If LQG now satisfactory, how to add matter?

[marcus]

A question always in the background in this thread is "how to add matter".
http://owpdb.mfo.de/show_workshop?id=783
I will quote excerpts from the organizers of the February 2010 Oberwolfach workshop:

==quote MFO document http://www.mfo.de/programme/schedule/2010/06b/OWR_2010_09.pdf ==
Noncommutative Geometry and Loop Quantum Gravity: Loops, Algebras and Spectral Triples
Organised by Christian Fleischhack (Paderborn) Matilde Marcolli (Pasadena) Ryszard Nest (Copenhagen)
February 7th – February 13th, 2010
Abstract. Spectral triples have recently turned out to be relevant for different approaches that aim at quantizing gravity and the other fundamental forces of nature in a mathematically rigorous way. The purpose of this workshop was to bring together researchers mainly from noncommutative geometry and loop quantum gravity –-two major fields that have used spectral triples independently so far–- in order to share their results and open issues.Introduction by the Organisers
The workshop “Noncommutative Geometry and Loop Quantum Gravity: Loops, Algebras and Spectral Triples” has been organized by Christian Fleischhack (Paderborn), Matilde Marcolli (Pasadena), and Ryszard Nest (Copenhagen). This meeting was attended by 23 researchers from 8 countries, including several younger postdocs and two PhD students. We enjoyed 16 talks lasting about 50 to 75 minutes plus discussions. As there were no “official” talks after lunch until 4 pm and also no talks in the evening, there was a large amount of time left for informal discussions.

The task of defining both a consistent and mathematically rigorous theory of quantum gravity is one of most challenging undertakings in modern theoretical physics. It is widely expected that at Planck scale the usual notions of smooth geometries have to be replaced by something different. Various arguments point towards geometric notions becoming noncommutative, so that geometric measurements should correspond to noncommuting operators.

In fact, noncommutative geometry (NCG) provides a remarkably successful framework for unification of all known fundamental forces. Mathematically, it mainly grounds on the pioneering work of Connes, who related Riemannian spin geometries to a certain class of spectral triples over commutative C∗-algebras. Extending this formalism, Chamseddine and Connes demonstrated that the standard model coupled to gravitation naturally emerges from a spectral triple over an almost commutative C∗-algebra together with a spectral action. This way they even entailed experimentally falsifiable predictions in elementary particle physics. However, although fully implementing the idea of unification, this approach has remained essentially classical. Moreover, as the theory of spectral triples has only been developed for Riemannian manifolds, full general relativity needing Lorentzian geometries has not been tackled.

Loop quantum gravity (LQG), on the other hand, is one of the most successful theories to quantize canonical gravity. Resting on a generalization of Dirac quantization by Ashtekar and Lewandowski, its decisive idea is to break down the quantization to finite-dimensional problems on graphs and then to reconstruct the continuum theory using projective/inductive limits over all graphs. Although the kinematical part of LQG is nicely understood, the dynamical part is vastly open territory – both mathematically and conceptually. This concerns mainly three, related issues: First of all, the spectral analysis of the quantum Hamiltonian constraint, responsible for time evolution, is very immature. Secondly, it is completely unknown how to reconstruct classical general relativity as a semiclassical limit of loop quantum gravity. And, instead of an emergent unification, matter has to be included by hand.

Although NCG and LQG use very similar mathematical techniques –- e. g., operator algebras in general, or spectral encoding of geometry to be more specific -–, their conceptual problems are rather complementary. Nevertheless, only recently, first steps to join the strengths of both approaches have been made. In several papers since 2005, Aastrup and Grimstrup, later with one of the organizers (RN), have outlined how to construct a semifinite spectral triple for the full theory out of spectral triples based on a restricted system of nested graphs.

One of the main tasks of the meeting was to bring together researchers from different fields – first of all, noncommutative geometry and loop quantum gravity, but also other fields like spectral triples on its own and axiomatic quantum field theory. For this, there were several introductory talks:

• Hanno Sahlmann and Thomas Thiemann gave an overview on the origins and the current status of loop quantum gravity. Sahlmann focused on physical and kinematical issues, Thiemann on open issues concerning dynamics.

• Giovanni Landi and Walter van Suijlekom presented introductions into noncommutative geometry. Whereas Landi spoke on general issues, Walter van Suijlekom showed how one can encode the standard model of particle physics within the language of spectral triples.

• Johannes Aastrup and Jesper Grimstrup demonstrated how spectral triples can fruitfully transfer ideas from noncommutative geometry into loop quantum gravity.

