Time, space and entanglement are the main characters of this book. Their nature is still a great mystery in physics and we study the possibility that these three phenomena are connected. In the book we show indeed how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit, generalize and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static Universe. In the theory, the total Hilbert space is divided in two entangled subsystems and one of them is equipped with an appropriate clock observable: through a constraint on total energy, the temporal dynamics (namely the Schrödinger equation) is recovered for the rest of the Universe with respect to the eigenvalues of the clock observable. Time is not considered here as an abstract, external coordinate, but as "what is shown on a clock" where the clock is a quantum system taken as a time reference. In the book, after providing a complete review of the salient aspects of the theory, we revisit, generalize and extend this time operational approach (which appears to be very promising considering experimental techniques involving quantum clocks) and we connect it with recent research on the foundations of statistical mechanics, providing a new understanding of the thermalization phenomenon.   The Page and Wootters theory can be interpreted as an internalization of the temporal reference frame. A natural evolution is therefore to extend the approach to the spatial degree of freedom. The central point of the book is therefore the generalization of the framework to the spatial degree of freedom where we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. With our fully relational approach, space and time are treated on equal footing within quantum theory and the dynamics of quantum systems is derived with respect to a 3+1 dimensional quantum reference frame. This seems very important to us, considering that, in the quantum gravity literature, it has been suggested that the use of quantum reference frames is a promising road to the formulation of a quantum theory of gravity. This is the reason why, as a last point, we study the evolution of quantum clocks interacting with a gravitational field produced by a spherical mass. In the framework of Page and Wootters approach, we derive a time dilation effect for clocks in agreement with the gravitational time dilation obtained from the Schwarzschild metric. We think the work can be considered as an excellent effort in the attempt to merge the theory of Relativity with Quantum Mechanics. Extending the Page and Wootters theory through a novel approach we open the road for further research and development. Indeed, the topic of quantum spacetime may have many future applications that will open new roads for theoretical investigations and experimental applications.   The book is targeted for doctoral students or young researchers, passionate about the foundations of quantum theory, who want to discover how entanglement may be at the basis of the emergence of space and time.weiterlesen