Unifying Gravity, Gauge Interactions & Higgs Bosons: Smolin & Speziale

[marcus]

A. Garrett Lisi, L. Smolin and S. Speziale, “Unification of gravity, gauge interactions and
Higgs bosons in the extended Plebanski formalism,” to appear.

This interesting reference was found in a recently posted preprint by Smolin and Speziale.
(thanks to MTd2 for spotting the Smolin Speziale paper)

 

[marcus]

The Lisi et al paper was reference [9] cited in:

http://arxiv.org/abs/0908.3388
A note on the Plebanski action with cosmological constant and an Immirzi parameter
Lee Smolin, Simone Speziale
14 pages
(Submitted on 24 Aug 2009)
"We study the field equations of the Plebanski action for general relativity when both the cosmological constant and an Immirzi parameter are present. We show that the Lagrange multiplier, which usually gets identified with the Weyl curvature, now acquires a trace part. Some consequences of this for a class of modified gravity theories recently proposed in the literature are briefly discussed."Very interesting there should be interaction between the Immirzi parameter of LQG and the cosmological constant Lambda. I had never seen a mathematical connection arise between these two---or don't recall one anyway. Curious. Here is a brief quote from page 3: "With the preparation from section 2 we are able to untangle the interplay of the Immirzi parameter and cosmological constant. We find a perhaps surprising result, which is that the field equations are only consistent for a few special values of the Immirzi parameter.

Once chosen, we show that deSitter (or AdS) spacetime is a solution to the full equations of motion. This result is essential to recent studies of the extension which includes a unification with Yang-Mills and Higgs fields.

Beyond this, the full structure of the equations of motion will be elucidated in [8], where it is shown that the gravitational sector of the theory is a bimetric theory."

 

[MTd2]

Damn, I haven't seen this thread!

I was watching TV, and just pressed the "post" button when the show was over.

 

[atyy]

Maybe they can get a prediction for Lorentz invariance violation for this - I think bimetric theories can do that?

 

[marcus]

Somehow it reminds me of when I took my driver's test, to get a license.
You get 3 points if you predict Lorentz violation and 5 points if you run over an old lady
at a stop sign.

 

[atyy]

Somehow it reminds me of when I took my driver's test, to get a license.
You get 3 points if you predict Lorentz violation and 5 points if you run over an old lady
at a stop sign.

I think I can live down 3 points ...

 

[marcus]

Seriously I'd be interested to hear whatever you want to say about bimetric theories, or anything related to these articles. You come at these things from a different point of view.

 

[atyy]

Seriously I'd be interested to hear whatever you want to say about bimetric theories, or anything related to these articles. You come at these things from a different point of view.

I just vaguely have in mind Mattingly's http://relativity.livingreviews.org/Articles/lrr-2005-5/ , say Section 2.1.

 

[marcus]

I just vaguely have in mind Mattingly's http://relativity.livingreviews.org/Articles/lrr-2005-5/ , say Section 2.1.

I hadn't looked at that paper earlier. It strikes me as excellent. Maybe he will update it.
Thanks for the reference.

 

[Coin]

At first I was a little surprised to see Lisi working on something that didn't seem to be related to his E8 proposal. But--

I was trying to see what I could deduce about the upcoming Lisi/Smolin/Speziale paper from this one. The first thing that strikes me is that it seems like this paper (as far as I know) could have been written much earlier, since it is unclear to me whether it relies on anything that was not known when say the Plebanski action was first worked on (the paper says the Plebanski action dates to the 70s but was not widely noticed until 1991). It doesn't seem to particularly be an LQG paper. They discuss the importance of the Plebanski action in relation to Ashtekar/loop variables but don't seem to actually use loops or any other LQG machinery (again unless I missed something-- most of the math is over my head). They seem to just be doing straightforward gauge theory and GR type stuff here.