• Klaus Fredenhagen and Rainer Verch introduced axiomatic quantum field theories as functors from the category of globally hyperbolic spacetimes into that of C∗-algebras. Fredenhagen concentrated on perturbation theory, i.e., such functors that are formal power series in . Verch used this framework to extend the notion of spectral triples to the Lorentzian case.

Beyond these talks there have been more specialized ones:
• Alan Carey described a generalization of spectral triples, so-called semifinite spectral triples. They arise naturally in the Aastrup-Grimstrup-Nest approach.

• Matilde Marcolli and Jerzy Lewandowski studied further noncommutative structures arising in loop quantum gravity. Marcolli described how extended spin foams define noncommutative coordinate algebras; Lewandowski replaced the underlying structure group SU(2) of LQG by the quantum group SUq(2).

• Victor Gayral and Thomas Krajewski spoke on quantum groups as well: Gayral from a more generalized perspective, Krajewski inspired by string theory.

• Fedele Lizzi described noncommutative lattices that may lead to emerging spacetime.

• Varghese Mathai and Raimar Wulkenhaar explained different types of deformation quantization. Mathai constructed noncommutative principal bundles and Wulkenhaar outlined why there should be non-perturbative quantum field theories over Moyal deformed R4.

The atmosphere within the workshop benefited very much from the liveliness of the discussions and questions, which occurred frequently before, during, and after the talks. From this point of view the meeting was very successful, on the one hand for enabling a significant exchange of ideas between researchers in the two major fields, and on the other side for presenting the results of the few scientists that work in the intersection of LQG and NCG. In particular the fact that for every talk usually at least half the audience was no specialist in the field covered in it, resulted in a very effective exchange of knowledge, from which both sides gained profit.
==endquote==

If you scroll down further you will find descriptive summaries of many of the talks. In the group photo here:
http://owpdb.mfo.de/detail?photo_id=12390
Fleischhack, one of the organizers, is on the far left, and I believe it is Thiemann third from the left. Richard Nest, another of the organizers, is on the far right. I don't recognized Sahlmann--perhaps he is third from the right with jeans and a black pullover.

 

[MTd2]

What is the largest GUT that Conne`s NCG predicts?

 

[marcus]

You could start a thread called "Connes GUT?" and get various people's opinions on that.

Connes' recent paper allows for the finite space F to change at very high energies. I gather that his predictions are about what one can eventually see with LHC and conceivable extensions along the same lines. In that range, where prediction is practical and meaningful, he has already determined what the finite algebra F must be. So the predictions which he lists are based on that.

I would not advise anyone to suppose that Spectral Geometry simply consists of Connes version of it. I don't think that the question in this thread is addressed by focusing on Connes version NCG and imagining that one simply layers that (in its 2010 form) on top of LQG. So it's not clear how talking about Connes NCG specifically is relevant to the topic. But I'm happy to do so!

The current version is defined by three 2010 papers:

http://arxiv.org/abs/1008.3980
Noncommutative Geometric Spaces with Boundary: Spectral Action
Ali H. Chamseddine, Alain Connes
26 pages, J.Geom.Phys.61:317-332,2011

http://arxiv.org/abs/1008.0985
Space-Time from the spectral point of view
Ali H. Chamseddine, Alain Connes
19 pages. To appear in the Proceedings of the 12th Marcel Grossmann meeting

http://arxiv.org/abs/1004.0464
Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
Ali H. Chamseddine, Alain Connes
56 pages, Fortschritte der Physik,58:553-600, 2010

Here are the predictions/postdictions listed in 1004.0464:

==quote Ali and Alain==
...We re-derive the leading order terms in the spectral action. The geometrical action yields unification of all fundamental interactions including gravity at very high energies. We make the following predictions:

(i) The number of fermions per family is 16.

(ii) The symmetry group is U(1)xSU(2)xSU(3).

(iii) There are quarks and leptons in the correct representations.

(iv) There is a doublet Higgs that breaks the electroweak symmetry to U(1).

(v) Top quark mass of 170-175 Gev.

(v) There is a right-handed neutrino with a see-saw mechanism. Moreover, the zeroth order spectral action obtained with a cut-off function is consistent with experimental data up to few percent.

