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Isolated Horizons and Black Hole Entropy in Loop Quantum Gravity
We review the black hole entropy calculation in the framework of Loop Quantum
Gravity based on the quasi-local definition of a black hole encoded in the
isolated horizon formalism. We show, by means of the covariant phase space
framework, the appearance in the conserved symplectic structure of a boundary
term corresponding to a Chern-Simons theory on the horizon and present its
quantization both in the U(1) gauge fixed version and in the fully SU(2)
invariant one. We then describe the boundary degrees of freedom counting
techniques developed for an infinite value of the Chern-Simons level case and,
less rigorously, for the case of a finite value. This allows us to perform a
comparison between the U(1) and SU(2) approaches and provide a state of the art
analysis of their common features and different implications for the entropy
calculations. In particular, we comment on different points of view regarding
the nature of the horizon degrees of freedom and the role played by the
Barbero-Immirzi parameter. We conclude by presenting some of the most recent
results concerning possible observational tests for theory
Analytic Continuation of Black Hole Entropy in Loop Quantum Gravity
We define the analytic continuation of the number of black hole microstates
in Loop Quantum Gravity to complex values of the Barbero-Immirzi parameter
. This construction deeply relies on the link between black holes and
Chern-Simons theory. Technically, the key point consists in writing the number
of microstates as an integral in the complex plane of a holomorphic function,
and to make use of complex analysis techniques to perform the analytic
continuation. Then, we study the thermodynamical properties of the
corresponding system (the black hole is viewed as a gas of indistinguishable
punctures) in the framework of the grand canonical ensemble where the energy is
defined \'a la Frodden-Gosh-Perez from the point of view of an observer located
close to the horizon. The semi-classical limit occurs at the Unruh temperature
associated to this local observer. When , the entropy
reproduces at the semi-classical limit the area law with quantum corrections.
Furthermore, the quantum corrections are logarithmic provided that the chemical
potential is fixed to the simple value
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