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Feb27-12, 12:55 AM
Sci Advisor
P: 5,437
I am not sure whether Alexandrov and Engle are talking exactly about the same issue, but they seem to be rather close. What I do not understand (see also marcus' comment) is that they never mention each other in the citations; they seem to ignore each other ;-(

Alexandrov goes one step backward. He first explains in a series of papers why EPRL and FK is wrong (I discussed this in several threads over the last year); his observation is related to general rules to implement second class constraints and he therefore addresses issues not specific to QG

The problem is that the correct method a la Dirac has been studied (I can remember the delta functions in the measure in some Thiemann papers) but that it was not possible to get a final answer in terms of the physical spin network states and the vertex amplitude; b/c QG is terribly complicated Alexandrov decided to study something different, namely the top. sector and the CY model, for several reasons: it's simpler; it's quantization can be derived by different methods; so it serves as a consistency check for his approach.

Alexandrov finds that the EPRL and FK method results in a wrong vertex amplitude for CY, so he can demonstrate in a very specific example (where the final result is known) that his general objections are justified. Then he applies his (Dirac's) method to this specific example and finds the correct CY result, so he can conclude that his method is correct.

The final step - and this is where Alexandrov and Engle shall meet - is the application of the new measure to full QG. Alexandrov explains what he sees as the major obstacles to apply his method to full QG, namely not conceptual problems but technical difficulties due to the different structure:

"The main difference distinguishing it from our model is the form of the constraints ... In particular, the secondary constraints become explicitly dependent on the B-field ... Although the construction of section 3.3 is still well defined in the presence of such dependence, it gives rise to many complications. The most important one is that the
quantity (3.22) starts to depend on the bivectors ... and its interpretation as a vertex amplitude is not viable anymore. It is not clear whether this is a serious problem or just a minor obstacle.

It's hard for me to compare every step by Alexandrov with Engle's paper; but I'll do my best ...