# Low background quantum geom/grav (map and current situation)

1. Dec 12, 2008

### marcus

What I mean by QG (quantum geometry = quantum gravity) is an approach that provides some type of quantum universe model.
So at a minimum there is a mathematical model of the continuum, with geometric measurements (such as area, volume, the dimensionality at a given location) represented by operators, as quantum observables. The quantum state of the universe's geometry appears in some form, in the theory. In a relativist context the geometry is the same thing as the gravitational field which is where the (QG = QG) equation comes from.

Periodically we try to summarize and report what the current map of QG looks like and what the main directions are. There have been several past threads about this, but they are way out of date. The field has been moving along pretty fast and the job of scoping it out has been getting easier because there are signs of convergence. The main initiatives are Loops QG, Loll's Blocks QG, Reuter's Fixedpoint QG, and Connes' Noncommutative Geometry.

The LOW BACKGROUND idea is, I think, well understood. To make a mathematical model of the continuum, or to represent dynamically changing geometry, you need some mathematical ground to build on. There is a choice of what and how much structure you invoke at the start, to launch the construction.

For example you can start with a topological space like S3 that doesn't even come with coordinates: no differential structure, no implied smoothness! No geometry either. Geomety only comes if you further impose a metric, a distance function, or reasonable facsimile thereof.

So maybe your theory doesn't get off the ground with a mere topological space, and you require more---eg some smooth differential structure, coordinates, calculus tools.

Or you can assume even more---a fixed metric etc. (The approaches we're discussing here don't do that.)
=========================

I believe General Relativity was the first physics theory to be low background in the sense that it does not require a fixed geometry to get started. It builds on a differential manifold---there are indeed coordinates with some smooth structure allowing calculus---but the manifold is limp and shapeless. It has no metric--no geometry.

In GR the metric arises dynamically, as a solution to the main equation. So for the first time, a physical theory is constructed without predetermined geometry. The gravitational field is represented by the geometry itself. Since the only thing that matters then is the field, it is abstracted, freed of dependence on any particular manifold, and becomes the focus of attention. The manifold? a temporary convenience, downplayed and easily replaced.

Low background QG approaches follow the lead of GR in that none of them declare a metric. In some respects they may be even leaner than their role-model GR. In any case there is no background spacetime, in the sense of a geometric setup.

How could the initial background be made even leaner?--even more barren of structure. If you take a limp differential manifold (like what GR starts with) having, as we said, no metric geometry specified, and want to make it even more nondescript, you can strip off the coordinates and the differential calculus structure. Then all you have left is a topological space. That, for example, is what Loll's building block QG starts with. Typically spacetime is treated as S3 x R1, and a geometry is imposed on it by triangulating it with equal size blocks. A spacetime geometry is viewed as a path of evolution from some initial spatial configuration to a final one. Each path has an amplitude. The approach is closely analogous to a Feynman path integral.

Last edited: Dec 12, 2008
2. Dec 12, 2008

### marcus

Anyway what stands out currently is four main approaches to quantum gravity or quantum geometry (QG = QG). And I will list the best most up-to-date sources I know of, and also try somewhat to see how they stand in relation.

There's Alain Connes' Noncommutative approach
excellent recent comprehensive statement http://arxiv.org/abs/0812.0165
The Uncanny Precision of the Spectral Action
a helpful simplified but somewhat older note http://arxiv.org/abs/0706.3690
Conceptual Explanation for the Algebra in the Noncommutative Approach to the Standard Model
Nelson et al application to cosmology http://arxiv.org/abs/0812.1657
Cosmology and the Noncommutative approach to the Standard Model
(Nelson's Noncom Cosmology is potentially very interesting, the paper is a first.)

