularites must effect the metric on the 4-MF so that the more mass there is, the more the spacetime metric curves? Isn't this what you have? Or have you only connected the QFT algebra with a 4-manifold?

Mike2 it is better to discuss this https://www.physicsforums.com/showpost.php?p=839078&postcount=29"

 

[CarlB]

Lisi's handle on the Standard Model comes from some 1999 work by Greg Trayling
http://arxiv.org/hep-th/9912231
A geometric approach to the standard model

I just found the above paper. It's very similar to what I'm doing. There are a lot of differences. Baylis and Trayling are assuming 4 hidden spatial dimensions while I'm assuming just one, and that one related to proper time (i.e. the Euclidean relativity). I put the electroweak symmetry breaking into the relationship between the tangent vectors and the Clifford algebra. And I assume a level or two of preons.

Carl

 

[garrett]

Hi Carl,
It's not so hard to get su(2) out of most any model. The trick is naturally getting su(3). And then it's even trickier to get a chiral su(2).

 

[CarlB]

Hi Carl, It's not so hard to get su(2) out of most any model.

Yes. Given X and Y any two distinct nontrivial canonical basis elements, if one has a complexified Clifford algebra one can assume that they square to unity. From there, if it is either the case that XY = YX or XY = -YX. The latter case produces an SU(2) with {X,Y,XY}. The former case produces a U(1) x U(1). Choosing canonical basis elements at random, you basically have a 50% chance of getting an SU(2).

The trick is naturally getting su(3).

That's where preons come in. If you assume that the elementary fermions are each composed of three subparticles, the SU(3) comes natural.

[edit]By the way, I just realized you're Lisa Garrett who's written papers in this area, so you'll understand what I mean when I explain how it is that I came to find out that SU(3) is hard to get (cleanly) in this method.

I wanted to get the charges of the quarks out of Hestenes' geometric algebra. So I looked for solutions to the eigenvector equation:

$$Q|\chi> = 1/3|\chi>$$

where Q and \chi are from the Clifford algebra. Eventually I spent about two weeks looking for idempotents $$\iota$$ with $$<\iota>_0 = 1/3$$. I used all kinds of methods to solve that damned equation but I kept coming up with idempotents that had a scalar element of the form n/2^m and never anything but.

So I basically discovered the spectral decomposition theorem for Clifford algebra idempotents the hard way. (And found it in Lounesto's book, if I recall.) It was only later that I realized that if I assumed preons, the 1/3 factors would come naturally, and that the way that the weak isospin and weak hypercharge quantum numbers work out the preon model falls right into your lap.[/edit]

And then it's even trickier to get a chiral su(2).

I don't see this at all, please explain.

Carl

 

[garrett]

I don't see this at all, please explain.

Chirality was the problem that more or less killed Kaluza-Klein theory in the 80's. Nature couples left-chiral fermions to the electro-weak gauge fields differently than it couples right-chiral fermions. It is hard to come up with a good geometric reason for why.

 

[CarlB]

Nature couples left-chiral fermions to the electro-weak gauge fields differently than it couples right-chiral fermions. It is hard to come up with a good geometric reason for why.

Yes, I did have trouble with this. The solution takes a bit of explaining.

First, you need to assume that the elementary fermions are the left-chiral and right-chiral ones. There is the problem that these are massless and you will later have to figure out a method of putting the masses in, but work is ongoing on that.

That done, the problem then becomes one of symmetry breaking. That is, the primitive idempotents of a Clifford algebra tend to be too danged symmetric.

I think the best way to break the symmetry is to modify the way that the tangent vectors (i.e. $$\partial_x, \partial_t$$, etc.) are connected up with the canonical basis vectors (i.e. $$\gamma_x, \gamma_y$$, etc.)

Use the standard Dirac matrices as an example. What we are going to do is to loosen the definition of the Dirac operator:

$$\nabla \psi = (\partial_x\gamma_x + \partial_y\gamma_y + ...) \psi$$

Instead of using the canonical basis VECTORS, $$\gamma_\mu$$, we will instead use arbitrary canonical basis ELEMENTS $$\Gamma_x, \Gamma_y, \Gamma_z, \Gamma_t$$ provided only that the usual Dirac relations are satisfied. That is,

$$\Gamma_x^2 = \Gamma_y^2 = \Gamma_z^2 = -\Gamma_t^2 = 1$$

and the $$\Gamma_\mu$$ anticommute.

