Discussing Section 8 of Marcolli's Paper: Spin Foams with Matter

[marcus]

Any comments you have on section 8 of http://arxiv.org/pdf/1005.1057
are welcome and could be helpful.

Section 8 "Spin Foams with Matter" begins on page 42 of Marcolli's latest paper.

==quote from ToC==
8. Spin foams with matter: almost-commutative geometries 42
8.1. The noncommutative geometry approach to the Standard Model 43
8.2. Spectral triples and loop quantum gravity 45
8.3. Spectral correspondences 46
8.4. Future work
==endquote==

Abstract and context for Marcolli's paper, Spin Foams and Noncommutative Geometry, is provided here
https://www.physicsforums.com/showthread.php?t=401394 in the brief Oberwolfach thread.

 

[marcus]

One practical consequence is that Caltech has joined the select list of North American institutions where a grad student has the chance of finding an advisor and doing research in LQG. There aren't many such places, so it adds intellectual degrees of freedom.

As a basis for comment, I will copy excerpts from section 8.

==quote==
One of the most appealing features of general relativity is its geometric nature. That is, in general relativity the gravitational force manifests as the evolution of the metric structure of spacetime itself. In contrast, quantum field theories are decidedly non-geometric, with their dynamical variables being located in an abstract Hilbert space and interacting according to various representation-theoretic data that specify “particles.”

Traditionally, one places quantum field theories on a background spacetime M by postulating the existence of a bundle structure over M. In this way, one creates an a priori distinction between the base manifold and fibers over it. The former is a geometric object, invariant under diffeomorphisms, while the latter are quantum field-theoretic, and invariant under gauge transformations.

While this is more or less the most straightforward way of combining these two theories into one with the desired symmetry group C^∞(M,G) × Diff (M), in the end what we have is a rather unmotivated and inelegant fusion.

The idea that noncommutative geometry could provide a purely geometric interpretation for quantum field theories goes back to the beginnings of the former subject. Since noncommutative geometry gives us access to more general spaces beyond our ordinary “commutative manifolds,” the hope is that we could find some noncommutative space X such that our quantum field theory is given by the evolution of the geometry on this space, just as general relativity is a theory of the evolution of the geometry of a commutative manifold M.

Quickly summarizing the current state of the art on this approach, we will simply say that this hope was indeed born out. In [30], Chamseddine, Connes, and Marcolli were able to reproduce the Standard Model minimally coupled to gravity and with neutrino mixing in a fully geometric manner—that is, as a theory of pure gravity on a suitable noncommutative space. This space is a product of an ordinary four-dimensional spacetime manifold with a small noncommutative space, which is metrically zero-dimensional but K-theoretically six-dimensional, and which accounts for the matter content of the particle physics model.

In particular, the analogue of the diffeomorphism group for the noncommutative space in question reproduces the desired symmetry structure C^∞(M,G)×Diff(M), with G = SU(3) × SU(2) × U(1) as desired.

The elegant success of noncommutative-geometry techniques in coupling quantum field-theoretic matter to classical spacetime geometry then suggests a possible approach for doing the same with the quantum spacetime geometry of loop quantum gravity. In what follows, we give a brief flavor of how gauge theories on a classical manifold are constructed in the framework of noncommutative geometry, before surveying existing work on incorporating the spin networks and foams of loop quantum gravity into this same framework. Once we have an idea of how quantum spacetime and matter are separately accounted for by noncommutative geometry, we suggest an approach to combining them, thus adding matter to loop quantum gravity and achieving a unified theory of quantum matter on quantum spacetime.
==endquote==

 

[ensabah6]

Once we have an idea of how quantum spacetime and matter are separately accounted for by noncommutative geometry, we suggest an approach to combining them, thus adding matter to loop quantum gravity and achieving a unified theory of quantum matter on quantum spacetime.
==endquote==

Would a completely successful LQG + NCG be considered a TOE? Perhaps NCG can supply the lagrangian needed for chiral twists in LQG'-Bilson braiding.

 

[marcus]

Nothing said about Bilson braiding---that might or might not turn out to be the right approach to realizing matter in Lqg. This is a different approach. Let's look at another passage of the Marcolli paper, starting on page 45:

==quote==
In this way, our original hope of realizing particle physics as an entirely geometric phenomenon is borne out: the Standard Model Lagrangian has been reproduced as a natural spectral action over the specified noncommutative space.

8.2. Spectral triples and loop quantum gravity.

The Noncommutative Standard Model, despite its success, still produces an essentially classical conception of gravity, as seen by the Einstein–Hilbert action embedded in eq. (8.2). Indeed, the authors of [36] comment on this directly in the context of their discussion of the mass scale Λ, noting that they do not worry about the presence of a tachyon pole near the Planck mass since, in their view, “at the Planck energy the manifold structure of spacetime will break down and one must have a completely finite theory.”

Such a view is precisely that embodied by theories of quantum gravity, including of course loop quantum gravity—a setting in which spin networks and spin foams find their home. The hope would be to incorporate such existing work toward quantizing gravity into the spectral triple formalism by replacing the “commutative part” of our theory’s spectral triple with something representing discretized spacetime. Seen from another point of view, if we can find a way of phrasing loop quantum gravity in the language of noncommutative geometry, then the spectral triple formalism provides a promising approach toward naturally integrating gravity and matter into one unified theory.