We discuss a number of open issues. We prepare the ground for computing higher order corrections since the predicted mass of the Higgs field is quite sensitive to the higher order corrections. We speculate on the nature of the noncommutative space at Planckian energies and the possible role of the fundamental group for the problem of generations.
==endquote==

The Connes model is what they call "almost commutative" where the relevant object is the product of a conventional commutative algebra C(M) with a small finite noncommutative F.
The blue highlight suggests that F can change at Planckian energies! This leaves the model open to new physics. It says that the geometry of spacetime can change radically as you increase the magnification.

The red highlight is how Connes recovers from his pre-2008 bad estimate of Higgs mass. He prepares the ground for higher order corrections, but at this time he does not calculate those corrections.

If you think of Connes "almost commutative" space as a sandwich of |F| different colored copies of ordinary 4D space---a finite sandwich of layers determined by F---then as you zoom into Planckian magnification the number of layers and the coloring can change.

The basic object, as I see it, is still an ordinary 4D manifold M, which we treat via the algebra of continuous functions C(M) defined on M. And then drink a little Connes kool-aid and we see that the right algebra is not simply C(M) but is, in fact, C(M) x F,

the cartesian product of the functions on the manifold M, with a little finite matrix algebra.

Pictorially it is as if M has changed to a sandwich of layers each of which looks like M but has an "F-color".

This is a radical oversimplification of course. If you don't like it then make up your own radical oversimplification.

Now Connes, in the next paper, the one presented at the 2009 Paris Marcel Grossmann, takes the bold step of speculating that if you go to REALLY high energies then even C(M) which you thought was the conventional algebra of functions on a classical 4D manifold becomes, itself, a large but finite algebra of matrices! This is something they didn't tell you when you bought your ticket and walked into the crystal palace.
http://www.icra.it/MG/mg12/en/
http://www.icra.it/MG/mg12/en/invited_speakers_details.htm#connes

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√∧± ÷←↓→↑↔~≈≠≡≤≥½∞⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅

 

[marcus]

As you know I consider that LQG-with-simple-matter has reached a satisfactory level of development because it is now TESTABLE. The theory's consequences for the geometry of the early universe have been investigated for over 10 years and the bounce prediction is generic and robust. The visible consequences have been worked out by experts in early universe phenomenology, whose interest is in testing, not in promoting this or that QG.
(see papers by Wen Zhao et al or by Aurelien Barrau et al.)

So it is high time to think about adding a richer variety of matter to LQG. And it's fairly clear that the community is doing just that. So what clues do we see about how that is going?

One clue is the makeup of the 2011 Zakopane QG school. This is a two week school around the beginning of March 2011, when the ski is good at Zakopane. The signs are that people think of including matter by NCG and maybe also GFT. There is to be a series of lectures by Steinacker and also possibly by Krajewski (not yet confirmed) and also possibly by Connes.
Let's look at this in context by reviewing the list of lecture series.
https://www.physicsforums.com/showthread.php?t=457381

 

[marcus]

Here is the list of lecture series around which the two-week school is structured. What can we learn from it about how people in the community think matter might be incorporated?
==quote==

Core lectures

The heart of the program will be a series of core lectures by leading researchers which will give students a solid introduction to the topics of the school.

Hanno Sahlmann/Kristina Giesel - Loop quantum gravity
The field of loop quantum gravity is the technically highest developed construction in quantum gravity. As in the last two schools there will be a thorough introduction into the underlying ideas and mathematical methods. The lectures will cover the basic construction of the kinematical hilbert space and some simple operators, working up to the dynamical Hilbert spaces and physical Hamiltonians following from the deparametrization models.

Carlo Rovelli - Spin foams
The most active field in the network in the last years has been spin foam models, starting with the development of the graviton propagator and the new models, to coherent state techniques and recent asymptotic results, the generalisation to arbitrary 2-complexes and cosmological applications. The lectures will present the current perspective on the construction of these models in terms of 2-complexes.

Harold Steinacker - Non-commutative geometry and matrix models
Non-commutative geometry is a natural extension of geometry in the context of quantum theories that potentially, may also include gravity.. NCG naturally occurs in particle physics, as shown by Alain Connes, and also appears naturally in the context of three-dimensional quantum gravity via Chern-Simons theory. It is also used as a technical tool in state sum models, particularly via quantum groups, which provide deformations of the usual spin network calculus which can be used to construct quantum gravity models. The lectures will cover the definition and construction of non-commutative spaces as well as the construction of QFTs on them. Another theme will be the relationship to matrix models.

