Thermally correlated states in Loop Quantum Gravity

This project takes place within the framework of Loop Quantum Gravity (LQG), one of the current attempt of building a quantum theory for the gravitational field. This theory is not a common Quantum Field Theory on a spacetime: in it, the spacetime itself emerges as a macroscopic property of quantum states. The reason of this feature, is that LQG undertakes, in the quantum setting, the merging of field and spacetime, which characterizes classical General Relativity. One of the main consequences of this aspect of the theory, is the fact that the space of quantum states is a well defined mathematical construction, but the relation with the states of the gravitational field describing a classical macroscopic spacetime (both flat and curved) is unclear. The natural ``vacuum'' of the theory, in fact, does not correspond to a flat spacetime, but rather to a zero-volume state (due to the ``background independence'', one of the main hypoteses of LQG). The precise description of a given spacetime is therefore a complex problem, comparable to the (simple) problem that one encounters in quantum optics, when trying to build up QED states corresponding to classical solutions of the Maxwell equations, or to the (difficult) problem of studying the vacuum of QCD. In the very last years, many progresses have been made in the attempt of understanding the ``semiclassical'' states of LQG, describing a given geometry. For example a construction similar to the one of coherent states used in quantum optics has been developed in great detail. This one allows for a definition of a quantum state of spacetime approximating a given (intrinsic and extrinsic) geometry in terms of mean values: that is the expectation values of the gravitational field and its momentum are those describing a classical geometry, with minimal uncertainty. These states have been used to study the classical limit of the theory and play a fundamental role in formulating the theorems which tie LQG with classical General Relativity. Nevertheless, the real semiclassical states of the theory cannot be determined by the mean value of the observables only. And, as a matter of fact, it is not unique the class of states which admit a semiclassical interpretation in this sense. In order to further characterize them, we need to fix how they fluctuate around those classical values. These quantum fluctuations are expressed by the field correlations at spacelike distance. It is not clear which is the specific shape to be required for these correlations, but we do know that in a certain approximation perturbative quantum gravity gives the correct predictions and therefore at distances large with respect to the Planck length, we want to recover the known behaviour. On the other hand, at very short distances, of the order of the Planck length, we know that those predictions will necessarily be corrected in order to be adapted to the full theory. Anyway, the difficulties of connecting to the perturbative case lie in the fact that most of the properties characterizing the quantum states of conventional field theories are formulated in terms of notions which cease to exist when the spacetime is dynamical. The short scale correlations, for example, depend on suitable powers of the (invariant) distance. But in a quantum theory of gravity, the distance is in turn a function of the field, making the usual expressions untranslatable. The idea behind this project, instead, is to explore a recent suggestion, which seems to elude this obstacle. The suggestion is the following. One of the consequences of the existence of spacelike correlations in quantum states is that a global Lorentz invariant state becomes mixed when restricted to the algebra of fields defined on a subspace, for example, on the region x >0 on the right of the origin. More precisely, it is a KMS state at inverse temperature 2π with respect to the flux generated by Lorentz boosts in the x direction. If K is the generator of such transformations, the density matrix takes thus the form: e^(-2πK). (1) This mathematical fact is at the root of the Unruh effect (an observer undergoing a constant acceleration in a Lorentz invariant quantum field theory vacuum will measure a temperature proportional to its acceleration). This property characterizes (at least partially) the behaviour of the correlations. And what makes it interesting is the fact it follows from few hypoteses only, which are the Lorentz invariance of the state and the positivity of energy: this is a remarkable fact, captured by the celebrated theorem by Bisognano and Wichmann. And it is crucial, for technical reasons, that all the quantities appearing in Equation (1) are well defined in LQG. Given a state of the theory, it is possible in fact to locally identify distinct regions and restrict the space to a subspace, and the Lorentz generator K is well defined. The property (1) is a global property in a theory on Minkowski space, but, by virtue of the equivalence principle, it is reasonable to expect a similar property to hold in an arbitrary spacetime, in a region small with respect to the expectation value of its curvature radius. In this thesis, the possibility of using (1) as a (partial) characterization of kinematical semiclassical states of LQG is explored. The same possibility has been lately brought forward in two articles. Bianchi e Myers suggested these thermal correlations to be the peculiar ones of semiclassical states of any quantum theory of spacetime. Rovelli and coworkers, instead, proposed (1) as a candidate to replace in the LQG framework the ``Hadamard condition'' characterizing instead the physical states of a field theory in curved space. It is therefore natural to investigate whether the coherent states of the theory, or a simple modification of them, have this property at local level