Towards self dual Loop Quantum Gravity

In this PhD thesis, we introduced a new strategy to investigate the kinematical and physical predictions of self dual Loop Quantum Gravity (LQG) and by-passed the old problem of implementing quantum mechanically the so called reality conditions inherent to the self dual phase space. Since our first motivations come from black holes thermodynamics, we first review, in the third chapter, the loop quantization of spherically isolated horizon based on the SU(2) Ashtekar-Barbero variables leading to the micro-canonical entropy. This approach is based on the effective model of a Chern-Simons connection coupled to point-like particles living on the horizon. Then we present the so called “gas of punctures” model for the isolated horizon, which provides the framework to go beyond the micro-canonical ensemble and compute the canonical and grand-canonical entropy. In the context of this model, we investigate how the assumption of a Bose-Einstein statistic for the punctures impacts the semi-classical result and underline a relation between the condensation phenomenon and the presence of purely logarithmic quantum corrections to the entropy. The presentation of this model enable us to point the different limits and drawbacks of the SU(2) loop quantization of spherically isolated horizon. First, one needs to proceed to an unnatural fine tuning on the Immirzi parameter to obtain the right semi-classical limit in the micro-canonical ensemble. Moreover, this quantization does not predict a holographic behavior for the degeneracy of the hole, as one could expect. Therefore, if one wants to avoid to precedent fine tuning on γ, one is led to assume the so called holographic hypothesis to obtain the right semi classical limit in the context of the gas of puncture model. The fourth chapter is devoted to studying to what extend the loop quantization based on the self dual variables could cure those problems. Obviously, since no one knows how to quantize the self dual Ashtekar phase space of General Relativity, because of the so called reality conditions, we are led to introduce a new strategy, based on an analytic continuation of the degeneracy from γ ∈ R to γ = ±i. We review in details the construction of the procedure, and present the results. The self dual degeneracy turns out to be naturally holographic, supplemented with some power law corrections which conspired, at the semi classical limit, to provide the expected logarithmic quantum corrections to the entropy. At the leading term, we recover the Bekenstein-Hawking area law. The discrete real area spectrum is turned into a continuous real area spectrum, even if we are now working with γ = ±i. Finally, we recognize that our procedure is a well known map which send the Casimir and the character of the SU(2) compact group into the Casimir and character of the continuous representations of the SU(1,1) non compact group. A detailed discussion on the status of our procedure is ￼provided at the end of the chapter. The fifth chapter is devoted to understanding more precisely the interplay between the disappearance of γ in the physical predictions of the quantum theory, the appearance of the SU(1,1) group and the relation with the self dual variables. To to so, we introduce a new toy model of three dimensional gravity admitting a Barbero-immirzi like parameter and SL(2,C) as its symmetry group. The canonical analysis of the toy model in two different gauges, one selecting the compact group SU(2), the second one selecting the non compact SU(1,1) group, allows us to conclude that at least in three dimensional gravity, the presence of γ in the SU(2) phase space is a pure gauge artifact. Finally, the loop quantization of the two phases spaces results in two different kinematical area spectrums, which turn out to be related through our analytic continuation procedure. At the end, we show that it is possible to reformulated the SU(2) phase space in terms of new one, based on a complex SL(2,C) connection, supplemented with some reality constraints. Once solved, the reality conditions reduces the complex SL(2,C) phase space to the SU(1,1) phase space as expected. This chapter allows us to exhibit an interesting mechanism concerning the disappearance of the Immirzi parameter in the predictions of three dimensional loop quantum gravity. Finally, the sixth chapter is devoted to applying our procedure to the simplest Loop Quantum Cosmology model. By first constructing the LQC dynamics for any arbitrary spin j and then implementing our analytic continuation, we show that our procedure preserves the key features of the LQC models, i.e. we obtain a bouncing universe which admits the right semi classical limit after the bounce.