This idea of expressing LQG in terms of noncommutative geometry has been investigated recently in a series of papers by Aastrup, Grimstrup, Nest, and Paschke [37, 38, 34, 35, 39, 40, 41, 42, 43, 44]. Their starting point is to construct a spectral triple from the algebra of holonomy loops. The interaction between this algebra and the spectral triple’s Dirac operator reproduces the Poisson bracket of both Yang–Mills theory and of the Ashtekar formulation of general relativity upon which LQG is based. Later papers illuminate the situation by making contact with the semiclassical limit, where they retrieve the Dirac Hamiltonian for fermions coupled to gravity in [44]. Their approach also resolves a long-standing obstruction to naïve transference of the LQG Hilbert space into a spectral triple: namely, that the Hilbert space in question is nonseparable, which creates problems for the spectral triple construction.

==endquote==

The "authors of [36]" are Chamseddine and Connes. Here at PF Beyond forum we discussed some of Aastrup Grimstrup's work back in 2006-2007, like the paper that is Marcolli's reference [38]
[38] J. Aastrup and J. M. Grimstrup, “Intersecting Connes noncommutative geometry with quantum
gravity,” Int. J. Mod. Phys. A22 (2007) 1589, arXiv:hep-th/0601127.
It seemed like a distant longshot then. I think the first Loops conference where Grimstrup gave a paper was Loops 2008 ("QG-squared" at Nottingham). Since then this approach to building matter into LQG has gradually become more visible. It looks completely distinct from Bilson braiding to me, though some subtle connection might eventually emerge.

 

[ensabah6]

Nothing said about Bilson braiding
gravity,” Int. J. Mod. Phys. A22 (2007) 1589, arXiv:hep-th/0601127.
It seemed like a distant longshot then. I think the first Loops conference where Grimstrup gave a paper was Loops 2008 ("QG-squared" at Nottingham). Since then this approach to building matter into LQG has gradually become more visible. It looks completely distinct from Bilson braiding to me, though some subtle connection might eventually emerge.

The two probably are distinct, the reason I suggest it is that
what are elementary particles in NCG? There isn't a lagrangian derived for braiding in LQG. I'm suggesting that NCG could supply an interpretation and lagrangian.

btw

http://www.fuw.edu.pl/~jpa/qgqg3/JesperGrimstrup.pdf

 

[Coin]

So to be clear: Whereas something like braid matter uses the mechanics of LQG itself to construct something that acts like matter, what these guys did is just show that they have a mathematically reasonable way of affixing matter content to "points" in LQG space by describing this matter using the formalism of the noncommutative standard model. Do I have that right?

Two things that stand out to me:

- This seems like extremely early-stage exploratory work. They say at the end:

One of the more pressing questions is the relation between the kinematical Hilbert space of LQG, and that constructed in the above theory. The fact that the spectral triple encodes the Poisson bracket of general relativity implies that it carries information on the kinematical sector of quantum gravity. However, due to differences in construction, it is clear that despite similar starting points with regards to the importance of holonomy variables, in the end the Hilbert space produced is not the same as that of LQG.

...and say they want to explore exactly what that means in future work. In other words, it sounds like they have shown that their construction is mathematically sensible (which is exciting by itself) but they don't yet have any idea what their construction actually does, or if when you run the dynamics it still has any of the properties we want LQG to have. Again, correct?

- This may not be a substantive criticism, but: I find it a little bit troubling that the construction is so incredibly pure-mathematics oriented. Vanilla LQG, or braid matter, or string theory we have some sort of story we can tell to explain to someone not familiar with the mathematical formalisms what the theory is "doing". You don't need to necessarily understand tensor algebra to "get" what is happening in GR or with kaluza-klein electromagnetics, you don't need to understand holonomy groups to understand basic spin network dynamics, you don't need to understand the horribly complicated math happening in string theory to sort of get how matter emerges from that theory. NCG, which is after all about rigorously describing unintuitive geometric structures, seems fairly resistant to intuitive explanations, and so when we wed NCG to spin networks we are combining one extremely abstract construct with another extremely abstract construct to produce something that lives way out in the land of pure math even by mathematical physics standards. I don't think I could explain what the "product M × F of a commutative spin manifold and a finite noncommutative space" means (maybe I'd have more luck if I understood spectral triples) in a sort of intuitive sense, or what justification we have for the attached noncommutative space, except that when we run the numbers it gives us the lagrangian we want. Can anyone? Now given if what I'm describing here is a problem it probably is more a problem with how the NCG theory is commonly explained and taught than a problem with the theory's fundamentals themselves, but I would worry that the high degree of abstraction and the resulting fact that you basically have to learn two relatively obscure mathematical branches and then run numbers to say much of anything about this theory could make it easy for big conceptual problems with the idea to linger unnoticed for awhile...

Anyway it should be very interesting to see where future work in this direction goes. Are there signs any of the established people working in either lqg or ncg such as Smolin or Connes might try to work with this new construction?

 

[ensabah6]

Anyway it should be very interesting to see where future work in this direction goes. Are there signs any of the established people working in either lqg or ncg such as Smolin or Connes might try to work with this new construction?

