Abstract Dynamical evolution can be reconstructed within stationary, closed quantum systems by employing the Page-Wootters “timeless approach”. When conditioning upon the state of a “clock” subsystem, the rest of the system regains its time dependence. This mechanism, involving entanglement between the above subsystems has gained much attention during the last few years. After a brief introduction to the topic we will elaborate on a few recent results: The derivation of new time-energy uncertainty relations, emergence of non-Hermitian dynamics when utilizing non-inertial quantum clocks and dynamical nonlocality in quantum time.
Abstract Dynamical evolution can be reconstructed within stationary, closed quantum systems by employing the Page-Wootters “timeless approach”. When conditioning upon the state of a “clock” subsystem, the rest of the system regains its time dependence. This mechanism, involving entanglement between the above subsystems has gained much attention during the last few years. After a brief introduction to the topic we will elaborate on a few recent results: The derivation of new time-energy uncertainty relations, emergence of non-Hermitian dynamics when utilizing non-inertial quantum clocks and dynamical nonlocality in quantum time.
In quantum theory it is possible to explain time, and dynamics, in terms of entanglement. This is the timeless approach to time, which assumes that the universe is in a stationary state, where two noninteracting subsystems, the ``clock'' and the ``rest,'' are entangled. As a consequence, by choosing a suitable observable of the clock, the relative state of the rest of the universe evolves unitarily with respect to the variable labeling the clock observable's eigenstates, which is then interpreted as time. This model for an ``evolution without evolution'' (Page and Wootters, 1983), albeit elegant, has never been developed further, because it was criticized for generating severe ambiguities in the dynamics of the rest of the universe. In this paper we show that there are no such ambiguities; we also update the model, making it amenable to possible new applications.
The concept of entanglement is at the heart of quantum physics.It plays a central role in all quantum phenomena involving composite systems.Interestingly, there is an intriguing idea that has attracted considerable attention recently, according to which quantum entanglement may also be essential for understanding the very emergence of time and dynamical evolution.Within this point of view, sometimes referred to as the timeless picture of quantum dynamics, the Universe is regarded as consisting of a clock and a system (or "rest of the Universe") that are jointly in a stationary quantum state, and time evolution arises as an emergent phenomenon rooted at the entanglement between the clock and the system.Here we provide a pedagogical and self-contained exposition, at the upper undergraduate level, of the role of entanglement in this timeless evolution approach to quantum mechanics.In particular, we give a detailed explanation of how the entanglement between the clock and the system is directly and quantitatively related to the average distinguishability between the states of the system at different times.
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.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
The problem of describing the dynamics of quantum systems relative to quantum clocks with which they interact is relevant to the physics of quantum reference frames and quantum gravity alike. There have been different approaches that derive a relational time-dependent Schrödinger equation for the system relative to the clock from a time-independent Hamiltonian constraint. We show that some of these approaches, due to Smith and Ahmadi, Gemsheim and Rost, as well as the so-called Born-Oppenheimer approach, can be unified in a single framework for describing conditional quantum states. In particular, the Born-Oppenheimer approach to quantum cosmology is seen to correspond to a type of interacting Page-Wootters formalism. We discuss the issue of unitarity and linearity, or lack thereof, in the relational dynamics, and we conclude with a brief outlook to further developments.
Physical systems are usually assumed to evolve relative to an external time parameter, which is problematic because in quantum theory that parameter is not a physical observable. Page and Wootters (1984) solved this by proposing that the Universe is in a stationary state, eliminating the need for the external time parameter. Instead, their model contains an isolated subsystem, a clock, with which other subsystems are entangled, making the latter appear to evolve relative to different states of the clock. While this resolves the problem of the time parameter, the assumption that the clock is isolated prevents it from being measured, as this requires an interaction with another system. We prove that the clock can be measured while preserving the core features of the Page-Wootters construction. We also discuss clock synchronization.