There's Loop QG
Of course Rovelli's book, but just sampling a few recent papers we have
Claudio Perini, Carlo Rovelli, Simone Speziale http://arxiv.org/abs/0810.1714
Self-energy and vertex radiative corrections in LQG
"We consider the elementary radiative-correction terms in loop quantum gravity. These are a two-vertex "elementary bubble" and a five-vertex 'ball'; they correspond to the one-loop self-energy and the one-loop vertex correction of ordinary quantum field theory. We compute their naive degree of (infrared) divergence."
Simone Speziale http://arxiv.org/abs/0810.1978
Background-free propagation in loop quantum gravity
"I review the definition of n-point functions in loop quantum gravity, discussing what has been done and what are the main open issues. Particular attention is dedicated to gauge aspects and renormalization."
Laurent Freidel and Florian Conrady http://arxiv.org/abs/0806.4640
Path integral representation of spin foam models of 4d gravity
To cover the Cosmology angle, Ashtekar http://arxiv.org/abs/0812.0177 is good
Loop Quantum Cosmology: An Overview

Then there's Blocks QG (Loll's triangulation approach)
I have to recommend their SciAm article---for a non-math introduction it's remarkably enlightening. Link's in my sig.
For a little heavier going, but still accessible, there's http://arxiv.org/abs/0711.0273
The Emergence of Spacetime, or, Quantum Gravity on Your Desktop
and http://arxiv.org/abs/0807.4481
The Nonperturbative Quantum de Sitter Universe

To complete the list of what seem to be the current main contenders, there's Asymptotically Safe QG
Martin Reuter and Roberto Percacci are major proponents. Here's a recent Reuter http://arxiv.org/abs/0811.3888
Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity
(You can see that Reuter may have settled on ASQG as a name for this approach, earlier called QEG.)
Here are two earlier papers, a survey http://arxiv.org/abs/0708.1317
Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity
and one covering the cosmology angle http://arxiv.org/abs/0803.2546
Primordial Entropy Production and Lambda-driven Inflation from Quantum Einstein Gravity
Roberto Percacci also has a valuable survey: http://arxiv.org/abs/0709.3851
Asymptotic Safety

In addition to listing these sources, I want to cover some recent progress, what has happened recently in these approaches, and also say something about how some of them relate. I'm running out of time for today though.

Last edited: Dec 13, 2008
3. Dec 14, 2008

### atyy

Another way to think about LQG, CDT and Asymptotic Safety for people like me who don't understand why background independence is important is that these approaches consider that no additional degrees of freedom are needed, and no supersymmetry. Maybe related is Bern's speculation that the plain old unsupersymmetric Einstein-Hilbert action may be better behaved than previously expected.

Unexpected Cancellations in Gravity Theories
Z. Bern, J.J. Carrasco, D. Forde, H. Ita, H. Johansson
http://arxiv.org/abs/0707.1035

Last edited: Dec 14, 2008
4. Dec 15, 2008

### marcus

Thanks for the Zvi Bern et al reference. I had seen other thngs about finiteness of e.g. D=4, N=8 Supergravity, but hadn't registered the interesting speculation that ordinary non-super gravity might also be better behaved than was thought. I sampled more recent papers by the authors---impressive and prolific people---but couldn't find followup on this particular theme.

No-frills QG Maybe that is the point. And not going background lite.
I'd be interested if you have a way of explaining why background independence doesn't seem important to you. I might learn something.

We should explore the idea that reduced frills is what unites some of these alternative QG approaches. A moral lesson one can draw from string theoretics is that frills cause trouble. The great variety of possible ways one can set up a manifold with curled extra dimensions is the direct cause of the Landscape calamity which string has suffered in recent years---the Umpteen Vacuums predicament ---and according to Steinhardt the extra dimensions may be incompatible with both past inflation and current accelerated expansion. The approaches of Loll, Rovelli, and Reuter all share a no-frills character---as you pointed out.

I just have trouble fitting Connes in. This is a chronic difficulty. It's difficult for me to fit the Noncomm approach into any broader picture.

Last edited: Dec 16, 2008
5. Dec 16, 2008

### MTd2

There is also this article from Dirk Kreimer (2007) http://arxiv.org/abs/0705.3897v1 has a very interesting conclusion:
"Our analysis suggests that gravity, regarded as a probability conserving but perturbatively
non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson–Schwinger equations."

This is Urs Schreiber's comment on a talk of this guy:

"Kreimer is making heavy use of a reformulation of perturbative quantum field theory using the Hopf algebra structure on the space of Feynman graphs which he developed together with Alain Connes. I once mentioned the definition of that at the end of Graphs, Operads and Renormalization.

This language, which relates many computations in perturbative quantum field theory to certaimn Hochschild cohomologies, apparently allows him to see some patterns otherwise not as easily visible.

