Understand hep-th/0502106: Get Help From Those Who Know It Better

[marcus]

the question on my mind is whether, if we start trying to understand
hep-th/0502106 will we occasionally get help from people who understand it better and who might look in sometimes?

this is something we cannot tell in advance and it depends to some extent on how capable we are on our own

so here is the situation. it is an important paper (as is the whole Ponzano Regge series I, II, III, by Freidel et al) and it has things I totally do not comprehend like "Non-Commutative Braided Field Theory". but there are some hopeful aspects of the situation:

1. Freidel et al are clear writers----look at the table of contents and the introduction of hep-th/0502106---it is very organized. Maybe we should call the paper P3 as short nickname for "ponzano regge model revisited III feynman diagrams and effective field theory"

2. It begins to include some matter----look at the end.

3. it makes things work out OK in 2+1 dimensions. the exciting possibility is that they might get the same program to work in 3+1 D.*

4. etera dropped in and posted a couple of very helpful clarifying posts in other threads.
==========

what I don't understand: I have not even begun to inventory this, there is too much. Like in section 4 there is the "star product". So instead of jumping around let's start at the the beginning: section 1 Introduction.
If anyone else wants to try reading this (and possibly getting some help from more knowledgeable folks) please post if you don't understand something about section 1.

* if they can get the same program to work in 3 + 1 D then curiously enough that seems to wrap up Quantum Gravity to a considerable extent.
something to think about.
(also remember the Freidel/Starodubtsev paper attempting a perturbative setup compatible with this)

 

[marcus]

here is paragraph one of the introduction:

"Spin Foam models offer a rigorous framework implementing a path integral for quantum gravity [1]. They provide a definition of a quantum spacetime in purely algebraic and combinatorial terms and describe it as generalized two-dimensional Feynman diagrams with degrees of freedom propagating along surfaces. Since these models were introduced, the most pressing issue has been to understand their semi-classical limit, in order to check whether we effectively recover general relativity and quantum field theory as low energy regimes and in order to make physical and experimental predictions carrying a quantum gravity signature. A necessary ingredient of such an analysis is the inclusion of matter and particles in a setting which has been primarily constructed for pure gravity. On one hand, matter degrees of freedom allow to write physically relevant diffeomorphism invariant observables, which are needed to fully build and interpret the theory. On the other hand, ultimately, we would like to derive an effective theory describing the propagation of matter within a quantum geometry and extract quantum gravity corrections to scattering amplitudes and cross-sections."

I bolded something that might need clarification or further elaboration. My understanding is that matter and space are so enmeshed that one cannot even define an AREA or a VOLUME in a purely abstract way, and have it be diffeo-invariant. One has to have some matter, like a desk, so that one can talk about the area of the desk-top.

In the familiar LQG setting the area and volume operators are defined before diffeos are factored out, so you don't have to have matter at that point. But here the authors are talking about "diffeomorphism invariant observables".

 

[etera]

I agree with you, there is no (simple) notion of area or volume at the diffeomorphism invariant level in a pure gravity theory, simply because points are moved around diffeomorphisms and the geometrical quantities are then moved around accordingly. Let's consider the length between two points. You need to be able to define the "points" in a diffeomorphic invariant way. It's like in the old hole paradox in general relativity described by Einstein... this is also well explained in Rovelli's review paper/book. Including matter in the theory makes things simpler. Consider two particles and a clock. The distance between these two particles at some given time as measured by an observer (you) is a diffeomorphic invariant object (basically because one have completely gauge-fixed the diffeomorphisms). Now the concept of physical particle in 3+1 gravity is problematic.. they are some attempts (by Smolin and Rovelli) of constructing some clock and coordinate systems using scalar fields or more general matter fields (some "old" papers between 90-95 and a more recent paper by Rovelli on GPS observables!). Also in 2+1d gravity where everything is more simple, particles are well-defined in quantum gravity and one can try to compute the spectrum of the length operator as defined between two particles. This is done in a paper last year by Noui&Perez and they find that the spectrum is as expected the same as the unphysical (non-diffeomorphism invariant) length operator in LQG. No doubt that this will be much harder to implement in 3+1d gravity...
i'm not sure whether i am making things clearer or more confused..

 

[marcus]

i'm not sure whether i am making things clearer or more confused..

clearer (by a considerable amount)

 

[marcus]

... Also in 2+1d gravity where everything is more simple, particles are well-defined in quantum gravity and one can try to compute the spectrum of the length operator as defined between two particles. This is done in a paper last year by Noui&Perez and they find that the spectrum is as expected the same as the unphysical (non-diffeomorphism invariant) length operator in LQG...

that is reassuring, and I do not imagine that it was trivial to show even in 2+1d.