They do use the "Immirzi parameter", which Marcus points out above is an LQG concept. However wikipedia seems to be saying that the Immirzi parameter can be derived in a straightforward fashion just from looking at the gauge theory related to GR, and that it has been adopted by the LQG program because it "turns out" to be proportional to the spectrum of the LQG area operator. Marcus, do you think you can explain a bit more what the Immirzi parameter is and why Smolin is likely to find it useful?

Reading this Smolin+Speziale paper it sort of seems like it exists for no reason other than to be a lemma of sorts for the upcoming paper with Lisi-- they define a variant of the Plebanski action in this paper, which they make use of in the next paper. They say in the conclusion:

The result is used in [6, 9] to support the extended Plebanski action as a possible action for a unification of gravity and Yang-Mills
theory.

[9] is the upcoming paper with Lisi, [6] however is this Smolin paper from 1997, which may be worth taking a look at-- it apparently uses the Plebanski machinery described in this paper, and it is titled "The Plebanski action extended to a unification of gravity and Yang-Mills theory", which looks to be roughly the same subject as the upcoming paper with Lisi. Opening it up one finds a couple of interesting things. First off, it is written in a much clearer and more explanatory style than the fairly dense mathematical paper Smolin+Speziale just released. Second off, it turns out to be an E8 paper!

We study a unification of gravity with Yang-Mills fields based on a simple exten-
sion of the Plebanski action to a Lie group G which contains the local lorentz group.
The Coleman-Mandula theorem is avoided because the dynamics has no global space-
time symmetry. This may be applied to Lisi’s proposal of an E8 unified theory, giving
a fully E8 invariant action. The extended form of the Plebanski action suggests a new
class of spin foam models.

(The paper goes on to discuss a way that, once you weld E8 to LQG, an alternate way of describing fermions within the E8 model arises. As I understood the fermion generations were a major difficulty with the original E8 proposal.)

So do you suppose we can also assume the upcoming Lisi+Smolin+Speziale paper will continue this work of studying E8 using the tools of LQG? And I guess somehow tell us something about the Higgs field in the process...?

 

[MTd2]

[9] is the upcoming paper with Lisi, [6] however is this Smolin paper from 1997,

That paper is from 2007, and it was written after Lisi's E8 paper, and somewhat based on it.

It is interesting how the abstract changed from v1, to v2, when Garrett had already starting working with Smoling: the part "the theory necessarily has a non-zero cosmological constant" was deleted.

 

[marcus]

...
...
http://arxiv.org/abs/0908.3388
A note on the Plebanski action with cosmological constant and an Immirzi parameter

...Very interesting there should be interaction between the Immirzi parameter of LQG and the cosmological constant Lambda...

I should not have said Im. P. of LQG. The ImP is not unique to LQG. It occurs already in classical General Relativity----in the Ashtekar variables formulation with gamma = sqrt(-1) and in the modification proposed by Barbero which has gamma a real number.

But although the ImP occurs elsewhere besides Loop, it is certainly important in Loop.

They do use the "Immirzi parameter", which Marcus points out above is an LQG concept. However wikipedia seems to be saying that the Immirzi parameter can be derived in a straightforward fashion just from looking at the gauge theory related to GR, and that it has been adopted by the LQG program because it "turns out" to be proportional to the spectrum of the LQG area operator. Marcus, do you think you can explain a bit more what the Immirzi parameter is and why Smolin is likely to find it useful?

Coin I think the Wiki article is poor. Often stuff about LQG in Wiki has been written in large part by people who don't know much about it, or have complicated motives. Maybe someday the Wiki Loop stuff will be competently and reliably rewritten. In the meantime it can be valuable if taken with occasional grano salis.

My personal opinion is that----well I'm not an authority so why should I have an opinion? Etera Livine comes here sometimes, he should tell you what is gamma. But I will tell you what I think.

Basically LQG is on the brink of accomodating renormalization---the running of Newton's G. Maybe the running of the cosmo constant also. This will shake Loop up and it will change the meaning of things like the Im. P.

