Coin, you asked what is the Immirzi Parameter. It might be fun to take a very uptodate nerdy look at how it arises in the very latest (spinfoam) version of Loop.
What I mean is, there was already this ImP in Ashtekar's formulation of classical GR and Barbero's variant of that. And Rovelli gives the straightforward discussion of it in his review, and then it goes on from there and gets into the canonical LQG of the 1990s. And various papers have speculated about possible physical significance etc. But it is still an unexplained ambiguity, I think, and no good reason to try to pin it down by hawking talk. (As Jacobson argues pretty convincingly, I think.)
So let's take the attitude that this is going to be one of the things we are going to find out more about in the next couple of years.
And so we have this question mark parameter, and let's see how it arises in latest Lorentzian spinfoam formulation, where it is forced to exist. Barrett doesn't put it in by hand, he
derives that this constant real number proportionality has to be there. The ImP comes out at him of its own accord.
And it turns out that it is the same ImP that Rovelli's spinfoam paper put in by hand.
But it is still an ambiguity! It is a real number ratio that all the simplices or triangles in the spinfoam must exhibit. In their irrep labels. It has to be the same for all, but it can be anything. It is not pinned down. This paper is surely going to be on our third quarter VIP poll.
http://arxiv.org/abs/0907.2440
Lorentzian spin foam amplitudes: graphical calculus and asymptotics
John W. Barrett, Richard J. Dowdall, Winston J. Fairbairn, Frank Hellmann, Roberto Pereira
30 pages
(Submitted on 14 Jul 2009)
"The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics."
To watch the video of Barrett's Planck Scale conference talk, go here:
http://www.ift.uni.wroc.pl/~rdurka/planckscale/index-video.php?plik=http://panoramix.ift.uni.wroc.pl/~planckscale/video/Day2/2-6.flv&tytul=2.6%20Barrett
He explains page 3 that the Lorentz group irreps are labeled (k,p) where k is a half integer and p is a real number.
In a spinfoam all the triangles get labels (k, p).
Then on page 15 right before equation (22) he proves that all the labels of all the triangles have to share a global proportionality p = gamma k
or else there is no stationary point of the action. But in a Lagrange setup you are looking for stationary points. All the other cases get washed out---don't count. At that point he cites Engle Pereira Rovelli (the famous "EPR" spinfoam paper) and observes that this is just like in EPR. Except with them it was partly a conjecture which they implemented manually, I think, and then Barrett et al proved.
And then later on he comes out and calls it the Immirzi parameter. For example on page 25, the Conclusions section:
==quote from Conclusions==
In this work we have defined a graphical calculus for the unitary representations of Lorentz group, and used it to give a systematic definition of the 4-simplex amplitude in the case of Lorentzian quantum gravity. The asymptotic analysis of the amplitude has some surprising features.
In the corresponding Euclidean quantum gravity problem analysed in [21], there was a puzzling superposition of terms with the Regge action multiplied by the Immirzi parameter and terms with the Regge action not multiplied by the Immirzi parameter.
In the Lorentzian quantum gravity analysed here, these two phenomena are separated out. The terms with the Immirzi parameter occur for boundary data of a Lorentzian metric and involve the Lorentzian Regge action. For this case, the result is much cleaner than for the Euclidean theory, as these are the only terms for this boundary data...
...Another result of this work is that a condition for the existence of stationary points of the action is that p
ab = γ k
ab for some constant γ . This is exactly the same restriction on the representation labels derived in [7], by different methods, where γ is the Immirzi parameter.
==endquote==
[7] is the Engle Pereira Rovelli paper.
One simple moral I get from this story is that it is better to stick with the Lorentzian case if you can manage to because then things make better sense.
