Near-Horizon Radiation and Self-Dual Loop Quantum Gravity

We compute the near-horizon radiation of quantum black holes in the context of self-dual loop quantum gravity. For this, we first use the unitary spinor basis of $\text{SL}(2,\mathbb{C})$ to decompose states of Lorentzian spin foam models into their self-dual and anti self-dual parts, and show that the reduced density matrix obtained by tracing over one chiral component describes a thermal state at Unruh temperature. Then, we show that the analytically-continued dimension of the $\text{SU}(2)$ Chern-Simons Hilbert space, which reproduces the Bekenstein-Hawking entropy in the large spin limit in agreement with the large spin effective action, takes the form of a partition function for states thermalized at Unruh temperature, with discrete energy levels given by the near-horizon energy of Frodden-Gosh-Perez, and with a degenerate ground state which is holographic and responsible for the entropy.Comment: 6+2 page

A note on the Holst action, the time gauge, and the Barbero-Immirzi parameter

In this note, we review the canonical analysis of the Holst action in the time gauge, with a special emphasis on the Hamiltonian equations of motion and the fixation of the Lagrange multipliers. This enables us to identify at the Hamiltonian level the various components of the covariant torsion tensor, which have to be vanishing in order for the classical theory not to depend upon the Barbero-Immirzi parameter. We also introduce a formulation of three-dimensional gravity with an explicit phase space dependency on the Barbero-Immirzi parameter as a potential way to investigate its fate and relevance in the quantum theory.Comment: 22 pages. Published version. Choice of gauge at the begining of section II.B. clarified. Published in Gen. Rel. Grav. (2013

A new look at Lorentz-Covariant Loop Quantum Gravity

In this work, we study the classical and quantum properties of the unique commutative Lorentz-covariant connection for loop quantum gravity. This connection has been found after solving the second-class constraints inherited from the canonical analysis of the Holst action without the time-gauge. We show that it has the property of lying in the conjugacy class of a pure \su(2) connection, a result which enables one to construct the kinematical Hilbert space of the Lorentz-covariant theory in terms of the usual \SU(2) spin-network states. Furthermore, we show that there is a unique Lorentz-covariant electric field, up to trivial and natural equivalence relations. The Lorentz-covariant electric field transforms under the adjoint action of the Lorentz group, and the associated Casimir operators are shown to be proportional to the area density. This gives a very interesting algebraic interpretation of the area. Finally, we show that the action of the surface operator on the Lorentz-covariant holonomies reproduces exactly the usual discrete \SU(2) spectrum of time-gauge loop quantum gravity. In other words, the use of the time-gauge does not introduce anomalies in the quantum theory.Comment: 28 pages. Revised version taking into account referee's comment

A Lorentz-Covariant Connection for Canonical Gravity

We construct a Lorentz-covariant connection in the context of first order canonical gravity with non-vanishing Barbero-Immirzi parameter. To do so, we start with the phase space formulation derived from the canonical analysis of the Holst action in which the second class constraints have been solved explicitly. This allows us to avoid the use of Dirac brackets. In this context, we show that there is a "unique" Lorentz-covariant connection which is commutative in the sense of the Poisson bracket, and which furthermore agrees with the connection found by Alexandrov using the Dirac bracket. This result opens a new way toward the understanding of Lorentz-covariant loop quantum gravity

Statistical Entropy of a BTZ Black Hole from Loop Quantum Gravity

We compute the statistical entropy of a BTZ black hole in the context of three-dimensional Euclidean loop quantum gravity with a cosmological constant $\Lambda$. As in the four-dimensional case, a quantum state of the black hole is characterized by a spin network state. Now however, the underlying colored graph $\Gamma$ lives in a two-dimensional spacelike surface $\Sigma$, and some of its links cross the black hole horizon, which is viewed as a circular boundary of $\Sigma$. Each link $\ell$ crossing the horizon is colored by a spin $j_\ell$ (at the kinematical level), and the length $L$ of the horizon is given by the sum $L=\sum_\ell L_\ell$ of the fundamental length contributions $L_\ell$ carried by the spins $j_\ell$ of the links $\ell$. We propose an estimation for the number $N^\text{BTZ}_\Gamma(L,\Lambda)$ of the Euclidean BTZ black hole microstates (defined on a fixed graph $\Gamma$) based on an analytic continuation from the case $\Lambda>0$ to the case $\Lambda<0$. In our model, we show that $N^\text{BTZ}_\Gamma(L,\Lambda)$ reproduces the Bekenstein-Hawking entropy in the classical limit. This asymptotic behavior is independent of the choice of the graph $\Gamma$ provided that the condition $L=\sum_\ell L_\ell$ is satisfied, as it should be in three-dimensional quantum gravity.Comment: 14 pages. 1 figure. Paragraph added on page 7 to clarify the horizon conditio