Thomas Krajewski (to be confirmed) - Group field theories
Recently group field theories, generalisations of matrix models to higher dimensions, have received renewed attention. In the last year work has begun to take them serious as quantum field theories and analyse their properties using the tools of QFT. The lectures will cover the general structure of GFTs and introduce the QFT tools used to study their renormalisation theory.

Stefan Hollands - Exact QFT in curved backgrounds
QFT on curved backgrounds is the formulation of QFT which does not require the symmetries of Minkowski or (Anti) de Sitter space times. From the mathematical point of view this is the highest development of QFT. It is also an important intermediate step between the standard QFT and quantum gravity. As an approximation to quantum gravity it supplies some of the most potent intuitions of the field (holography, black hole entropy). The lecture will cover the recent results and successes in the exact construction of these quantum field theories.

Alain Connes (to be confirmed) - Non-commutative geometry
===endquote===

There will also be individual auxilliary lectures to follow up on the introductory core lecture series in most cases by presenting some more advanced or specialized ideas suitable for research. Here are a few of these, as a sample (not the full list.)

Singh - Loop quantum cosmology
Jurkiewicz (an Ambjorn Loll co-author) on CDT
Barrett on the large j limit of spinfoam amplitudes
Rivasseau on EPRL-GFT
Noui on SL(2,C)_q the quantum deformation of the Lorentz group
http://www.fuw.edu.pl/~kostecki/school3/

Basically it looks to me as if the whole Loop contingent has wheeled around to confront the MATTER issue. Because in the introductory lecture series that in a sense define the field for an entry-level researcher, there is a big representation of NCG and GFT and curved background QFT. This is what somebody guesses you likely need to move ahead on the matter issue.

 

[marcus]

More on the including matter front. This was posted today:

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."

My first take, on reading portions and glancing through the rest: this paper is great.

 

[MTd2]

Are these YM already quantized? Where are the bosons, then?

 

[marcus]

Are these YM already quantized? Where are the bosons, then?

At this point I can only suggest read section VII. It is short.

BTW I think in a dynamically curved spacetime the idea of a particle is non-essential and poorly defined. Particles are more at home in flat, or other prearranged geometries.

 

[genneth]

At this point I can only suggest read section VII. It is short.

BTW I think in a dynamically curved spacetime the idea of a particle is non-essential and poorly defined. Particles are more at home in flat, or other prearranged geometries.

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]

BTW I think in a dynamically curved spacetime the idea of a particle is non-essential and poorly defined. Particles are more at home in flat, or other prearranged geometries.

So what is responsible for the transmission of forces? There is no gauge particles now?

 

[marcus]

So what is responsible for the transmission of forces? There is no gauge particles now?

fields. I consider a field is a more general concept than particle.
(I mean something specific by particle. A particle is something that can be counted. One can say unambiguously how many there are. A field does not always resolve into a definite number of particles---it may depend on the viewpoint etc---but in come cases on a fixed background such as Minkowski space it may usefully be so treated.)

This is probably just a semantic misunderstanding. You know very well what you are talking about and you are not confused---but we could be using words differently. I believe that fields really exist and live in geometry (which is itself a field). I don't believe particles exist as a real part of my fundamental ontology---instead their fields exist. The appearance of a particle is just a temporary mathematical device for describing a field, which may or may not be appropriate in the given situation.

I'm not saying anything that is unusual or out of the mainstream, I hope! I vaguely recall someone, Feynman I think, remarking that an electron could be as big as the Empire State building in New York City. Because the building has a steel frame.

 

[marcus]

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).

Genneth, I strongly agree. This year has been one of very rapid development in LQG. Probably the best overall perspective/review is the one I call "December 4707" to help me remember the arxiv tag:
http://arxiv.org/abs/1012.4707.

This review takes the contents of several interesting (series of) LQG papers, including the one you referred to, the Spinfoam Fermions paper, and puts them all together in a balanced coherent picture---with some motivation and history. December 4707 is the one thing I'm trying to find time to study in the midst of Holiday festivities.

In case anyone just dropped in, the Spinfoam Fermions paper is December 4719.
In other words http://arxiv.org/abs/1012.4719

 

[space_cadet]

fields. I consider a field is a more general concept than particle. (I mean something specific by particle. A particle is something that can be counted. One can say unambiguously how many there are. A field does not always resolve into a definite number of particles---it may depend on the viewpoint etc---but in come cases on a fixed background such as Minkowski space it may usefully be so treated.)

So what about wave-particle duality then? Also it is true that there is ambiguity in the definition of a particle, but there is also ambiguity in the definition of a field. That's what gauge and diffeomorphism invariance are all about, right?