Now in the usual spinor model of quantum mechanics, this modification leaves the Dirac equation unchanged. That is, $$\Gamma_\mu$$ makes just as good a set of generators of a Clifford algebra as $$\gamma_\mu$$ does. So the change doesn't alter any of the usual physics.

But in Trayling's model of the elementary particles, the change from $$\gamma_\mu$$ to $$\Gamma_\mu$$ amounts to a remapping of the elementary particles.

The remapping by the above naturally preserves all addition and multiplication in the Clifford algebra so all the remapped particles preserve their quantum numbers with respect to the remapped operators. But the remapping does not preserve the squared magnitude of the Clifford algebra. Since the squared magnitudes are what we associate with probabilities, this means that while the particle propagation is unchanged (same Dirac equation), the particle interactions are altered. This is exactly the kind of symmetry breaking you need.

I've been assuming just one small cyclic hidden dimension and a complexified Clifford algebra. This gives 8 primitive idempotents. Before breaking symmetry, these 8 idempotents all have squared magnitude 1/8. The symmetry breaking can only raise the squared magnitude of a primitive idempotent. But on the other hand, the primitive idempotents still sum to one so the squared magnitude of the sum of all the idempotents is still going to be 1.

Another way of saying this is to say that the primitive idempotents become unstable. Typically, four of them become slightly unstable (tending to form up in pairs), while the other four become extremely unstable (also tending to form up in pairs). A typical set of squared magnitudes is:

$$8|\iota_{---}|^2 = 1.31$$
$$8|\iota_{--+}|^2 = 1.31$$
$$8|\iota_{-+-}|^2 = 1.31$$
$$8|\iota_{-++}|^2 = 1.31$$
$$8|\iota_{---} + \iota_{--+}|^2 = 2.000$$
$$8|\iota_{-+-} + \iota_{-++}|^2 = 2.000$$
$$8|\iota_{+--} + \iota_{+-+}|^2 = 2.000$$
$$8|\iota_{++-} + \iota_{+++}|^2 = 2.000$$
$$8|\iota_{+--}|^2 = 231.7$$
$$8|\iota_{+-+}|^2 = 231.7$$
$$8|\iota_{++-}|^2 = 231.7$$
$$8|\iota_{+++}|^2 = 231.7$$

Without the symmetry breaking, all 8 of the primitive idempotents in the above would have squared magnitude 1/8, and the four pairs would have squared magnitude 1/4.

For example, you've now reduced your set of 8 primitive idempotents as potential elementary particles to 4 idempotents (the ones that have squared magnitude of 2.000). These idempotents can interact by making small changes, but the probabilities are very different between the two pairs. Hence, the symmetry is broken in probability / amplitude.

To match up with the weak hypercharge, weak isospin structure of the standard model, so far I've had to assume that the elementary particles correspond to certain combinations of four primitive idempotents each.

I presented a poster at the PANIC05 conference in Santa Fe last month on this subject that goes into more detail on this:
http://brannenworks.com/PPANIC05.pdf

The detailed calculations for the symmetry breaking are on pages 14-16 of this unfinished paper:
http://brannenworks.com/long_PANIC_Not_Complete_.pdf

An older paper with a less complete derivation of the particle symmetry breaking is here:
http://brannenworks.com/PHENO2005.pdf

Carl

 

[garrett]

Hey Carl,

I looked over the stuff you linked to, and you do have a lot of good ideas there. We seem to share a lot of thinking on the importance of Clifford algebras in constructing pretty particle multiplets. (I also first got into the subject by reading Hestenes.) And I agree the quantum mechanical density matrix gets short shrift in most QFT treatments. Its importance only becomes clear when tackling quantum optics. It's a good structure to play with when trying to understand QM geometrically. Also, it makes sense to try a preon model if you're determined to use a low dimensional Clifford algebra, since you need a bigger one if you're going to fit in the known fermions as fundamental fields. On the other hand, your promotion of the speed of light constant, c, to a multi-component variable is a terrible idea. ;)

But, as you point out, your model is incomplete. What I've managed to pull off is to put together a complete model using many of these same elements. And I've kept as close as possible to the conventional formalism, using only the most elementary differential geometry, and only things that already have a fundamental place in the standard formalism of QFT and GR. The key was to see a Clifford algebra as a big Lie algebra. I had that idea in my head when I saw Freidel's papers on restricted BF gravity -- and bang, it all came together at once. When I combined the Clifford version of BF gravity with Trayling's model, it was as if they were made for each other. When the Higgs multiplet popped out, multiplying the vierbein, I nearly fell out of my chair! So I had to write it all up.

