BF theory is basic to LQG so we should learn about it

[marcus]

Broda is an important figure in the history of BF theory
and Broda has just written an encyclopedia entry on BF
which he posted today
BF system—encyclopedic entry
B. Broda
http://arxiv.org/hep-th/0502045

I don't know if this is good or not but it is short and I am going on Broda's reputation, and altho I'm not qualified to judge, it looks OK to me

He give four references, one of which is to a John Baez lecture notes published in 2000.

Near the end Broda says
"One of the most exciting applications of BF systems is gravity [4]. 3D gravity is directly identified with a BF system, and the cosmological term plays the role of the cosmological term in gravity. 4D BF system can be a starting point for a novel formulation of 4D gravity (direction pioneered by Plebanski). "

his reference [4] is to the Baez lecture notes.

 

[marcus]

the Baez lecture notes that Broda references is this

http://arxiv.org/abs/gr-qc/9905087

An Introduction to Spin Foam Models of Quantum Gravity and BF Theory
John C. Baez
55 pages, 31 figures
Journal-ref: Lect.Notes Phys. 543 (2000) 25-94

"In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a 'spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory."

 

[Chronos]

I just don't see how Broda is making a case with this. BF theory is a 2D+ toy model as far as I can see.

 

[marcus]

I just don't see how Broda is making a case with this. BF theory is a 2D+ toy model as far as I can see.

that's what I thought too and so until recently I didnt pay much attention to it
and there is at least this much truth in what you say: if you restrict to pure BF theory and only have one term in the action (B wedge F)
then in 4D there are no degrees of freedom
(the field can't wiggle, it is not a field in the usual sense but rather a topological invariant)

I am new to this BF discussion, you may know more about it, but I am repeating what I've read and agreeing with you:

in 4D pure BF is toy, or not interesting.

what is happening tho is that ever since Plebanski (1977) they add extra terms to the action. this makes it no longer a "topological" ft and it becomes "constrained topological" ft.
the first term they add is quadratic in B and is called "constraint" term.

the action becomes not just B wedge F,
but
B wed F - (some coeff)B wed B

I have not seen Plebanski 1977 paper but I understand that shows that by including this quadratic term you get General Relativity with a cosmological constant Lambda...(the Lambda is related to the "some coefficient")...
that is when you vary the action and extremize the "equation of motion" turn out to be the Einstein equation.
the Baez paper has something about this on pages 41 and 43.

the Baez (2000) paper that Broda cites is certainly not perfect, I have had trouble finding more recent and complete stuff at an introductory level tho.
In reading it you have to mentally substitute E for B and sometimes
notice that he is equating E = e wed e
so some equations only look familiar if you substitute B for e wed e.

I think one can say that the Plebanski 1977 action formulation is classical and reproduces the einstein equation (or something experimentally indistinguishable from General Relativity). I may be wrong about this but I think when Smolin on page 12 of "Invitation" says

S = S_topological + S_constraint + S_matter

he is just repeating the Plebanski action version of Gen Rel. It is something that is known to work, has as much experimental evidence for it as GR itself, and where the constraint term is just this B wed B with coefficient which is related to cosm. const. Lambda.

very interested in this right now. if you find out more about it please keep me posted.

 

[selfAdjoint]

Yes indeed 2-D BF theory is a toy model, but that doeesn't mean it isn't worthwhile to study. Phi-4 is also a toy model, in Quantum Field Theory, and physicsits have used it for decades to try out and discover new ideas. Just so BF-theory is regarded by Baez, for example, as a laboratory to test ideas of LQG.

Pure BF theory doesn't go anywhere, but we must learn it well enough to see how the various things they add on to it work with it. This is the language the big guys speak and it's no use to complain that all you want to do is order a beer in Barcelona.