He commented on how renormalizable and non-renormalizable QFTs look like from this Hopf algebraic point of view. He emphasized that while (say 4-dimensional, pure) perturbative Einstein gravity is of course non-renormalizable – but very special among all non-renormalizable field theories.

The upshot was that gravity is, that’s how Kreimer put it, dual to a renormalizable theory. It behaves like a renormalizable field theory if we think of amplitudes and loop number interchanged.

When asked if this is related to recent results which found unexpected hidden cancellations in perturbative quantum gravity, he said he didn’t know the precise connection, but that one would expect this to be related."

http://golem.ph.utexas.edu/category/2007/03/recent_developments_in_quantum.html#c010903

Last edited: Dec 16, 2008
6. Dec 16, 2008

### marcus

That's really interesting. Thanks. It may be difficult for me to understand but I will certainly have a look at the links you gave.

7. Dec 17, 2008

### MTd2

Hi Marcus. I don't understant that either. But I posted here so that we can learn together. :)

8. Dec 17, 2008

### marcus

This puts Dirk Kreimer on the map for me. I've been aware of the name for some time but hadn't tried reading any of his papers. This is definitely something to keep an eye on, and try to keep track of its development. We can follow up by seeing what other papers cite this one
http://arxiv.org/abs/0705.3897v1
http://arxiv.org/cits/0705.3897v1

Just now I tried "cits" and found
http://arxiv.org/abs/0805.3438
Perturbative Gravity in the Causal Approach
50 pages
(Submitted on 22 May 2008)
"Quantum theory of the gravitation in the causal approach is studied up to the second order of perturbation theory. We prove gauge invariance and renormalizability in the second order of perturbation theory for the pure gravity system (massless and massive). Then we investigate the interaction of massless gravity with matter (described by scalars and spinors) and massless Yang-Mills fields. We obtain a difference with respect to the classical field theory due to the fact that in quantum field theory one cannot enforce the divergenceless property on the vector potential and this spoils the divergenceless property of the usual energy-momentum tensor. To correct this one needs a supplementary ghost term in the interaction Lagrangian.

and a follow-up by Kreimer himself:
http://arxiv.org/abs/0805.4545
Not so non-renormalizable gravity
Dirk Kreimer
7 pages
(Submitted on 29 May 2008)
"We review recent progress with the understanding of quantum fields, including ideas how gravity might turn out to be a renormalizable theory after all."

I see that on page 5 of "Not so non-", Kreimer cites Martin Reuter ASQG (asymptotically safe quantum gravity)---references [15][16].
==quote Kreimer==
...In particular, if we compute in a space of constant curvature and conformally reduced gravity which maintains many striking features
of asymptotic safe gravity [15, 16], the above identities should hold for suitably deﬁned characters: indeed, in such circumstances we can renormalize using a graviton propagator which is eﬀectively massive..."
==endquote==

Reuter is another one who is arguing that gravity is not so nonrenormalizable as people used to think. I will take a look at the Urs Schrieber comment that you mentioned.

Last edited: Dec 17, 2008
9. Dec 17, 2008

### atyy

Well, I guess the "hole argument" which Rovelli uses to motivate "background independence" doesn't seem very profound to me. In Newtonian or special relativistic physics, the metric is a system of clocks and rulers which we set up "independent" of our experiment, and Galilean or Lorentz transforms are special because they are isometric. The difference in GR is that the metric is dynamic, so a diffeomorphism applied to dynamic objects automatically generates an isometry, whereas in Newton and SR, the metric is static, so a diffeomorphism applied to dynamic objects need not generate an isometry. But the requirement that isometric situations have the same physics doesn't seem special to GR.

The second reason is that Rovelli's solution to the hole argument as preserving point coincidences seems to me very background dependent - since the point coincidences are those of test particles propagating on a fixed background.

I guess the annoying thing (in a fabulous way ) about GR is that the configuration of matter and metric (which being rods and clocks is also matter) must be determined self consistently. So in this sense, condensed matter (Sakharov, Visser, Volovik, Wen) or AdS/CFT approaches in which metric and matter emerge together are fascinatingly "background independent", whereas LQG which quantizes pure spacetime is very "background-dependent".

