Favalli, Tommaso (2023) On the Emergence of Time and Space in Closed Quantum Systems. [Tesi di dottorato]

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Abstract

In this work we examine, revisit and generalize the Page and Wootters (PaW) theory which was originally introduced in order to describe the emergence of time from quantum correlations. PaW theory consists in dividing the total Hilbert space of a global quantum system (satisfying a total energy constraint) into two entangled subsystems and considering one of them as a clock. The "flow of time" then emerges in the other subsystem, with respect to the clock values, thanks to entanglement. After giving a complete overview of the PaW theory, we start by extending the theory in order to consider any generic Hamiltonian with discrete and finite spectrum as a clock Hamiltonian by using POVMs in describing the time states. We show that, even if the time states are not fully distinguishable, the rest of the Universe still evolves with respect to clock time according to the Schrödinger equation. In addition, we recover a continuous flow of time still maintaining a discrete and finite clock Hamiltonian. We emphasize that the framework we develop allow us to use every generic Hamiltonian as clock Hamiltonian. This made it possible for us to merge the Canonical Typicality and PaW quantum time. Indeed, considering a closed quantum Universe consisting of a small subsystem weakly interacting with a large environment, it has been demonstrated that for the vast majority of randomly chosen wave-functions of the Universe satisfying a total energy constraint, the reduced density matrix of the small subsystem is given by the canonical statistical distribution. At the same time, through the PaW mechanism, we find a Schrödinger-like evolution (even if corrected by a non-local term) for the relative state of the small subsystem with respect to environment which serves as a clock. These two approaches seem contradictory but can coexist considering that, in this mixed framework the action of tracing out the environment is equivalent to tracing over all times: the trace over the environment coincides with a temporal average. Still, by calculating the conditioned state of the small subsystem and thus looking at a single time, we find a pure state evolving with a Schrödinger-like equation. Then we introduce the spatial degree of freedom in the discussion and we provide a description of a model of non-relativistic quantum spacetime. We start giving our own version of the PaW mechanism for space and we show that, in a closed quantum system satisfying a global constraint on total momentum (and therefore with the absolute position totally indeterminate), a well-defined relative position emerges between entangled subsystems, where one of the two subsystems is taken as quantum spatial reference frame. Introducing in the Universe an additional subsystem acting as a clock, we then consider the Universe satisfying a double constraint: both on total momentum and total energy. We show how this framework can be implemented without contradiction in the simple case of one spatial degree of freedom (considering also the case of multiple time measurements) and in the “more physical” case of three spatial degrees of freedom, thus providing a 3 + 1 dimensional quantum spacetime emerging from entanglement. The dynamics of relativistic particles with respect to a non-relativistic, quantum reference frame is also considered. Finally, we investigate the behavior of quantum clocks when interacting with a Newtonian gravitational potential produced by a non-rotating spherical mass, again through PaW theory. As a purely quantum phenomenon, we find that the clock placed inside the gravitational field remains delayed in ticking with respect to a far-away, non-perturbed clock and we show that this time dilation effect has the same form as the gravitational time dilation obtained from the Schwarzschild metric.

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