Aastrup and Grimstrup has had papers from 2006 but so far as I know, it hasn't gotten much traction (i.e from Smolin or Connes) -- there appears some differences in detail (i.e the hilbert space in this is separable vs non-separable in LQG)


arXiv:hep-th/0601127


Intersecting Connes Noncommutative Geometry with Quantum Gravity
Authors: Johannes Aastrup, Jesper M. Grimstrup
(Submitted on 18 Jan 2006)

Abstract: An intersection of Noncommutative Geometry and Loop Quantum Gravity is proposed. Alain Connes' Noncommutative Geometry provides a framework in which the Standard Model of particle physics coupled to general relativity is formulated as a unified, gravitational theory. However, to this day no quantization procedure compatible with this framework is known. In this paper we consider the noncommutative algebra of holonomy loops on a functional space of certain spin-connections. The construction of a spectral triple is outlined and ideas on interpretation and classical limit are presented.

1. arXiv:1003.3802 [pdf, other]
Title: On a Derivation of the Dirac Hamiltonian From a Construction of Quantum Gravity
Authors: Johannes Aastrup, Jesper M. Grimstrup, Mario Paschke
Comments: 13 pages, two figures.
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
2. arXiv:0911.4141 [pdf, other]
Title: Lattice Loop Quantum Gravity
Authors: Johannes Aastrup, Jesper M. Grimstrup
Comments: 3 figures
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)
3. arXiv:0911.2404 [pdf, other]
Title: Emergent Dirac Hamiltonians in Quantum Gravity
Authors: Johannes Aastrup, Jesper M. Grimstrup, Mario Paschke
Comments: one figure
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
4. arXiv:0907.5510 [pdf, other]
Title: On Semi-Classical States of Quantum Gravity and Noncommutative Geometry
Authors: Johannes Aastrup, Jesper M. Grimstrup, Mario Paschke, Ryszard Nest
Comments: 31 pages, 10 figures
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)
5. arXiv:0902.4191 [pdf, other]
Title: Holonomy Loops, Spectral Triples & Quantum Gravity
Authors: Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest
Comments: 24 pages, 7 figures, based on talk given by J.M.G. at the QG2 conference, Nottingham, juli 2008; at the QSTNG conference in Rome in sept/oct 2008; at the AONCG conference, Canberra, dec. 2008
Journal-ref: Class.Quant.Grav.26:165001,2009
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
6. arXiv:0807.3664 [pdf, ps, other]
Title: A new spectral triple over a space of connections
Authors: Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest
Comments: 14 pages, 5 figures
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
7. arXiv:0802.1784 [pdf, ps, other]
Title: On Spectral Triples in Quantum Gravity II
Authors: Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest
Comments: 43 pages, 1 figure
Journal-ref: J.Noncommut.Geom.3:47-81,2009
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
8. arXiv:0802.1783 [pdf, ps, other]
Title: On Spectral Triples in Quantum Gravity I
Authors: Johannes Aastrup, Jesper M. Grimstrup, Ryszard Nest
Comments: 84 pages, 8 figures
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc)

 

[marcus]

... Are there signs any of the established people working in either lqg or ncg such as Smolin or Connes might try to work with this new construction?

Astute ("sociological") question. I think it's clear that sociological cues are an important way we get to understand leading edge research.

The big news from NCG 2006-2008 is their realizing the Standard Model geometrically. Constructing a model of spacetime inwhich the particles and forces of physics were engrained in the spacetime. Intrinsic to to it, built in.

The names on the papers presenting that result were
Chamseddine
Connes
Marcolli

What do you mean by "established people", Coin? In 2008 after she and co-authors put the Standard Model into NC geometry, Marcolli took a tenured full professorship in the Math department at Caltech.

Before that, from 2003-2008, she was a tenured associate professor at the Max Planck Institute for Mathematics at Bonn.

She was born in November 1969, still only 40. To me she looks established. I rate Caltech Math even higher than Caltech Physics. I wish her well. Who knows? (I just looked up Alain Connes, he was born 1 April 1947. Great guy. Wish him well too.)

 

[ensabah6]

Astute ("sociological") question. I think it's clear that sociological cues are an important way we get to understand leading edge research.

The big news from NCG is their realizing the Standard Model geometrically. Constructing a model of spacetime inwhich the particles and forces of physics were engrained in the spacetime. Intrinsic to to it, built in.

The names on the papers presenting that result were
Chamseddine
Connes
Marcolli

What do you mean "established people", Coin? After she and co-authors put the Standard Model into NC geometry, Marcolli took a tenured position in the Math department at Caltech.

Since Smolin et al aren't publishing papers on qg-qg as he did on lqg-verlinde perhaps that's a sociological cue?

I know NCG predicts Higgs mass and SM lagrangians but does it also explain some of the experimentally unexplained parameters of the SM like masses and 3 generations? Does it offer a DM candidate?

 

[marcus]

...
Anyway it should be very interesting to see where future work in this direction goes. Are there signs any of the established people working in either lqg or ncg such as Smolin or Connes might try to work with this new construction?

Coin, John Barrett is established. He runs the show at Nottingham. He directs a QG funding agency of the ESF (european science foundation). He has a prominent bunch of grad students and postdocs at Nottingham. He has Kirill Krasnov. Rovelli sends his PhDs to Nottingham to postdoc. Barrett organizes QG schools and organized Loops 2008. His group has supplied Rovelli's program with key results, proved key theorems. Barrett is a major LQG figure. ("barrett-crane spin foam"?)

