- #1

tom.stoer

Science Advisor

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## Main Question or Discussion Point

There's a new paper dealing with constraint algebra and Gupta-Bleuler quantization in LQG and SF models.

http://arxiv.org/abs/1012.1738

Authors: Wolfgang Wieland

(Submitted on 8 Dec 2010)

Abstract: From the Holst action in terms of complex valued Ashtekar variables additional reality conditions mimicking the linear simplicity constraints of spin foam gravity are found. In quantum theory with the results of You and Rovelli we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our result perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.

The idea is to relate Ahtekar's variables plus reality conditions of canonical LQG to simplicity constraints of SF models.

Whoever is interested in these topics and has some experience regarding constraint quantization a la Dirac: let's discuss if this can be correct; I still think that they 'cheat' when trying to avoid second class constraints and Dirac quantization.

So my conclusion is that the paper does not fix the problems of simplicity constraints in the SF formulation, but that it translates the errors made there to the canonical framework. This is a remarkable result not b/c the problems are resolved but b/c they become explicit in the canonical framework.

http://arxiv.org/abs/1012.1738

**Complex Ashtekar variables and reality conditions for Holst's action**Authors: Wolfgang Wieland

(Submitted on 8 Dec 2010)

Abstract: From the Holst action in terms of complex valued Ashtekar variables additional reality conditions mimicking the linear simplicity constraints of spin foam gravity are found. In quantum theory with the results of You and Rovelli we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our result perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.

The idea is to relate Ahtekar's variables plus reality conditions of canonical LQG to simplicity constraints of SF models.

Whoever is interested in these topics and has some experience regarding constraint quantization a la Dirac: let's discuss if this can be correct; I still think that they 'cheat' when trying to avoid second class constraints and Dirac quantization.

- (24f) means that the algebra does not close; the following lines of reasoning do not seem to be convincing
- I do not see how they prove consistency of (30) with (24)
- (40) is the main result which shows the equivalence between reality conditions and simplicity constraints; but (41) shows that C is not compatible with other constraints
- Then unfortunately they stop calculating secondary and tertiary constraints which is required to complete the analysis
- In (47) and (48) they explain what they want to do - and it simply looks wrong; they basically neglect that the two sets of constraints do not commute and are therefore not consistent;
- afaik Gupta-Bleuler fails in QCD b/c the method is not able to deal with ghosts; here they have a similar situation but they apply Gupta-Bleuler w/o worrying about ghosts; why?
- in section 5.2 they admit that they did not treat T which is second class correctly and that for H there is still no conclusive and consistent expression available

So my conclusion is that the paper does not fix the problems of simplicity constraints in the SF formulation, but that it translates the errors made there to the canonical framework. This is a remarkable result not b/c the problems are resolved but b/c they become explicit in the canonical framework.