Spinfoam Fermions, Lqg now has matter

[marcus]

https://www.physicsforums.com/showthread.php?p=3048498#post3048498

This should have its own thread:


http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
8 pages
(Submitted on 21 Dec 2010)
"We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form."

 

[marcus]

The more I read this one the higher my opinion of it. A game-changer.
I guess this would be my nomination (if I had to pick just one) for the Loop-and-related
2010 "Paper of the Year".

 

[marcus]

I was interested in what Genneth said about this "Spinfoam Fermions" paper, when it came up in another thread. Judging from his other posts Genneth is a physics theory PhD doing research in some other field--not QG. So I think of him as not having a personal stake in any particular type of QG. I could be wrong---don't have any facts to go on besides his posts---but tend to hear him as unbiased. Not a partisan of this or that.

So his reactions to this paper carried some weight. I'll quote:

Actually, reading the paper, it seems the concept is not so hard --- after all, the current spinfoam incarnation is conceptually a quantum (i.e. linearly superposable) discretised geometry. As has been known for a while, on the classical level fermions + YM can be written as a theory of gauge strings connected by fermions; this paper I believe simply (!) implements that idea. Thus, particle states are localised --- but at the same time slightly delocalised --- to spacetime vertices, which means that at each vertex you get a set of fermion states (0, +/- or 2).

I'm not entirely sure at the moment what they mean by using the gravitational radiative corrections to generate the YM action, but I suspect they mean by a Einstein-KK-esque argument, on the quantum level.

I find all this development to be massively exciting, though in the end the proof will be in the form of concrete calculations (and of course, experimental verification of said calculations).

 

[MTd2]

So, there isn`t much to do now. LQG is totally done with matter!

 

[marcus]

Heh, heh, you are joking.
I think there is more work left to do than was done in the first 25 years---you can read the history of that in Rovelli's review http://arxiv.org/abs/1012.4707.

But there are many more people working on LQG now, so the work may go faster.

I suppose LQG could fail to pass the observational tests. Observations of the CMB may be incompatible with Loop bounce model of early universe. Then there would be no work to do. But if it passes the CMB tests, there is still a lot to work out. It should be fascinating!

Defining Quantum Fields on spinnetwork and spinfoam version of spacetime. A truly General Relativistic QFT.

=============================

I got interrupted and had to go out for an hour or so. I was about to quote this interesting post by Genneth. Someone, I think you MTd2, had asked "how about non-gauge bosons?" Yes! It was you who asked!
In any case, the topic of non-gauge bosons came up, and his comment was:

Actually, one might hope there aren't any. Renormalisation arguments show that plain ol' self-interacting bosons tend to have Landau poles, which strongly suggest that something (i.e. phase transition) must occur before the Planck scale. After all, the current dominant Higgs mechanism in electroweak breaking is just the simplest possible one of its type --- one could (and many have) plausibly imagine quite complicated bosonic structures which arise out of some higher energy phase transition instead. The only real constraint is that there should be a massless boson beforehand --- and Goldstone's theorem provides plenty of those in abundance.

 

[MTd2]

"Loop quantum gravity is resolutely a theory of quantum gravity that does not address the unification problem. In this, it is like QED, or, more precisely, QCD: a quantum field theory for a certain interaction, which can be coupled to other interactions, but is also consistent by itself. The philosophy underlying loop gravity is that we are not near the end of physics, we better not dream of a final theory of everything, and we better solve one problem at the time, which is hard enough."

page 7

 

[marcus]

What is your point? All that page 7 quote means is that the philosophy governing the research program SO FAR HAS BEEN INCREMENTAL. The gradualistic, step-by-step, solve one problem at a time approach.

And probably this tradition will persist in the foreseeable future and the guiding of the program WILL CONTINUE TO BE gradual step-by-step, solving one problem at a time.

Rather than leap to a bold guess, hoping it will solve all problems at once in a final unique "ToE".

What you quoted means that whatever the future holds, whatever partial successes and partial successes, the program will probably not go running after the dream of an ALL-AT-ONCE final "ToE".

But that does not mean it cannot proceed gradually to add particle fields and enlarge the LQG action. Thiemann was already trying that kind of thing in the 1990s.

Rovelli has never put limits on the theory's growth in future. He only defined a way of proceeding gradually with limited objectives at each stage.

 

[MTd2]

I didn`t make any point. I just think it was an important remark by itself.

 

[genneth]

My intuition (which is really not worth much) is that the matter unification potentially offers two potentially interesting outcomes (amongst others which I cannot conceive with my limited falculties...):

1. Matter coupling actually tells us how to go about measuring the quantum gravity effects --- direct quantum GR effects on normal matter; e.g. scattering modification signatures, or astrophysical signatures.

2. A complete moon-shot: predict a grand unification for the forces which differ in a crucial way from the standard model, e.g. Higgs sector

As Marcus quite correctly notes, I have no vested interests apart from being in the peanut gallery, so my best wishes are for the 2nd case :-D

Since it's Christmas and all that, I'm feeling the force of my forefathers --- we might be living in interesting times...