However, a way to argue for "pure spacetime" is that in SR we can use orthogonal EM plane wave solutions to set up our background metric for a non-EM experiment. What GR would say is that this is impossible, since these different plane waves must interact gravitationally, and Maxwell's equations should become nonlinear. In EM (no gravity), the usual way to make the Maxwell field nonlinear so that "plane waves" interact is to introduce the Dirac field. So it makes sense that that gravity is just another field which makes Maxwell's equations nonlinear. The difference is that the "pure" Dirac field is linear, whereas the "pure" gravitational field is nonlinear. From this point of view, LQG, CDT and Asymptotic Safety which quantize "pure" spacetime do make sense to me.

10. Dec 19, 2008

### marcus

Thanks, seeing how you are thinking about it gives me some new perspective. It hadn't occurred to me that Rovelli might somehow be using the hole argument as motivation for loop's BI premise. (That's only one thing. You made several unfamiliar points.)

The hole argument is interesting in its own right, I think. Einstein discovered it in 1913 and it played a role in his eventual formulation of GR in 1915. The Stanford website SEP gives some detail. Up to now I've considered the hole argument as having no direct connection with BI but rather involving general covariance and the contingency of space. This is what the SEP explains, and also the wikipedia article on hole argument.
http://plato.stanford.edu/entries/spacetime-holearg/
http://en.wikipedia.org/wiki/Hole_argument

The brief account is that Newton attributed objective reality to space. Thought if you took away all matter it would still be there. He substantiated space, and time. The hole argument seems to indicate that one should not substantiate space or time. The hole argument is based on general covariance. As AE put it in 1916, the principle of general covariance deprives space and time of "the last shred of physical reality".

This is not all that different from something you were saying---seems in line with it actually. The idea that one focuses attention not on space and time but on the metric and that operationally-defined the metric involves matter---rulers and clocks.

But I haven't sorted out how this relates to what I understand by BI, which is simply that the theory be developed without relying on a fixed metric manifold. No metric, no geometry put in by hand.

I think Loop gravitists actually require several things of theory, related but slightly different--perhaps three main requirements are these:

being nonperturbative
general covariance (also called "diffeomorphism invariance")
background independence (no metric put in by hand)

Nonperturbative almost says BI and makes the requirement redundant. Because the gravitational field is the metric, so if you deal with it perturbatively you are fixing on a background metric and perturbing around it. The metric becomes a large fixed piece (e.g. flat Minkowski) plus a small dynamic piece. So any perturbative approach necessarily has a fixed metric put in by hand, and is not BI. Therefore BI implies nonperturbative, but not conversely. It may be that in all or most interesting cases the converse holds---that nonperturbative almost implies BI.
Just have to spell this out. I'm sure I'm not telling you anything new, atyy, but there may be other readers.

BTW this year Rovelli posted an essay called Loop Quantum Gravity at MaxPlanckInstitute Living Reviews.
http://relativity.livingreviews.org/Articles/lrr-2008-5/
It is a completely revised update of a review article he wrote for them in 1998.
It is the clearest, most authoritative short account of the LQG program that I know.
This would be a good way to get clear about basic features, requirements, and underlying thought.

Last edited: Dec 19, 2008
11. Dec 19, 2008

### marcus

I want to focus on five areas and identify representative recent work. I'll start with the earlier list and remove all papers that are not from this year. So this is like the earlier list but 2008 only. Where possible I'll indicate applications to cosmology, since that's apt to be the crucial test-bed for theory.
==================================

Noncom geometry/gravity:

The Uncanny Precision of the Spectral Action
http://arxiv.org/abs/0812.0165

Cosmology and the Noncommutative approach to the Standard Model
http://arxiv.org/abs/0812.1657
=========

Loops:

Loop Quantum Gravity (Rovelli's 2008 review in 45 pages plus bibliography)
http://relativity.livingreviews.org/Articles/lrr-2008-5/

Self-energy and vertex radiative corrections in LQG
http://arxiv.org/abs/0810.1714

Background-free propagation in loop quantum gravity
http://arxiv.org/abs/0810.1978

Path integral representation of spin foam models of 4d gravity
http://arxiv.org/abs/0806.4640

