A gravitationally collapsed object can bounce-out from its horizon via a tunnelling process that violates the classical equations in a finite region. Since tunnelling is a non-perturbative phenomenon, it cannot be described in terms of quantum fluctuations around a classical solution and a backgroundfree formulation of quantum gravity is needed to analyze it. Here we use Loop Quantum Gravity to compute the amplitude for this process, in a first approximation. The amplitude determines the tunnelling time as a function of the mass. This is the key information to evaluate the relevance of this process for the interpretation of Fast Radio Bursts or high-energy cosmic rays. The calculation offers a template and a concrete example of how a background-free quantum theory of gravity can be used to compute a realistical observable quantity.

In previous work, the Lorentzian proper vertex amplitude for a spin-foam model of quantum gravity was derived. In the present work, the asymptotics of this amplitude are studied in the semi-classical limit. The starting point of the analysis is an expression for the amplitude as an action integral with action differing from that in the EPRL case by an extra 'projector' term. This extra term scales linearly with spins only in the asymptotic limit, and is discontinuous on a (lower dimensional) submanifold of the integration domain in the sense that its value at each such point depends on the direction of approach. New tools are introduced to generalize stationary phase methods to this case. For the case of boundary data which can be glued to a non-degenerate Lorentzian 4-simplex, the asymptotic limit of the amplitude is shown to equal the single Feynman term, showing that the extra term in the asymptotics of the EPRL amplitude has been eliminated.

Bell-network states are loop-quantum-gravity states that glue quantum polyhedra with entanglement. We present an algorithm and a code that evaluates the reduced density matrix of a Bell-network state and computes its entanglement entropy. In particular, we use our code for simple graphs to study properties of Bell-network states and to show that they are non-typical in the Hilbert space. Moreover, we investigate analytically Bell-network states on arbitrary finite graphs. We develop methods to compute the Rényi entropy of order p for a restriction of the state to an arbitrary region. In the uniform large-spin regime, we determine bounds on the entanglement entropy and show that it obeys an area law. Finally, we discuss the implications of our results for correlations of geometric observables.

We use the requirement of diffeomorphism invariance in the Bianchi I context to derive the form of the quantum Hamiltonian constraint. After imposing the correct classical behavior and making a certain minimality assumption, together with a certain restriction to "planar loops", we then obtain a unique expression for the quantum Hamiltonian operator for Bianchi I to both leading and subleading orders in . Specifically, this expression is found to exactly match the form proposed by Ashtekar and Wilson-Ewing in the loop quantum cosmology (LQC) literature. Furthermore, by using the projection map from the quantum states of the Bianchi I model to the states of the isotropic model, we constrain the dynamics also in the homogeneous isotropic case, and obtain, again to both leading and subleading order in , a quantum constraint which exactly matches the standard 'improved dynamics' of Ashtekar, Pawlowski and Singh. This result in the isotropic case does not require a restriction to planar loops, but only the minimality assumption. Our results strengthen confidence in LQC dynamics and its observational predictions as consequences of more basic fundamental principles. Of the assumptions made in the isotropic case, the only one not rigidly determined by physical principle is the minimality principle; our work also shows the exact freedom allowed when this assumption is relaxed.

The proper spin-foam vertex amplitude is obtained from the EPRL vertex by projecting out all but a single gravitational sector, in order to achieve correct semi-classical behavior. In this paper we calculate the gravitational two-point function predicted by the proper spin-foam vertex to lowest order in the vertex expansion. We find the same answer as in the EPRL case in the 'continuum spectrum' limit, so that the theory is consistent with the predictions of linearized gravity in the regime of small curvature. The method for calculating the two-point function is similar to that used in prior works: we cast it in terms of an action integral and to use stationary phase methods. Thus, the calculation of the Hessian matrix plays a key role. Once the Hessian is calculated, it is used not only to calculate the two-point function, but also to calculate the coefficient appearing in the semi-classical limit of the proper vertex amplitude itself. This coefficient is the effective discrete "measure factor" encoded in the spin-foam model. Through a non-trivial cancellation of different factors, we find that this coefficient is the same as the coefficient in front of the term in the asymptotics of the EPRL vertex corresponding to the selected gravitational sector.