Our current understanding of the physical world is based on the framework of the quantum theory
of fields (QFT) and on the classical theory of general relativity (GR). Both of them have received countless experimental confirmations, however, the two theories are at odds with each other: QFT is formulated on a fixed, nondynamical spacetime, while in GR the spacetime is a dynamical entity, affected by the presence of matter. In this scenario, one possibility to reconcile GR and QFT is to construct an encompassing quantum theory of the gravitational field, in which the latter couples with the matter fields by means of a quantum version of the Einstein equations.
This Master thesis fits into the theoretical framework of Loop Quantum Gravity (LQG) [1], which is one of the most promising, tentative theories of quantum gravity (QG). LQG is based on two pillars. The first is the background independence, which is assumed as a guiding principle of the quantization procedure, instead as something to recover a posteriori. The second is a minimalist point of view, in which the only elements of QG are quantum mechanics and GR, merged according to mathematical consistency. The main result of the theory is the appearance of an inherent notion of 3D space with quantum mechanical, geometrical properties and fundamentally discrete at the Planck scale.
In this context, our work is an attempt of linking LQG with the semiclassical level of a QFT on a classical background, in a particular case. After a thorough review of the path leading from GR to LQG, we consider the quantum state studied numerically in [2]. This is a particular truncation of the so-called covariant vacuum, therefore, it is an approximate solution of the quantum Einstein equations.
In the first part, we argue that this truncation of the covariant vacuum can be interpreted as the quantum state of a closed Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime [3], filled with a sort of background of gravitational radiation, in its early expansion phase. This prediction relies on the quantum geometrical properties of the state and on some strong assumptions, which are carefully discussed.
On the basis of the previous result, in the second part we consider a massless spin 2 quantum field (the graviton) on a closed FLRW background. We find the expression for the leading behaviour of the entanglement entropy of the field inside a closed 2-surface [4], which is a measure of the quantum correlations existing between the points inside the surface and the points outside. We conclude by discussing the possible match of this expression with the numerical results of [2], concerning the entanglement entropy of a quantum of space with respect to the rest of the system.
[1] Rovelli, C., and Smolin, L. Loop Space Representation of Quantum General Relativity. Nucl.
Phys. B 331 (1990), 80–152.
[2] Gozzini, F., and Vidotto, F. Primordial fluctuations from quantum gravity. Front. Astron.
Space Sci. (Jun 2019).
[3] Weinberg, S. Cosmology. Cosmology. OUP Oxford, 2008.
[4] Solodukhin, S. N. Entanglement entropy of black holes. Living Reviews in Relativity 14, 1
(oct 2011)