A Lagrangian approach to the Barrett-Crane spin foam model
http://arxiv.org/abs/0812.3456

Loop Quantum Cosmology: An Overview
http://arxiv.org/abs/0812.0177
=================

Blocks:

The Nonperturbative Quantum de Sitter Universe
http://arxiv.org/abs/0807.4481

SciAm article
Link's in my sig.
=================

Asymptotically Safe:

Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity
http://arxiv.org/abs/0811.3888

Primordial Entropy Production and Lambda-driven Inflation from Quantum Einstein Gravity
http://arxiv.org/abs/0803.2546
==================

Reformed gravity:

Motion of a “small body” in non-metric gravity
http://arxiv.org/abs/0812.3603

Plebanski gravity without the simplicity constraints
http://arxiv.org/abs/0811.3147

Modified gravity without new degrees of freedom
http://arxiv.org/abs/0812.3200

see MTd2's recent posting on the Krasnov non-metric gravity thread https://www.physicsforums.com/showthread.php?t=158899

Last edited: Dec 20, 2008
12. Dec 22, 2008

### atyy

Well, my present perspective is influenced by several sources. My personal favourite is Wen, but that's just because I cannot resist his hypothesis that we live in noodle soup! :rofl: The other bit of Wen that I like, that Rovelli also mentions is - no final theory - "no end to the richness of nature". To be a bit more serious, among LQG researchers, I very much like the points of view that Markopoulou and Yidun Wan have been evolving. I found Rovelli's review interesting for his discussion of the subtly different emphases within LQG, and see that the speculations particularly aesthetic to me should also be associated with Smolin and Bilson-Thompson.

13. Dec 30, 2008

### marcus

Atyy, I see you are interested in Yidun Wan's work. He wrote several interesting papers this year. Partly on the strength of your interest I have included one of them in this list of candidate papers for this year's MIP poll (most influential/valuable QG paper)

The Uncanny Precision of the Spectral Action
Ali H. Chamseddine, Alain Connes
http://arxiv.org/abs/0812.0165

Loop Quantum Cosmology: An Overview
Abhay Ashtekar
http://arxiv.org/abs/0812.0177

Loop Quantum Gravity
Carlo Rovelli
(new review)
http://relativity.livingreviews.org/Articles/lrr-2008-5/

LQG propagator: III. The new vertex
Emanuele Alesci, Eugenio Bianchi, Carlo Rovelli
http://arxiv.org/abs/0812.5018

"In the first article of this series, we pointed out a difficulty in the attempt to derive the low-energy behavior of the graviton two-point function, from the loop-quantum-gravity dynamics defined by the Barrett-Crane vertex amplitude. Here we show that this difficulty disappears when using the corrected vertex amplitude recently introduced in the literature. In particular, we show that the asymptotic analysis of the new vertex amplitude recently performed by Barrett, Fairbairn and others, implies that the vertex has precisely the asymptotic structure that, in the second article of this series, was indicated as the key necessary condition for overcoming the difficulty."

Self-energy and vertex radiative corrections in LQG
Claudio Perini, Carlo Rovelli, Simone Speziale
http://arxiv.org/abs/0810.1714

Background-free propagation in loop quantum gravity
Simone Speziale
http://arxiv.org/abs/0810.1978

On the semiclassical limit of 4d spin foam models
Florian Conrady, Laurent Freidel
http://arxiv.org/abs/0809.2280

Path integral representation of spin foam models of 4d gravity
Florian Conrady, Laurent Freidel
http://arxiv.org/abs/0806.4640

A Lagrangian approach to the Barrett-Crane spin foam model
Valentin Bonzom, Etera R. Livine
http://arxiv.org/abs/0812.3456

The Nonperturbative Quantum de Sitter Universe
J. Ambjorn, A. Goerlich, J. Jurkiewicz, R. Loll
http://arxiv.org/abs/0807.4481

Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity
Elisa Manrique, Martin Reuter
http://arxiv.org/abs/0811.3888

Motion of a “small body” in non-metric gravity
Kirill Krasnov
http://arxiv.org/abs/0812.3603

Effective Theory of Braid Excitations of Quantum Geometry in terms of Feynman Diagrams
Yidun Wan
http://arxiv.org/abs/0809.4464

Last edited: Dec 30, 2008