This is probably just a semantic misunderstanding. You know very well what you are talking about and you are not confused---but we could be using words differently. I believe that fields really exist and live in geometry (which is itself a field). I don't believe particles exist as a real part of my fundamental ontology---instead their fields exist. The appearance of a particle is just a temporary mathematical device for describing a field, which may or may not be appropriate in the given situation.

Couldn't we switch the words "particle" and "field" in the above paragraph and still have it make sense? If you think about geometry, at a quantum level, you expect it to be discrete or at least not smoothly continuous (C^\infty). There should be all kinds of defects and singularities, and you might naturally want to designate such structures as "particles".

I'm not saying anything that is unusual or out of the mainstream, I hope! I vaguely recall someone, Feynman I think, remarking that an electron could be as big as the Empire State building in New York City. Because the building has a steel frame.

I don't get that last part.

To sum up, IMHO, you need both - fields and particles. Particles are what source fields and fields are what shuttle back and forth between particles helping them communicate. And in the end everything is geometry ;-) !

ps: I've been following this discussion and thought this might be a good time to jump in :-)

 

[marcus]

Always have to be careful with wikipedia, not the most authoritative source! however
http://en.wikipedia.org/wiki/Quantum_field_theory
"...Fermions, like the electron, can also be described as ripples/excitations in a field, where each kind of fermion has its own field. In summary, the classical visualisation of "everything is particles and fields," in quantum field theory, resolves into "everything is particles," which then resolves into "everything is fields." In the end, particles are regarded as excited states of a field (field quanta)..."

http://arxiv.org/abs/gr-qc/0409054
What is a particle?
Daniele Colosi, Carlo Rovelli
"Theoretical developments related to the gravitational interaction have questioned the notion of particle in quantum field theory (QFT). For instance, uniquely-defined particle states do not exist in general, in QFT on a curved spacetime..."

 

[MTd2]

Well, by coupling fermions and YM fields now, LQG is done, right? Although, I wonder about the non gauge bosons...

 

[genneth]

Well, by coupling fermions and YM fields now, LQG is done, right? Although, I wonder about the non gauge bosons...

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.

 

[marcus]

The major QG story these days is the convergence of different approaches into a settled form of the theory. Satisfactory as theory-building goes, ready for testing.

Adding matter to the theory is an important step---earlier in this thread we mentioned the "Spinfoam Fermions" paper (http://arxiv.org/abs/1012.4719). Here is another piece of the puzzle. You may get the idea from the abstract that it is only about 3D gravity, but look at the conclusions. The authors' focus is on the merger of two research lines in the full 4D theory.

MTd2 spotted this one, and added it to the bibliography today.

http://arxiv.org/abs/1101.3524
The Hamiltonian constraint in 3d Riemannian loop quantum gravity
Valentin Bonzom, Laurent Freidel
(Submitted on 18 Jan 2011)
"We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. This Hamiltonian has a clear interpretation in terms of discrete geometries: it computes the extrinsic curvature from dihedral angles. The Wheeler-DeWitt equation takes the form of difference equations, which are actually recursion relations satisfied by Wigner symbols. On the boundary of a tetrahedron, the Hamiltonian generates the exact recursion relation on the 6j-symbol which comes from the Biedenharn-Elliott (pentagon) identity. This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity."

What we see happening on a lot of fronts is what could be called "tying up loose ends."

What has become the main LQG thrust (latest review paper: http://arxiv.org/abs/1012.4707 ) is not "derived" from anyone thing by some "quantization" procedure. It is a kind of synthesis inspired by several QG directions. So it is buttressed from several independent developments and an important one is the older Hamiltonian approach. That older program was never fully completed. So it looks like Bonzom Freidel are investigating how to complete it and how it will, when completed, support the prevaling spinfoam covariant approach.

 

[marcus]

To save the reader trouble, here is the portion of the conclusions of the Bonzom Freidel paper relevant to the 4D program:
==quote http://arxiv.org/abs/1101.3524 ==

...Our result shows that it is possible to identify very precisely the recurrence relations satisfied by spin foam amplitudes with a quantum implementation of the classical symmetries in loop quantum gravity. Thus, this suggests two ways to apply this programme to 4d LQG.

The first idea is to derive recurrence relations for the amplitudes of the most promising spin foam models, and then try to produce them from an operator in LQG. Since these models describe geometry with areas and normals to triangles, we expect the corresponding operator to produce both differences on spins and some differential parts on the normals.