 

[marcus]

We do in fact live in interesting times! We'll just have to make the best of it, I guess. But coupling geometry to matter could be quite fruitful, for the reason you mention.

...
1. Matter coupling actually tells us how to go about measuring the quantum gravity effects --- direct quantum GR effects on normal matter; e.g. scattering modification signatures, or astrophysical signatures.
...

 

[marcus]

This paper, Spinfoam Fermions, led the MIP poll. Five out of nine respondents picked it.
Likely one of the more important QG papers of 2010, and could start a followup wave of papers in 2011.

http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
8 pages
(Submitted on 21 Dec 2010)
"We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form."

Here is the fourth quarter MIP poll, in case anyone wants to check out what other papers were in the race:
https://www.physicsforums.com/showthread.php?t=458853

 

[marcus]

It might help clarify the issues to emphasize that the matter+geometry ansatz here is based manifoldless Lqg. This is the radically new formulation of Lqg that Rovelli and others have been working out during 2010.

There is a nice concise development on page 4 of the new paper (Spinfoam fermions) where they begin by recalling the partition function for pure gravity, citing http://arxiv.org/abs/1010.1939 equation (4) at bottom of page 1.
where the partition function is derived from four Feynman rules.

So they take that pure gravity partition function, on page 4 of the new paper, as their equation (35) and they adjoin a fermion term to it, to get equation (36).

That becomes the partition function of the combined geometry + matter theory.

It is manifold-free — based on the 2-complex (the minimal combinatorial structure that 4D curvature can live on) — with matter world-lines running through the 2-complex. This represents our finite information about the geometry and matter in question. No extra baggage.

The development here obviously depends on the new manifoldless formulation of Lqg. BTW they cite here the recent paper "Lorentz covariance of loop quantum gravity" as their reference [29]. This comes soon after the combined matter+geometry partition function (36):

...the choice of a specific SU(2) subgroup at each edge, but this dependence drops from the total expression, because of the SL(2,C) integrations, precisely as discussed in [29]. Therefore Lorentz invariance is implemented in the bulk.

This expression defines a quantum theory of gravity interacting with fermions.

In the next section we express this amplitude in a more conventional local form, in the spin network basis, and we write explicitly the fermion vertex amplitude. For this, we need to discuss the fermion states.​

 

[marcus]

The only place in the new paper (December4719, matter+geometry paper) where they drag in a manifold is at the beginning for MOTIVATION. Since people have done field theory this way for generations, and it's the language they expect, the authors provide a manifold-based discussion at the start. This makes intuitive the matter term they then add to the partition function.

They use a manifold to find it, as heuristic, but then they manipulate it so that it no longer depends on the coordinates, and is expressed in the same terms as the pure geometry partition function. Then the treatment of matter merges right in with the manifoldless treatment of pure gravity, or geometry.

After equation (36) they focus on a local, vertex-by-vertex formulation. They restate the theory in terms of vertex amplitudes, and show in equations (43)-(46) that the original matter+geometry partition function can be recovered from vertex amplitudes.

The net effect of the fermions is simply to add fermion world-lines over the foam. The weight of each world-line is a contraction of spin-connection group elements along the world-line, taken in the fundamental representation. The worldlines that run over the foam carry a j = 1/2 representation, and couple to the intertwiners at the edges. They overlap at most twice and where they overlap, they run in the j = 0 representation.​

 

[marcus]

So, there isn`t much to do now. LQG is totally done with matter!

Even though there is a huge amount to do, further work that, in fact, this "Spinfoam Fermions" paper reveals and spotlights for us, there is also truth in what you say. This paper is a major landmark.

It marks, for the Loop program, "the end of the beginning."

The goal of the program, stated clearly in the 1990s, has been a backgroundless quantum field theory.

Now, with the spinfoam fermion ansatz we have a cornerstone of that reformulation of QFT, the inclusion of geometry with matter fields in a full QFT.

It is also a moment of high risk.

 

[MTd2]

Wow, it's the 2nd time you answer my post! You also change a bit your mind.

 

[marcus]

I don't think I change my mind about the main thing. It is not true that "there isn't much to do".

Even the installation of fermion matter in the 2-complex (i.e. foam) setting is not complete. There are cases to be worked out that they skipped over in this first paper, and mentioned only in footnotes.

But more than that, the physics has to be derived from this model.

Like they say on page 6 in the concluding remarks:

"The physics it yields will be analyzed elsewhere."

This is why I said it is a moment of risk...the ansatz of matter is a natural and fitting one for spinfoam quantum geometry and it may be that it yields wrong physics. What we are looking at is a major test of LQG. Can its 2-complexes (foams) serve as a basis for the successful manifoldless reformulation of quantum field theory?

Not on Euclidean spacetime are the fields to be defined, not on Minkowski spacetime, and not on any manifold's pre-arranged smooth continuum, but only on the 2-complex, a minimal skeleton of geometric and material information.