It's a nice unification. The fields of the standard model and gravity are in a single Clifford valued connection,
A = phi e + omega + W + B + G + nu + e + u + d
and the dynamics are in its curvature.

But it doesn't work perfectly. I don't see where generations come from, and the masses don't come out. I'm clearly biased, but it feels to me as if it's in the stage string theory was in during the late 80's. Back when people were saying "Wow, look how well this little model works -- I'll bet if we play with this we'll be getting particle masses in no time!" Of course, they were wrong then... and I could well be wrong now. But I've only written up the most conservative approach possible that fits the standard model. In the paper, I've described in detail a minimal set of building blocks, and I do wish others like you would go play with them, as I'm only one guy and I'm very slow. There are hundreds of different ideas to try: preon models in smaller Clifford algebras, su(5) GUT's in bigger ones, Kaluza-Klein versions, Cartan geometry, sigma models, modified actions, etc. And those paths don't even yet include quantization. The motivation for my model was recent work in Loop Quantum Gravity -- in which the pre-quantized field variables are the chiral (self-dual) spin connection and vierbein. And since my model only involves the same connection variables in a bigger algebra, the same path to a spin-network formulation should be accessible, as well as the recent work in topological perturbation based on BF theory. A good quantum gravity theory is going to reproduce standard quantum field theory when this bigger connection is plugged in.

There are just so many things to do it boggles my mind. But please do take what I've done and use the pieces you like. They really are all the most conventional bits of differential geometry. And there's a decent chance you or me or someone will put a scheme together that connects a whole bunch of these 31 dots.

 

[CarlB]

On the other hand, your promotion of the speed of light constant, c, to a multi-component variable is a terrible idea. ;)

Well, as a non academic who presents papers at APS meetings, it is traditional to start with an "Einstein Was Wrong!" foil. The multivariable speed of light is a natural generalization for the variable speed of light that has been used recently to explain the inflation problem in cosmology. (See "VSL" in arxiv.org). But what got me into it was that I really wanted to see the elementary particles fall out as the natural small oscillations in a media.

If I gave you a glass jar filled with a liquid that treated right and left polarized light differently you would likely conclude that it was filled with a liquid that was not right-left symmetric. The same thing should apply to spacetime. Of course a jar is easy because we are outside the jar.

Hestenes has an interesting argument in favor of a flat metric (which therefore requires a variable speed of light). If I recall, it amounts to requiring, from a sort of mathematical point of view, that the tangent vectors be embedded in the manifold, and therefore that the manifold be flat. He didn't say it quite like that.

What I've managed to pull off is to put together a complete model using many of these same elements.

I look forward to reading your papers. I just copied them off the web and will read them at home probably tomorrow night. But I'm really not a gravitation guy and I bet that a lot of that is just going to go over my head.

I don't see where generations come from, and the masses don't come out.

I'm going to assume that mass is essentially an interaction where a left-handed particle turns itself into a right-handed one and at the same time reverses its direction. There's a way of writing this out in Feynman diagrams that can be resummed to get the massive propagator from the massless propagators. The bad news is that it is highly Lorentz asymmetric. But my guess is that this is a clue on how nature does it. By the way, Feynman hints at this technique in a footnote in his popular book "QED: The strange theory of matter and light".

Anyway, this method of giving masses to the particles implies that you have to allow the left and right handed particles to change back and forth to one another. Since you have three preons, this implies that you have some (hidden from the standard model) freedom in how to choose their relative phases. By the rules of Feynman diagrams, the preon phase is preserved, or multiplied by a constant, in this change, and the amplitude gives the branching ratio. This implies that you can write a matrix that gives the branching ratios and phase changes between the three preons.