LQG has a lot of vitality and resiliancy. It has been undergoing radical changes every 3 or 4 years, my vague impression, since I have been watching. This is an imprecise statement.

If you want a careful review of the status quo, look at Rovelli's 2008 Review. It explains clearly how gamma arises in classical GR and is carried over into Loop. See section 6.1 The Classical Theory. As I read it, as of 2008 the physical significance of this parameter is not understood. Some people explain it so and so but then, Rovelli says,see also Jacobson.

Ted Jacobson is extremely good. If you go look at the Jacobson 2007 paper that Rovelli points to it will say that
1. Immirzi is an ambiguity in Loop. We do not know what it is. We cannot confidently pin it down by Hawking talk.
2. Loop people need to take renormalization into account and this will affect the meaning and treatment of the Immirzi parameter among other things.

So Jacobson (who sometimes plays challenger/troublemaker at Loop meetings: the indispensable internal critic) is pointing to a potential for change. Or at least for clarification. Maybe my hunch is wrong and nothing will happen about this over the next couple of years. Maybe the ImP will stay what it is (an adjustable mixing parameter) and violent revolution will be averted

I guess I am being completely unhelpful. Maybe I will post the links to Rovelli's review and Jacobson 2007 and come back later and try to say something more useful.

Review of LQG as of May 2008:
http://relativity.livingreviews.org/Articles/lrr-2008-5/

http://arXiv.org/abs/0707.4026
Renormalization and black hole entropy in Loop Quantum Gravity
Ted Jacobson
8 pages; Class.Quant.Grav.24:4875,2007
(Submitted on 26 Jul 2007)

"Microscopic state counting for a black hole in Loop Quantum Gravity yields a result proportional to horizon area, and inversely proportional to Newton's constant and the Immirzi parameter. It is argued here that before this result can be compared to the Bekenstein-Hawking entropy of a macroscopic black hole, the scale dependence of both Newton's constant and the area must be accounted for. The two entropies could then agree for any value of the Immirzi parameter, if a certain renormalization property holds."

Coin, how much am I saying that is even new to you? You have read widely in the QG literature. I think you may have already read Jacobson's paper and called attention to it here. But I can't recall for sure.

 

[atyy]

From marcus's bibliography: http://arxiv.org/abs/0902.0957 . Relevant?

 

[marcus]

Coin, you asked what is the Immirzi Parameter. It might be fun to take a very uptodate nerdy look at how it arises in the very latest (spinfoam) version of Loop.

What I mean is, there was already this ImP in Ashtekar's formulation of classical GR and Barbero's variant of that. And Rovelli gives the straightforward discussion of it in his review, and then it goes on from there and gets into the canonical LQG of the 1990s. And various papers have speculated about possible physical significance etc. But it is still an unexplained ambiguity, I think, and no good reason to try to pin it down by hawking talk. (As Jacobson argues pretty convincingly, I think.)

So let's take the attitude that this is going to be one of the things we are going to find out more about in the next couple of years.

And so we have this question mark parameter, and let's see how it arises in latest Lorentzian spinfoam formulation, where it is forced to exist. Barrett doesn't put it in by hand, he derives that this constant real number proportionality has to be there. The ImP comes out at him of its own accord.

And it turns out that it is the same ImP that Rovelli's spinfoam paper put in by hand.

But it is still an ambiguity! It is a real number ratio that all the simplices or triangles in the spinfoam must exhibit. In their irrep labels. It has to be the same for all, but it can be anything. It is not pinned down. This paper is surely going to be on our third quarter VIP poll.

http://arxiv.org/abs/0907.2440
Lorentzian spin foam amplitudes: graphical calculus and asymptotics
John W. Barrett, Richard J. Dowdall, Winston J. Fairbairn, Frank Hellmann, Roberto Pereira
30 pages
(Submitted on 14 Jul 2009)
"The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics."

