Newton's Law starting from no background spacetime

[marcus]

Here's a bit from the Conclusion section of the April 2006 Rovelli et al paper
http://arxiv.org/abs/gr-qc/0604044

==quote==
7 Conclusion
We have computed a first and a second order term in the expansion in λ of the diagonal components of the graviton propagator, in a large distance regime, starting from a background–independent formulation of quantum gravity.

The result has the 1/|x−y|² dependence on the distance expected from the linearized quantum theory, the expected dependence on the physical constants, and the numerical
proportionality constants can be fixed as a condition on the semiclassical boundary state.

The main tool we have used is the definition of general covariant n-point function, given in (49). Many issues remain open. Among these are: the calculation of non-diagonal terms in the propagator [47];...the physical interpretation of the numerous subdominant terms, and their relation with the relativistic and the quantum corrections to the Newton law [9]; ...

In our opinion, the interest of the calculation is not so much in the final agreement with the linearized expression or in the details of the model used, but rather in the fact that it shows how some low-energy quantities with a transparent physical meaning can be computed, starting from the abstract context of a background independent formalism.

The specific choices of ingredients used for the calculation is therefore of lesser interest, in our opinion, that the display of the feasibility of calculations of this sort.
==endquote==

This is the result which Rovelli paraphrased informally sometime later by saying
We have calculated Newton's Law starting from a world with no space and no time.

I realized recently that this particular statement could sound extravagant to someone encountering it unprepared, but in fact this is exactly what was done by starting from a background independent formalism. Since there is no prior background geometry, there is no space and time as we know them. It is not immediately obvious how to describe even the simple situation where there are two points a certain distance apart.

However this was done, and much more. So since it is a recent development, which may strike those unfamiliar with the business as surprising, I will list some of the landmark papers. The paper just quoted was:

Graviton propagator in loop quantum gravity
Eugenio Bianchi, Leonardo Modesto, Carlo Rovelli, Simone Speziale
41 pages, 6 figures
published in Class.Quant.Grav. 23 (2006) 6989-7028
(Submitted on 10 Apr 2006)

"We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter."

 

[marcus]

To get Newton's law (starting from no space and no time) what one needs to get right are the DIAGONAL terms in the graviton propagator. Interestingly enough, the spinfoam model which Rovelli's team were using in 2006 gave the wrong off-diagonal terms so that was not the end of the story.

In 2007 they made a correction in the model and posted several more papers. Here is quite a recent one:

http://arxiv.org/abs/0711.1284
The complete LQG propagator: II. Asymptotic behavior of the vertex
Emanuele Alesci, Carlo Rovelli
(Submitted on 8 Nov 2007)

"In a previous article we have shown that there are difficulties in obtaining the correct graviton propagator from the loop-quantum-gravity dynamics defined by the Barrett-Crane vertex amplitude. Here we show that a vertex amplitude that depends nontrivially on the intertwiners can yield the correct propagator. We give an explicit example of asymptotic behavior of a vertex amplitude that gives the correct full graviton propagator in the large distance limit."

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

For historical interest, the first in this series of papers was a brief report which appeared in August 2005 and was published in Physical Review Letters in 2006
http://arxiv.org/abs/gr-qc/0508124
Graviton propagator from background-independent quantum gravity
Carlo Rovelli
6 pages
Phys.Rev.Lett. 97 (2006) 151301
(Submitted on 30 Aug 2005)

"We study the graviton propagator in euclidean loop quantum gravity, using the spinfoam formalism. We use boundary-amplitude and group-field-theory techniques, and compute one component of the propagator to first order, under a number of approximations, obtaining the correct spacetime dependence. In the large distance limit, the only term of the vertex amplitude that contributes is the exponential of the Regge action: the other terms, that have raised doubts on the physical viability of the model, are suppressed by the phase of the vacuum state, which is determined by the extrinsic geometry of the boundary."

I have to go, will try to get back to this and make some comments.
Some people may be interested how one can have a background-independent model of quantum gravity. How can one get started without some prior fixed choice of a background geometry? I'll try to think how to say that. Anybody else's helpful comments about background independence would be very welcome.

 

[marcus]

This result (Newton's Law starting from no space and no time) is something Loop has that Triangulations doesn't yet.

I'd like to see Triangulations get Newton's Law this year

Or another nice thing would be if Triangulations researchers would put matter in and get a cosmological bounce

The Loop approach (including Spinfoam and Group Field Theory) is much older, more highly developed, and has about 20 times as many people working on it----compared with Triangulations.

Triangulations only got started in 1998 and that was with a lower dimensional case. They didnt get 4D to emerge until 2004. So doing triangulations QG in 4D is only 3 years old.
It is a newer approach and in a certain sense it is more radical.

Radical because it starts with even less. Both start with a GEOMETRYLESS CONTINUUM but in the Triangulations case it doesn't even have enough structure to determine its dimension.

So when you get a quantum spacetime solution to happen, dimensionality is something you OBSERVE. It is a quantum observable you measure at some given location, at various scales. The continuum can have dimension that changes continuously with scale---be a 1.9D or 2.1D foam at small scale, while appearing smooth at larger. At sub-planckian scale it can be foamy and yet at a scale of 10 or 20 Planck lengths it can be smoothly 4D.

Smoothness is EMERGENT, and likewise the appearance of whole-number dimensionality.

Loop starts with more, in a sense, because even though it does not start with a geometry---there is no prior distance function---it does start with a smooth continuum of some specified whole-number dimensionality.

So Triangulations is behind Loop in almost every way. Loop cosmology is more developed---it has a bounce replacing the cosmological singularity. Loop black hole models are more developed. Loop has Newton's Law.

Triangulations lags behind because it has really only been in the 4D business since 2004, and because it has fewer people working on it, and because it is in certain respects more radical (starts with less structure on board.)

So I'd like to see Triangulations catch up to Loop in certain respects. 2008 seems like a good year for that to happen.

BTW Bianca Dittrich's 42 minute video called "Introduction to Quantum Gravity" is remarkably informative.
After watching it three times, I am seeing QG from a distinctly new perspective.

It was a Perimeter seminar talk around 6 December 2007, with Lee Smolin, Leonard Susskind, and several other persons of note plying her with questions. The talk was for non-specialists---people from other branches of physics who needed an introductory survey of QG. She made the talk clear and basic so that string theorists, for example, would have no trouble understanding. There seemed to be a lot of them in the room.

If anyone else wants the same kind of introduction, here is the link to the video:
http://pirsa.org/07120030
just click on "windows media" and wait a minute while it loads.