It is a bold move. "Ansatz" means "educated guess", or more exactly "educated supposition". It is when someone with exceptional insight says "let's suppose it's like this and this, now let's see..."

This move is too radical to be a "derivation" or "quantization" from some earlier established setup. That is why the authors refer to it as an ansatz.

 

[MTd2]

Each of the superstrings has a fixed quantity of internal symmetries. I wonder if there is something similar in this case...

 

[marcus]

I think you are looking at it from the right perspective. You asked about the symmetries in the spinfoam+matter theory. Here is a revelvant quote from the conclusions of http://arxiv.org/abs/1012.4719

A spinfoam amplitude can be seen as a definition of the gravitational path integral which takes into account the quantization of geometry: intermediate states along the evolution are properly basis elements of the LQG Hilbert space. To couple fermions, we have found a discretization of the fermion action which remains valid on a curved spacetime and which is expressed in terms of the same variables that appear in the spinfoam amplitudes. This discretization couples naturally with the spinfoam amplitudes.

As is always the case in quantum gravity, the appropriate way to view this calculation is not as a “derivation” or a “quantization”, but rather as a heuristic hint, yielding an ansatz for a definition of the coupled gravity-fermion theory. The theory appears to have the correct degrees of freedom and the proper symmetries.
The physics it yields will be analyzed elsewhere.​

BTW in one of your posts you asked why the words "October 1942" had come up in one of the papers we were reading. I don't know the answer to that, but this was an historical turning point which a famous commenter labeled "not the end, nor even the beginning of the end, but, perhaps, the end of the beginning." I will spare you the details . It's resonant if you know mid-20th century history.

In this paper we see, pretty much for the first time, the quantum spacetime of Lqg, namely a 2-complex, serve instead of a 4D manifold as a place where matter can live. The 2-complex, a natural extension of the idea of a graph to one higher dimension, is the natural vehicle to carry 4D geometry and play this role.
We see a 2-complex (colloquially a "foam") having fermion matter defined on it and we see it with fermion world-lines embedded in it.

We see the combined geometry+matter path integral partition function and we see that broken down locally into vertex amplitudes.

This is what I have been looking for since 2003 when I dropped in as an observer on this scene. I have a feeling now, a hunch, that it is being done correctly.

But of course it also might not be the right way! Happily, there is a way to find out. They can see what physics is yielded by the 2010 geometry+matter ansatz. It will yield the right physics or it will not.
So we'll see.

There is still a long way to go, in the program. But I think you hear me, MTd2, when I call it "the end of the beginning".

 

[MTd2]

Hmm, but I meant a certain fixed group of internal symmetries, like SU(32) or E(8)XE(8).

 

[marcus]

Hmm, but I meant a certain fixed group of internal symmetries, like SU(32) or E(8)XE(8).

We are both looking at http://arxiv.org/pdf/1012.4719
and we both see page 6 section 7

It is early to be talking of additional symmetry. Here's a slight hint in that direction, partly just put in by hand.

Suppose now that the fermion lives in fundamental representation of a compact group G. Then the above theory is invariant under global G trasformations. To make it invariant under local gauge transformations we can introduce a group element U_ve ∈ G associated to each wedge (v, e), and replace (9) by...
...
...
The quantum kinematics on the boundary is then evident: spinfoams carry representations of SL(2,C) and intertwiners at the nodes have a possible extra leg representing fermions in (antisymmetric products of) the fundamental representation of SL(2,C) × G.

What is the dynamics? One possibility of obtaining it is simply to keep only the gravity and fermion terms in the action. The Yang-Mills action is then generated by the one-loop radiative corrections to the fermion action...​

 

[MTd2]

I mean, for example, that only one G could be possibly used. All G, that were not this one, would yield inconsistent theories.

 

[marcus]

I understand what you are saying. It's probably too early to guess about what groups G might be tried and what the results might be. For my part, I'd be interested in any comment on the section at the end where they consider a way to include the particle's mass into the spinfoam picture, if you've looked at it.

 

[marcus]

Followup paper today:

http://arxiv.org/abs/1101.3264
Spinfoam Fermions: PCT Symmetry, Dirac Determinant, and Correlation Functions
Muxin Han, Carlo Rovelli
26 pages, 9 figures
(Submitted on 17 Jan 2011)
"We discuss fermion coupling in the framework of spinfoam quantum gravity. We analyze the gravity-fermion spinfoam model and its fermion correlation functions. We show that there is a spinfoam analog of PCT symmetry for the fermion fields on spinfoam model, where a PCT theorem is proved for spinfoam fermion correlation functions. We compute the determinant of the Dirac operator for the fermions, where two presentations of the Dirac determinant are given in terms of diagram expansions. We compute the fermion correlation functions and show that they can be given by Feynman diagrams on the spinfoams, where the Feynman propagators can be represented by a discretized path integral of a world-line action along the edges of the underlying 2-complex."

In case anyone is new, here's the topic paper:

http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
8 pages
(Submitted on 21 Dec 2010)
"We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form."

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