Barrett was already doing NC geometry in 2006. The first Connes paper achieving the Standard Model in NCG, it came out the same week at a paper by Barrett proving the same result.
http://arxiv.org/abs/hep-th/0608221

This indicates a major established LQG player is interested in NCG, to the point of nearly scooping a bigtime Connes result. Barrett also invited Chamseddine (Connes' co-author) to be a plenary speaker at Loops 2008. (officially known as the "QG-squared" conference). Barrett's ESF funding agency as well as supporting LQG research, has also funded NCG workshops and schools.

I actually don't KNOW how close the links are. But I think there is more than a enough platform for Marcolli's gambit to take off from. If it is Nature's will that it be a fruitful line of research :-D

Right now if you want to look at the sociology I think you have to look at what the people are doing (the very good ones) that were born 1965 or later, and also keep an eye on the funding sources and the hirings.
LQG is in a phase of rapid growth. Many strong younger people, increasing output. New faculty hires in places you wouldn't think of like Sydney Australia, LSU Baton Rouge, Karlsruhe, Haverford. Thomas Thiemann now has his own LQG team at Erlangen, another new center. Daniele Oriti recently set up his own team at Potsdam.

If you do a spires search with the obvious keywords you get lqg/sf publications by year that go
https://www.physicsforums.com/showthread.php?p=2708501#post2708501

2005 40
...
...
2009 140
For details see the post I linked to.
If you know another field of fundamental physical theory with comparable growth over the same period, please let me know.
If I wanted to estimate the future I would not look at what the people that an outsider happens to know are doing.
I would look at what the first-rate younger people are doing: Laurent Freidel, Kirill Krasnov, Etera Livine, Eugenio Bianchi, Simone Speziale, Florian Girelli, Emanuele Alesci, Florian Conrady, Jon Engle,... too many to try to be complete. All these young people, a few already advising their own graduate students but many of them just beginning to publish on their own.

 

[atyy]

I actually don't KNOW how close the links are. But I think there is more than a enough platform for Marcolli's gambit to take off from. If Nature so wills, that it be a fruitful line of research :-D

How about via group field theory, the link is pretty close, don't you think?

spin foam - gft - ncg

spin foam - gft is linked by Reisenberger and Rovelli

gft - ncg is linked by Girelli, Livine and Oriti

 

[marcus]

How about via group field theory, the link is pretty close, don't you think?

spin foam - gft - ncg

spin foam - gft is linked by Reisenberger and Rovelli

gft - ncg is linked by Girelli, Livine and Oriti

Heh heh, Atyy we both know that you have a sharp eye and an impressive memory for the web of research. I defer to you about the LQG/NCG roadmap. Yes! Florian Girelli! I had forgotten his name. Recently hired at University of Sydney, which starts another seedling Lqg research center.

 

[Coin]

Marcus/ensabah thanks for the background on the paper authors.

 

[ensabah6]

Marcus/ensabah thanks for the background on the paper authors.

what works needs to be completed in combining LQG + NCG and what would be the name of this newer synthesis.

What are the unresolved issues currently LQG + NCG and do they affect previous results from either loop or ncg

in LQG + NCG, what are elementary partices in terms of spin networks, spin foam.
Does LQG + NCG role in derifiving the candidate Hamiltonian for dynamics.

 

[francesca]

LQG map

One practical consequence is that Caltech has joined the select list of North American institutions where a grad student has the chance of finding an advisor and doing research in LQG. There aren't many such places, so it adds intellectual degrees of freedom.

I've added it to my list, cheers! F
http://maps.google.com/maps/ms?ie=U...85216139270436.0004843830d27f3e6c50e&t=h&z=0"

 

[marcus]

A world map of LQG centers!
I hadn't realized how many of the new generation of researchers have already gotten permanent faculty positions. Now there are way more universities (than there used to be or that I knew about) where a grad student has the chance of finding an advisor and doing LQG research.

I like the world map visual, Francesca. I will add it to the list of handy links in the "Intro to LQG" thread.
https://www.physicsforums.com/showthread.php?p=2714192#post2714192

 

[marcus]

Marcolli came out with another paper today. Taken together with her recent Spinfoam/NCG paper, this makes it look to me as if she is leading her Caltech group in the direction of quantum cosmology.

http://arxiv.org/abs/1005.2256
The spectral action and cosmic topology
Matilde Marcolli (Caltech), Elena Pierpaoli (USC), Kevin Teh (Caltech)
(Submitted on 13 May 2010)
"The spectral action functional, considered as a model of gravity coupled to matter, provides, in its non-perturbative form, a slow-roll potential for inflation, whose form and corresponding slow-roll parameters can be sensitive to the underlying cosmic topology. We explicitly compute the non-perturbative spectral action for some of the main candidates for cosmic topologies, namely the quaternionic space, the Poincaré dodecahedral space, and the flat tori. We compute the corresponding slow-roll parameters and see we check that the resulting inflation model behaves in the same way as for a simply-connected spherical topology in the case of the quaternionic space and the Poincaré homology sphere, while it behaves differently in the case of the flat tori. We add an appendix with a discussion of the case of lens spaces."