The second idea is to first derive difference equations from an operator and then interpret them as recurrence relations for some spin foam models. Typically, our operator Hv,f has a natural generalization in 4d. ...
...
Ultimately, one may expect these two approaches to coincide. This is actually what we achieved in the present article on the 3d model.

In four dimensions, some preliminary results have been obtained in [6]. There the topological Ooguri model for BF theory is revisited by lifting the Hamiltonian we have just used here, H_v,f , to the boundary of a 4-simplex. Classically, this provides the phase space of 4d loop quantum gravity with a Hamiltonian dynamics which can be interpreted in terms of twisted geometries. In the spin network basis, the Wheeler-DeWitt equation gives recursion relations which are actually satisfied by the Wigner 15j-symbol. We expect to extend these ideas for more realistic spin foam models ...

More generally, the relation between spin foams and Hamiltonian dynamics is investigated in [6] in the large spin limit through difference equations on the 4-simplex amplitude. In 3d, such an equation is obviously the recursion relation (4) on the 6j-symbol, simplified in the asymtotics. Such equations give criteria to know whether a model is semi-classically approximated by some quantum Regge calculus. For instance, it is clear from these results that the (naive) proposal (89) is solved in the asymptotics by exponentials of i times the Regge action of the 4-simplex with quantized areas.
==endquote==

 

[MTd2]

The problems were solved with a restriction that was done on the phase space of LQG. This is funny considering Freidel's previous paper.

 

[marcus]

The problems were solved with a restriction that was done on the phase space of LQG. This is funny considering Freidel's previous paper.

More comment by MTd2 on this topic:
https://www.physicsforums.com/showthread.php?p=3092118#post3092118

Reminder about the particle concept when space is curved:

http://arxiv.org/abs/gr-qc/0409054
What is a particle?
Daniele Colosi, Carlo Rovelli
"Theoretical developments related to the gravitational interaction have questioned the notion of particle in quantum field theory (QFT). For instance, uniquely-defined particle states do not exist in general, in QFT on a curved spacetime..."

The thread topic is ideas of how LQG can add matter. We already mentioned this December 2010 paper:

More on the including matter front. This was posted today:
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."
...

For completeness, we should add a followup paper that appeared in January 2011 along the same lines:

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."

 

[marcus]

How to add matter?

In view of everything we have been seeing happen in QG recently the question for 2011 does seem to boil down to the topic question of this thread: how to add matter?

I want to review the evidence (some quite recent and not yet covered in this thread) that this is the question that is "on everybody's mind" in the QG community.

This is interpretive, and I could be wrong. If you see a different prevailing focus in today's research, schools, workshops, conferences ...etc then please articulate it in opposition to my view.

I'll start reviewing the evidence and trying to integrate the different stuff we are seeing at this point, maybe tomorrow, or later this evening.

 

[atyy]

Try the last chapter of Hellman's thesis.

 

[marcus]

Atyy, thanks for the pointer. I will fetch your explanatory comment from the other thread and keep it in mind. What I'm thinking about now is my impression that a combined field of QG is taking shape consisting of maybe half-dozen different research lines. I quote from the
http://www.conferences.itp.phys.ethz.ch/doku.php?id=qg11:start
homepage of the June Zurich conference

Topics to be covered include:

• General quantum theory, relativistic quantum theory, emergence of space(-time)
• General quantum field theory, including deformations of QFTs
• QFT on curved and NC space-times
• Canonical quantum gravity and supergravity
• Regge calculus
• String theory and M-theory
• Loop gravity, spin foam
• Quantum cosmology

International Advisory Board

John Barrett
Harald Grosse
Hermann Nicolai
Carlo Rovelli
Roger Picken

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The message of the conference seems to be that QG (quantum geometry/gravity) is a single field and that to the extent that your research is about QG you ought to know the other people and what they are doing, and how it might relate to what you do.
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This is still an indefinite thought on my part. I am not sure of it. I have the idea that what it means for the LQG people is they will now focus on the question how to add matter.

LQG already has a pretty strong hand in quantum cosmology. (both theory and phenomenology). But I have the expectation that the community is going to focus on adding matter. I'm not sure why I think that. Some hint of it in recent papers of Rovelli and of Barrett, perhaps.

I guess it is part of a natural process of consolidation where you build bridges to other research contingents that belong to the same field.

BTW John Baez is one of the Zurich speakers. (He's the one who called our attention to the conference.) Also Lance Dixon. Also Alain Connes.