In case anyone else is reading, I believe that the main Noui&Perez article that etera refers to is
http://arxiv.org/abs/gr-qc/0402111
Three dimensional loop quantum gravity: coupling to point particles
Karim Noui, Alejandro Perez
38 pages

Abstract: "We consider the coupling between three dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovelli's generalized projection operator which can be represented as a sum over spin foam amplitudes. In addition we provide an explicit expression of the (physical) distance operator between two particles which is defined as a Dirac observable."

there are also some shorter papers last year by Noui&Perez which I think recapitulate and enlarge on the work of this main paper
http://arxiv.org/find/grp_physics/1/AND+au:+noui+au:+perez/0/1/0/all/0/1
in particular, about the length operator spectrum:

http://arxiv.org/abs/gr-qc/0402113
Observability and Geometry in Three dimensional quantum gravity
Karim Noui, Alejandro Perez
6 pages
To appear in the procedings of the Third International Symposium on Quantum Theory and Symmetries (QTS3), September 2003

Abstract: "We consider the coupling between massive and spinning particles and three dimensional gravity. This allows us to construct geometric operators (distances between particles) as Dirac observables. We quantize the system a la loop quantum gravity: we give a description of the kinematical Hilbert space and construct the associated spin-foam model. We construct the physical distance operator and consider its quantization."

 

[marcus]

... Let's consider the length between two points. You need to be able to define the "points" in a diffeomorphic invariant way. It's like in the old hole paradox in general relativity described by Einstein... this is also well explained in Rovelli's review paper/book. Including matter in the theory makes things simpler. Consider two particles and a clock. The distance between these two particles at some given time as measured by an observer (you) is a diffeomorphic invariant object (basically because one have completely gauge-fixed the diffeomorphisms)...

I am still explicating to myself what is in etera's post.
I downloaded the first two papers in the series "Ponzano 1" and
Ponzano 2". they are related to what is talked about here.

the introduction to "Ponzano 3" makes clear the connection:

---quote from P3 introduction---
... Recently, this model has been studied in much detail. The paper ["Ponzano 1"] has tackled particle insertions and shown how they can be understood as a partial gauge fixing of the state sum model. As a result, explicit quantum amplitudes for massive and spinning particles coupled to gravity were constructed. The work in ["Ponzano 2"] established an explicit link between the Ponzano-Regge quantum gravity and the traditional Chern-Simons quantization, and identified a kappa-deformation of the Poincaré group as the relevant symmetry group of spin foam amplitudes. In this article we follow the same line of thoughts focusing on the relation between spin foam amplitudes and usual Feynman graph evaluations...
---end quote---

All I did here was put in the nicknames for the papers so it would be obvious what prior work was put into what context. In case anyone wants to look at these two prior papers (which are by Freidel and David Louapre) here they are:

http://arxiv.org/abs/hep-th/0401076
Ponzano-Regge model revisited I: Gauge fixing, observables and interacting spinning particles

http://arxiv.org/abs/gr-qc/0410141
Ponzano-Regge model revisited II: Equivalence with Chern-Simons

for completeness, here is the link to the paper (hep-th/0502106) being discussed in this thread

http://arxiv.org/abs/hep-th/0502106
Ponzano-Regge model revisited III: Feynman diagrams and effective field theory

 

[marcus]

it is considered bad manners for scientists to speculate too much but
as a science-WATCHER, which is different, I don't have to obey that restriction

so I will speculate as to what it might look like if the program in this
hep-th/0502106 (nicknamed "Ponzano 3") works out, as it looks to me now that it will

I am going to copy from the 0502106 "Conclusions" section and try to see where it points.

but first I want to remark that their "Section 3: Particle insertions in Ponzano-Regge Spinfoam Gravity" is the closest thing in current quantum gravity research to seeming on-track to a watcher like me
because it offers a mechanism by which matter can curve space.

they put the feynman diagram for some matter into the spinfoam, and the MASS of the particle is associated with a certain ANGLE and, well isn't that exactly what we were all this time waiting for?, when you go around in the spin network, or (in spacetime) the spin foam, you get a DEFICIT ANGLE, and a deficit angle where the triangle vertices do not add up to 180 degrees is what we originally wanted for CURVATURE.

so Freidel/Livine put some mass, riding on a feynman diagram, into their spinfoam, and the mass makes some curvature.

this is what the einstein equation was telling us
"curvature = energy density"
but it never told us how it happens!

So, if you are like me, and you are curious about how it can happen, by what mechanism, that the mass-energy density can bend the geometry of space relations, then look at this Section 3, of Ponzano 3 paper, from page 9 to page 14.

and if you are like me you will understand almost nothing of it. but there is a mechanism for matter bending space that begins to show itself. of course the word mechanism is like an onion, that is to say that it is never used except in a provisional sense, but nonetheless it is a mechanism just as an onion is an onion

Now we should look at the Conclusion section starting page 37

[Freidel/Livine have carelessly forgotten to include Conclusion in their TOC on page 2, so you must flip pages to find this important section of the paper]

BTW maybe the most important thing about this paper, or one of the things, is that it offers to remove the ambiguity in DSR
this sounds technical, but DSR could really be much more interesting if there were only one way to implement it