I'm hoping someone here will explain what nonperturbative means in this context, so I've highlighted it.

 

[marcus]

Today a couple of new NCG papers came out, with a phenomenology angle. I put them on the bibliography thread and will get the abstracts here in case anyone wants to comment.

http://arxiv.org/abs/1005.4276
Gravitational Waves in the Spectral Action of Noncommutative Geometry
William Nelson, Joseph Ochoa, Mairi Sakellariadou
15 pages, 3 figures
(Submitted on 24 May 2010)
The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics on the same footing. The bosonic sector of the theory contains a modification to Einstein-Hilbert gravity, involving a nonconformal coupling of curvature to the Higgs field and conformal Weyl term (in addition to a nondynamical topological term). In this paper we derive the weak field limit of this gravitational theory and show that the production and dynamics of gravitational waves are significantly altered. In particular, we show that the graviton contains a massive mode that alters the energy lost to gravitational radiation, in systems with evolving quadrupole moment. We explicitly calculate the general solution and apply it to systems with periodically varying quadrupole moments, focusing in particular on the the well know energy loss formula for circular binaries.

http://arxiv.org/abs/1005.4279
Constraining the Noncommutative Spectral Action via Astrophysical Observations
William Nelson, Joseph Ochoa, Mairi Sakellariadou
5 pages
(Submitted on 24 May 2010)
The noncommutative spectral action extends our familiar notion of commutative spaces, using the data encoded in a spectral triple on an almost commutative space. Varying a rather simple action, one can derive all of the standard model of particle physics in this setting, in addition to a modified version of Einstein-Hilbert gravity. Thus, noncommutative geometry provides a geometric interpretation of particle physics coupled to curvature. In this letter we use observations of pulsar timings, assuming that no deviation from General Relativity has been observed, to constrain the gravitational sector of this theory. Thus, we directly constrain noncommutative geometry, a potential grand unified theory of physics, via astrophysical observations. Whilst the bounds on the coupling constants remain rather weak, they are comparable to existing bounds on deviations from General Relativity in other settings and are likely to be further constrained by future observations.

 

[marcus]

I've added it to my list, cheers! F
http://maps.google.com/maps/ms?ie=U...85216139270436.0004843830d27f3e6c50e&t=h&z=0"

That map is a really nice information resource, Francesca! I just updated the list of selected Lqg links here:
https://www.physicsforums.com/showthread.php?p=2812996#post2812996
and included your map.

There is also a list of LQG researchers worldwide in Wikipedia which I think may be in part your doing. Potential advisors for someone who wants to do a PhD thesis in the field.

 

[Orbb]

What's the status in NCG concerning the Higgs mass? The last model I've seen predicted a mass around 170 GeV, which is excluded by the Tevatron. How rigid is the Higgs mass prediction in Connes' model?

 

[marcus]

What's the status in NCG concerning the Higgs mass? The last model I've seen predicted a mass around 170 GeV, which is excluded by the Tevatron. How rigid is the Higgs mass prediction in Connes' model?

This bad prediction (and some mitigating rationale) is discussed in Connes Chamseddine latest paper, that came out this year. I will get the abstract and see if I can find where they mention it.
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
(Submitted on 3 Apr 2010)
"We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this notation is used to determine the spectral data of the standard model. The particle spectrum with all of its symmetries is derived, almost uniquely, under the assumption of irreducibility and of dimension 6 modulo 8 for the finite space. The reduction from the natural symmetry group SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that the two layers of space-time are finite distance apart but is non-dynamical. The square of the Dirac operator, and all geometrical invariants that appear in the calculation of the heat kernel expansion are evaluated. 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."

The mass of the Higgs is discussed on page 33.
I think you will want to see for yourself what they say.

 

[Orbb]

Thank you, marcus. I wonder why they didn't mention the sensitivity to higher-order corrections in the paper http://arxiv.org/PS_cache/hep-th/pdf/0610/0610241v1.pdf, where they first mentionned a prediction for the Higgs mass. Anyways, it will be interesting to see what Connes et al have to say and what will show up at the LHC.

 

[sbrothy]

Although it is way over my little head, I think the calculation (That's their Standard Model Lagrangian right?) they perform on page 36 is the meanest looking piece of mathematics I've ever seen. :)

 

[marcus]

This thread is about Marcolli's May paper, and I see now that I never actually copied in the abstract, to provide focus for discussion. Thanks to everyone who responded so far!

http://arxiv.org/abs/1005.1057
Spin Foams and Noncommutative Geometry
Domenic Denicola (Caltech), Matilde Marcolli (Caltech), Ahmad Zainy al-Yasry (ICTP)
48 pages, 30 figures
(Submitted on 6 May 2010)
"We extend the formalism of embedded spin networks and spin foams to include topological data that encode the underlying three-manifold or four-manifold as a branched cover. These data are expressed as monodromies, in a way similar to the encoding of the gravitational field via holonomies. We then describe convolution algebras of spin networks and spin foams, based on the different ways in which the same topology can be realized as a branched covering via covering moves, and on possible composition operations on spin foams. We illustrate the case of the groupoid algebra of the equivalence relation determined by covering moves and a 2-semigroupoid algebra arising from a 2-category of spin foams with composition operations corresponding to a fibered product of the branched coverings and the gluing of cobordisms. The spin foam amplitudes then give rise to dynamical flows on these algebras, and the existence of low temperature equilibrium states of Gibbs form is related to questions on the existence of topological invariants of embedded graphs and embedded two-complexes with given properties. We end by sketching a possible approach to combining the spin network and spin foam formalism with matter within the framework of spectral triples in noncommutative geometry."

