On the Emergence of Time and Space in Closed Quantum Systems

TL;DR

The work investigates how time and space can emerge from entanglement in closed quantum systems, extending Page–Wootters time emergence to a full spatial-relational framework. By treating clocks and spatial references as quantum subsystems with appropriate observables (including Pegg’s age and POVMs for bounded spectra), the thesis derives emergent Schrödinger dynamics, time observables, and a relational 1+1 and 3+1 dimensional spacetime, even in the presence of gravitational interactions. It connects PaW time with Canonical Typicality, showing how thermal equilibrium and dynamical evolution can coexist when the environment doubles as a clock, and demonstrates that time dilation arises in a PaW setting consistent with Schwarzschild predictions. The results collectively push toward a fully relational quantum spacetime where dynamics, locality, and relativistic effects emerge from entanglement across clocks and reference frames, with implications for quantum foundations and quantum gravity.

Abstract

Time, space and entanglement are the main characters in this work. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit 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, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we propose a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.

Figures (6)

• Figure 1: Schematic representation of the $p+1$ values of the energy spectrum (on the left) and the $s+1 > p+1$ values of $\alpha_{m}$ (on the right) in a system system with unequally-spaced energy levels. The red dashed lines in the energy spectrum indicate the minimum distance between two successive values in the eigenvalues of $\hat{H}_p$ so that all other energy values can be considered as multiples of this minimum step, they are not energy levels.
• Figure 2: Rappresentation of the probability $P(\alpha)$ when the system is in an energy eigenstate. The $\alpha$ quantity distribution is constant across the whole period $T$ according to the uncertainty relation.
• Figure 3: Schematic representation of the energy spectrum (on the left) and the $p+1$ values of $\tau_{m}$ (on the right) in a system with equally-spaced energy levels.
• Figure 4: The subsystem $S$ consists of two clocks, $A$ and $B$, and we study their evolution with respect to the clock $C$.
• Figure 5: The subsystem $S$ consists of two clocks $A$ and $B$ both placed within the gravitational potential. $B$ is at distance $x$ from the center of the mass $M$, while $A$ is at distance $x+h$.