In my preon theory, these three preons are what is giving you SU(3) in your quarks, so they must be treated cyclically. If you assume geometric 3-vectors that distinguish those three preons, you get that the branching ratios have to be in the ratio of 4 to 1 to 1. (You can show this by computing the usual (1+cos)/2 factors for three vectors equally distributed on a cone.) This implies a 3x3 matrix with entries as shown in equation (6) of this paper:
http://arxiv.org/PS_cache/hep-ph/pdf/0505/0505220.pdf

As it turns out, the above matrix, which uses the Cabibbo angle for some of the phase changes, postdicts the electron, muon and tau masses accurately to experimental error. You get the three generations and you get their masses.

I'm clearly biased, but it feels to me as if it's in the stage string theory was in during the late 80's.

Yes, I bet there's a lot of string theory people who wish it were back in the late '80s again.

I will certainly try to use what you've written.

By the way, you can't possibly understand how much of a relief it is to find someone who can actually understand what I'm working on.

Carl

 

[torsten]

Hey Carl,
I looked over the stuff you linked to, and you do have a lot of good ideas there. We seem to share a lot of thinking on the importance of Clifford algebras in constructing pretty particle multiplets. (I also first got into the subject by reading Hestenes.) And I agree the quantum mechanical density matrix gets short shrift in most QFT treatments. Its importance only becomes clear when tackling quantum optics. It's a good structure to play with when trying to understand QM geometrically. Also, it makes sense to try a preon model if you're determined to use a low dimensional Clifford algebra, since you need a bigger one if you're going to fit in the known fermions as fundamental fields. On the other hand, your promotion of the speed of light constant, c, to a multi-component variable is a terrible idea. ;)
But, as you point out, your model is incomplete. What I've managed to pull off is to put together a complete model using many of these same elements. And I've kept as close as possible to the conventional formalism, using only the most elementary differential geometry, and only things that already have a fundamental place in the standard formalism of QFT and GR. The key was to see a Clifford algebra as a big Lie algebra. I had that idea in my head when I saw Freidel's papers on restricted BF gravity -- and bang, it all came together at once. When I combined the Clifford version of BF gravity with Trayling's model, it was as if they were made for each other. When the Higgs multiplet popped out, multiplying the vierbein, I nearly fell out of my chair! So I had to write it all up.
It's a nice unification. The fields of the standard model and gravity are in a single Clifford valued connection,
A = phi e + omega + W + B + G + nu + e + u + d
and the dynamics are in its curvature.
But it doesn't work perfectly. I don't see where generations come from, and the masses don't come out. I'm clearly biased, but it feels to me as if it's in the stage string theory was in during the late 80's. Back when people were saying "Wow, look how well this little model works -- I'll bet if we play with this we'll be getting particle masses in no time!" Of course, they were wrong then... and I could well be wrong now. But I've only written up the most conservative approach possible that fits the standard model. In the paper, I've described in detail a minimal set of building blocks, and I do wish others like you would go play with them, as I'm only one guy and I'm very slow. There are hundreds of different ideas to try: preon models in smaller Clifford algebras, su(5) GUT's in bigger ones, Kaluza-Klein versions, Cartan geometry, sigma models, modified actions, etc. And those paths don't even yet include quantization. The motivation for my model was recent work in Loop Quantum Gravity -- in which the pre-quantized field variables are the chiral (self-dual) spin connection and vierbein. And since my model only involves the same connection variables in a bigger algebra, the same path to a spin-network formulation should be accessible, as well as the recent work in topological perturbation based on BF theory. A good quantum gravity theory is going to reproduce standard quantum field theory when this bigger connection is plugged in.
There are just so many things to do it boggles my mind. But please do take what I've done and use the pieces you like. They really are all the most conventional bits of differential geometry. And there's a decent chance you or me or someone will put a scheme together that connects a whole bunch of these 31 dots.

Hi Garrett,
did you know the work of Tolksdorf (math-ph/0503059). He is doing similar things: to derive the standard model including Gravity by using the Clifford bundles and its Dirac operator.

Torsten

 

[marcus]

Garrett, occasional visitor here, posted this today:
http://arxiv.org/abs/gr-qc/0511120
Clifford bundle formulation of BF gravity generalized to the standard model
A. Garrett Lisi
24 pages
"The structure and dynamics of the standard model and gravity are described by a Clifford valued connection and its curvature."
congratulations.

this was the original post on this thread
which started to be about Garrett's paper but got into an intense discussion of Torsten-Helge

selfAdjoint just noticed gr-qc/0511120 in a recent comment at Woit's blog and reminded us of it. I'd be happy if we could get more of this paper explained.