 

[marcus]

I was reminded of this paper by a post from Haelfix in the thread about what REALLY disappoints one about string theorizing.

... the landscape problem of quantum gravity (really high energy physics) is not going to go away and will be a generic problem for any approach, even if they haven't studied or appreciated it yet.

String theory does better than any other current alternative on the market in this regard since it restricts what can come out of the low energy physics (the swampland).

Whereas a high energy theory of gravity that can arbitrarily couple any matter without constraints, will automatically have an (infinitely) worse landscape problem.

My question about this is can one really say with confidence that a landscape problem similar to having no selection principle amongst the 10⁵⁰⁰ string vacua MUST inevitably arise in any approach whatever?

In this particular approach, Marcolli's, can someone explain specifically how such a severe landscape problem must arise?

 

[marcus]

Happily enough, Haelfix elaborated. And Atyy concurred. We also have an arxiv link to check out. I think the claim is interesting (any QG+matter theory would suffer from a landscape predicament), but it doesn't really fit into the "REAL string disappointment" thread, so I will copy the posts here:

I would hope that a reader would realize that what I am saying is that in fact it DOES apply broadly to any tentative theory of matter.

What we now know, which wasn't necessarily appreciated a few years ago, is that a landscape of meta stable vacua is not just a truism about KKLT in string theory, but also a *generic* property of quantum field theory (and not just supersymmetric ones) including the standard model.

That is to say, any theory (under a reasonable set of conditions like having a reasonable scalar spectrum) that wishes to include gravity, and that contains a small cosmological constant and that possesses the standard model as an effective field theory at low energies also necessarily has a large landscape of metastable vacua.

Nima, Michael Dine and a few others have convincingly shown this, and I emphasize again that it is completely independant of the nature or properties of any tentative UV completion at high energies.

See for instance arXiv:hep-th/0703067

In fact, Haelfix's claim is self-evidently correct.

I'm hoping someone will explain. Let's have a look at http://arxiv.org/abs/hep-th/0703067

It is possible that Haelfix has already waffled or retracted what he said, since what he now seems to be saying has been much qualified by new conditions.
The new conditions may, if examined closely, presuppose that the QG+matter theory is to be imagined in some straitjacket format not mentioned in the original claim.

 

[marcus]

In fact, Haelfix's claim is self-evidently correct.

which one? regarding the landscape issue?

Yes, the one about a theory of gravity which can couple to any form of matter.

I'd like to shift discussion of Haelfix generic landscape claim over to this thread so as not to distract from the original thread's focus on string.

The claim as originally stated is:
The landscape problem of quantum gravity (really high energy physics) is not going to go away and will be a generic problem for any approach, even if they haven't studied or appreciated it yet.
...
[Any] high energy theory of gravity that can arbitrarily couple any matter without constraints, will automatically have an (infinitely) worse landscape problem.BTW it should be pointed out that the string landscape problem, really I think better termed a predicament, is the absence of a selection principle and despair of finding one. It is not simply that there are a range of solutions. It is quite possible for a theory to have adjustable parameters and to come in an infinity of versions---but not to run afoul on that account, given a clear route to selecting the right version.Haelfix offered an arxiv link---but the article does not address the main issue. It is an article from back in 2007 by Arkani-Hamed and others. It shows that the Standard Model+gravity has a range of versions (in lower dimensions than 4D) which can be easily excluded. There is no landscape problem/predicament. The article itself points out that (effective) SM+gravity has a unique 4D vacuum.

 

[marcus]

So the question is, here is an approach to QG+M and does it necessarily face a severe landscape problem (worse than string). I'm not sure you can prove that it faces any landscape problem at all, but anyway that's the issue.

AFAIK here is the latest word on the string landscape predicament:
http://pcts.princeton.edu/pcts/bigbang/Program-11-21-08.pdf

There was a November 2008 mini-symposium at Princeton with a panel discussion on the landscape (Tom Banks, Nathan Seiberg, Nima Arkani-Hamed...) Peter Woit attended and we have his notes on what each of the participants said. I will quote excerpts. The different views expressed were, I think, illuminating.

==portions from http://www.math.columbia.edu/~woit/wordpress/?p=1285 ==
The Landscape at Princeton
The Princeton Center for Theoretical Science has been having a mini-symposium on the string theory Landscape, and as part of this today hosted a “panel discussion” on the topic. It turns out that there’s not a lot of support for the Landscape in Princeton.

Michael Douglas was the only real Landscape proponent in evidence. He gave a presentation on the state of Landscape studies, beginning by noting that landscapeologists keep finding more possible string vacua. Evidently the 10^500 number always quoted for the number of semi-realistic vacua is no longer operative, with latest estimates more like 10^(10^5) or higher. Douglas acknowledged that this pretty much removes any hope of making predictions by using experiment to fix this freedom and end up with non-trivial constraints...