To watch the video of Barrett's Planck Scale conference talk, go here:

http://www.ift.uni.wroc.pl/~rdurka/planckscale/index-video.php?plik=http://panoramix.ift.uni.wroc.pl/~planckscale/video/Day2/2-6.flv&tytul=2.6%20Barrett

He explains page 3 that the Lorentz group irreps are labeled (k,p) where k is a half integer and p is a real number.

In a spinfoam all the triangles get labels (k, p).

Then on page 15 right before equation (22) he proves that all the labels of all the triangles have to share a global proportionality p = gamma k
or else there is no stationary point of the action. But in a Lagrange setup you are looking for stationary points. All the other cases get washed out---don't count. At that point he cites Engle Pereira Rovelli (the famous "EPR" spinfoam paper) and observes that this is just like in EPR. Except with them it was partly a conjecture which they implemented manually, I think, and then Barrett et al proved.

And then later on he comes out and calls it the Immirzi parameter. For example on page 25, the Conclusions section:
==quote from Conclusions==
In this work we have defined a graphical calculus for the unitary representations of Lorentz group, and used it to give a systematic definition of the 4-simplex amplitude in the case of Lorentzian quantum gravity. The asymptotic analysis of the amplitude has some surprising features.

In the corresponding Euclidean quantum gravity problem analysed in [21], there was a puzzling superposition of terms with the Regge action multiplied by the Immirzi parameter and terms with the Regge action not multiplied by the Immirzi parameter.

In the Lorentzian quantum gravity analysed here, these two phenomena are separated out. The terms with the Immirzi parameter occur for boundary data of a Lorentzian metric and involve the Lorentzian Regge action. For this case, the result is much cleaner than for the Euclidean theory, as these are the only terms for this boundary data...

...Another result of this work is that a condition for the existence of stationary points of the action is that p_ab = γ k_ab for some constant γ . This is exactly the same restriction on the representation labels derived in [7], by different methods, where γ is the Immirzi parameter.
==endquote==
[7] is the Engle Pereira Rovelli paper.

One simple moral I get from this story is that it is better to stick with the Lorentzian case if you can manage to because then things make better sense.

 

[marcus]

From marcus's bibliography: http://arxiv.org/abs/0902.0957 . Relevant?

Atyy, thanks for pointing that out. It looks relevant. I don't understand the Immirzi parameter well enough to be sure what is and what isn't. Not necessarily of any significance but both authors were postdocs---Institute for Gravitation and the Cosmos---Ashtekar's outfit at Penn State. It looks interesting. You may have better antennae for this than I do. You could elaborate a bit on the paper, if you wished to.

What I am expecting is that unless Krasnov's new BF formulation is shot down very soon we will see a new spinfoam version based on it. Which he indicates he is working on. And then we will have to see if and how the Immirzi arises in this new spinfoam model.

Krasnov gravity could be called BCK (Bengtson, Capovilla, Krasnov.)
It has an odd history. Bengtson published first in 1996, and indicated that somehow the idea had occurred to Capovilla earlier, but wasn't published. And then Capovilla published in 1997, and the idea was completely forgotten! Then around 2006 Krasnov rediscovered it and posted a paper, and Bengtson recognized the idea, and replied with a paper in 2007.
I like Bengtson a lot. Something he is connected with has to be worth pursuing, however it turns out right or wrong. IMHO. He is a senior guy with a sense of humor, who thinks concretely and physically and with a certain philosophical depth as well. You might enjoy his essay about why the world is 3D.
I suppose it is even possible that there be some crossover between the Calcagni-Mercuri paper you mentioned (which replaces the ImP by a field) and the new formulation that Krasnov is investigating where there is a certain potential added to the action. He calls it V( .. ). He does not indicate that it has any connection with the Im.P. and it probably doesn't. But I can't exclude that possibility.
http://arxiv.org/abs/0907.4064 ("Gravity as BF theory plus potential").

 

[Coin]

My personal opinion is that----well I'm not an authority so why should I have an opinion?

Because you know more than I do Thank you for your responses.