Tom Banks began by claiming that the effective field theory picture used in the landscape is just not valid. He also pointed out that if the landscape arguments were valid, the landscape would be disconfirmed by experiment, since 10-20 of the Standard Model parameters are unconstrained by anthropics, but take unusually small values, not the random distribution one would expect. Banks takes the attitude that the CC probably has an anthropic explanation, but not particle physics or the SM parameters. He also attacked the usual claims that different vacua are all states of the same theory, arguing that they instead correspond to different theories. Finally, he pointed out that the one prediction that landscapeologists had claimed they would be able to make, the scale of SSYM breaking, hadn’t worked out at all (Douglas now acknowledges that this can’t be done).

Nati Seiberg then argued that, as one gets to deeper and deeper levels of understanding of particle physics, one might reach a level where the only explanations are environmental and have to give up. He sees no reason for that to be the case now, with the main problem that of EWSB, and nothing to indicate that anthropics has anything to do with the problem. Rather, the problem is there because we haven’t had high enough energy accelerators (the LHC should change that), and the problem is hard. He ended by saying that the appropriate response at the present time to anthropic arguments like the Landscape is to just ignore them.

The last speaker was Nima Arkani-Hamed, who I suppose was chosen as a proponent of anthropics. He didn’t live up to this, saying that he pretty much agreed with Seiberg... About the cosmological issues brought up by Douglas, his opinion is that there’s probably no point to thinking about these questions now, doing so might be like trying to come up with a theory of superconductivity in 1903. As far as EWSB goes, he believes the LHC will show us a non-anthropic explanation for its scale.

He explicitly attacked the discussion of measures that Douglas had engaged in as “not fruitful”, saying that he didn’t see any “endgame”, that it was wildly improbably that these could predict anything about particle physics... He went on to argue for the currently fashionable enterprise of studying S-matrix amplitudes, arguing that looking at the local physics embodied in Lagrangians was no longer so interesting, that instead one should be trying to understand questions where locality is not manifest.

Finally, Arkani-Hamed ended with the statement that string theory is useful as a way to study questions about quantum gravity, but “unlikely to tell us anything about particle physics”. This is an opinion that has become quite widespread among theorists, but news of this has not gotten out to the popular media, where the idea that string theory has something to do with the LHC keeps coming up.

So, all in all, I found myself in agreement with most of the speakers...
==endquote==

 

[atyy]

My guess is that what we call landscape problem is due to the fact that we have one great achievement in string theory, namely that it turned theories into solutions / vacua. That is nice, but it does not make all the theories go away. There seems to be not so much difference whether there is a landscape of theories or a landscape of vacua :-)

I had in mind the landscape of theories in any theory of pure gravity that can couple arbitrarily to matter. This is the "bewildering number of possibilities" Percacci mentions in http://arxiv.org/abs/0911.0386"Let us assume for a moment that this ambitious goal can be achieved, and that pure gravity can be shown to be asymptotically safe. Still, from the point of view of phenomenology, we could not be satisfied because the real world contains also dozens of matter fields ... More generally the asymptotic safety program requires that the fully interacting theory of gravity and matter has a FP with the right properties. Given the bewildering number of possibilities, in the search for such a theory one needs some guiding principle."

From a condensed matter point of view, the landscape is not a problem - since condensed matter is all about studying the landscape!

 

[MTd2]

From a condensed matter point of view, the landscape is not a problem - since condensed matter is all about studying the landscape!

We, just have 1 universe to test!

 

[marcus]

I had in mind the landscape of theories in any theory of pure gravity that can couple arbitrarily to matter. This is the "bewildering number of possibilities" Percacci mentions in http://arxiv.org/abs/0911.0386
"Let us assume for a moment that this ambitious goal can be achieved, and that pure gravity can be shown to be asymptotically safe. Still, from the point of view of phenomenology, we could not be satisfied because the real world contains also dozens of matter fields ... More generally the asymptotic safety program requires that the fully interacting theory of gravity and matter has a FP with the right properties. Given the bewildering number of possibilities, in the search for such a theory one needs some guiding principle."
...

But Atyy, you have not offered any evidence of a landscape problem! Your paper shows Percacci immediately applying a readily available selection principle to narrow down the (comparatively modest) range of possibilities.

He may have used the word "bewildering" at the start of the paper to highlight the challenge he is undertaking, but he does not act in any way bewildered. Nor does he desperately invoke "anthropics" as Susskind did for string in 2003. He forges right ahead and applies the Wetterich method of "effective average action" ---see the blue highlight in the quote below.

At least wait until the AsymSafe people cry "Help!" before you conclude they are in trouble. Or until you see them thrashing aimlessly around.

Many seem prone to an unfortunate tendency, whenever one hears of some snag or drawback in the string program, to project it on all the other approaches and believe (or pretend) they have it worse. Either it's not a problem, or string doesn't really have the problem, or all the other approaches have it worse.

You have probably picked up on this. I recall a couple of years back a discussion of the "no background geometry" feature where I was told seriously that LQG was not background independent and that string was much more independent than everybody else (LQG, CDT).
There is a kind of compulsion to bend the words so that it always comes out like that.

Here's the context of the Percacci passage quoted above, in case anyone is interested:

==quote Percacci 0911. ==
The original motivation for this work comes fromthe progress that has been made in recent years towards understanding the UVbehaviour of gravity. It seems that pure gravity possesses a Fixed Point (FP) with the right properties to make it asymptotically safe, or in other words nonperturbatively renormalizable [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 39] (see also [20] for reviews). Let us assume for a moment that this ambitious goal can be achieved, and that pure gravity can be shown to be asymptotically safe. Still, from the point of view of phenomenology, we could not be satisfied because the real world contains also dozens of matter fields that interact in other ways than gravitationally, and their presence affects also the quantum properties of the gravitational field, as is known since long[21].

Indeed, in a first investigation along these lines, it was shown in [22] that the presence of minimally coupled (i.e. nonself-interacting) matter fields shifts the position of the gravitational FP and the corresponding critical exponents. In some cases the FP ceases to exist, so it was suggested that this could be used to put bounds on the number of matter fields of each spin.

More generally the asymptotic safety program requires that the fully interacting theory of gravity and matter has a FP with the right properties. Given the bewildering number of possibilities, in the search for such a theory one needs some guiding principle. One possibility that naturally suggests itself is that all matter self-interactions are asymptotically free[33]. Then, asymptotic safety requires the existence of a FP where the matter couplings approach zero in the UV, while the gravitational sector remains interacting.

We will call such a FP a “Gaussian Matter FP” or GMFP. Following a time honored tradition,as a first step in this direction, scalar self interactions have been studied in [34, 35]. Here we pursue that study further. The tool that we use is the Wetterich equation, an exact renormalization group (RG) flow equation for a type of Wilsonian effective action Γ_k , called the “effective average action”. This functional, depending on an external energy scale k, can be formally defined by...
==endquote==

 

[marcus]

So I guess there are three approaches to QG+matter that stand out, where one could ask "do they necessarily lead to intractable landscapes?"

Atyy recently posted about one, citing papers by Livine.
Essentially the approach is combined spinfoam+Feynman diagram:
https://www.physicsforums.com/showthread.php?p=2809599#post2809599

Approaches to QG+matter that don't seem to suffer landscape pains.

1. Spinfoam+Feynman single unified package---papers Atyy pointed to.

2. Spinfoam+NCG---this thread, there was a whole workshop devoted to this organized by Marcolli, so it is lots of people. Jesper Grimstrup was one of the early ones e.g. 2007.

3. AsymSafe QG+matter---Atyy also pointed me to this, citing a paper by Percacci, where he first acknowledges the complexity of coupling matter to QG, and then proceeds to attack it with the tools at hand. No sense that the range of variation is intractable or that one lacks principles of selection in this case.

 

[Haelfix]

Marcus, I think I have wasted enough of my time on this topic for now, and will not continue the discussion.

The reason I won't continue the discussion is because it is glaringly obvious that you have no idea what we are talking about and clearly lack the correct background to even begin to have a sensible conversation about the specifics.

Unfortunately, there are a lot of specifics regarding eg moduli stabilization and AdS/CFT, and I don't expect a layman to understand everything, however I do expect some amount of civility and some intellectual integrity to admit when you are way out of your depth.

I post on these boards to try to help people learn something about physics, including bits and pieces of the cutting edge material found in modern research. I don't post on these boards to get into arguments with amateurs who obfuscate facts with a wall of text and a bunch of utter nonsense. I'll leave viewers with the following quote by the authors of the paper.

"The phenomenon that a unique action may give rise to an infinite number of vacua is not a special feature of Superstring/SUSY theories, it is also a feature of the minimal Standard Model!"

 

[marcus]

"The phenomenon that a unique action may give rise to an infinite number of vacua is not a special feature of Superstring/SUSY theories, it is also a feature of the minimal Standard Model!"

However one does not see field theorists concerned with the Standard Model resorting to appeals to the Anthropic Principle.

By itself a large range or landscape does not constitute a problem. In some circumstances you can get an infinite number of solutions just by varying one parameter. String, however, has encountered a severe landscape predicament, which to many appears intractable. Your claim that any other treatment of QG+matter would necessarily meet with the same or worse predicament is unsupported by logic or evidence. I've explained this already and won't repeat here.

The 2007 paper Haelfix mentioned (I think pointlessly) is:
http://arxiv.org/abs/hep-th/0703067
in case anyone wants to check it out.
The passage quoted is on page 3, at the start of section 2:
==quote==
2. The Standard Model Landscape
We will now show that the action of the minimal Standard Model (SM) plus General Relativity (GR) has more than one distinct vacuum, actually a true landscape of vacua.
==endquote==

However, so what? the existence of extra vacua does not cause us trouble since we know how to use the SM predictively. It is a somewhat contrived "landscape" but it does not represent a landscape problem. One shouldn't have to say this.

 

[atyy]

But Atyy, you have not offered any evidence of a landscape problem! Your paper shows Percacci immediately applying a readily available selection principle to narrow down the (comparatively modest) range of possibilities.

He may have used the word "bewildering" at the start of the paper to highlight the challenge he is undertaking, but he does not act in any way bewildered. Nor does he desperately invoke "anthropics" as Susskind did for string in 2003. He forges right ahead and applies the Wetterich method of "effective average action" ---see the blue highlight in the quote below.

At least wait until the AsymSafe people cry "Help!" before you conclude they are in trouble. Or until you see them thrashing aimlessly around.

I didn't say AS was in trouble. But I do imply that similarly, string theory is not in trouble.