Measuring time in a timeless universe (2406.14642v2)

Abstract: Physical systems are typically assumed to evolve relative to an external, real-valued time parameter. This seemingly innocent assumption is problematic because the time parameter is not physical. For instance, it is not an observable of any physical system. In quantum theory, this problem is solved by a construction due to Page & Wootters (1984), in which the universe is in a stationary state so that the external time parameter is irrelevant. Instead, a subsystem of the Page-Wootters universe will 'evolve' according to the Schr\"odinger equation because it is entangled with another system, aptly called a 'clock'. It is often assumed necessary for the clock to be a dynamically isolated system, as this is one of the assumptions that Page & Wootters make in their original formulation. This apparently prevents the clock from being measured, as a measurement would require it to interact with another system. In this work, we show that while isolation is a sufficient condition for the Page-Wootters construction, it is not a necessary one. Because interactions between the clock and other systems are allowed, it is possible to measure clock time. We also discuss clock synchronisation.

• The paper shows that emergent time arises from the entanglement between a clock subsystem and the rest of the universe.
• The paper relaxes the isolation requirement of clocks, demonstrating that interactions enable measurable clock dynamics in a timeless framework.
• The paper employs rigged Hilbert space formalism to validate its model, paving the way for integrating quantum mechanics with spacetime dynamics.

An Examination of "Measuring time in a timeless universe"

The paper "Measuring time in a timeless universe" by Samuel Kuypers and Simone Rijavec provides an insightful exploration into the conceptual foundations of time in physics, specifically within the framework of the Page–Wootters (PW) construction. The research challenges the traditional notion of an external, real-valued time parameter in physical systems, as it lacks physical observability, especially in quantum mechanics. Instead, the authors contribute to the discourse on timeless physics by examining how time can be an emergent property of static physical systems.

Overview of the Page–Wootters Construction

Kuypers and Rijavec delve into the PW construction, a timeless approach in quantum theory that explains time as an entanglement phenomenon. The construction suggests that while the universe itself can be considered a stationary state, subsystems within it can exhibit dynamical evolution relative to each other due to entanglement with a clock subsystem. This effectively negates the necessity for an external time parameter $\lambda$.

The construction assumes three key components: (1) the universe is in a zero-energy eigenstate, (2) the universe can be divided into two non-interacting subsystems—a clock $\mathcal{C}$ and the rest of the universe $\mathcal{R}$, and (3) the clock possesses a time observable $\hat{t}$ that is conjugate to its Hamiltonian $\hat{H}_\mathcal{C}$.

Key Contributions and Findings

A significant contribution of this paper is the relaxation of the assumption that the clock must be an isolated system. The authors demonstrate that while isolation suffices to uphold the PW model, it isn't necessary. By allowing for interactions between the clock and other systems, the paper finds that the clock time can indeed be measured—deftly addressing one of the critiques of the original PW construction. This revelation implies that the timeless universe can accommodate clock measurements, modeling behaviors of both naturally occurring and artificial clocks.

Moreover, the paper provides a simple model of clock synchronization, extending the framework to encompass interactions with additional clocks, showing that such synchronization can happen even if clocks are initially in a non-interacting state. This opens avenues for investigation in scenarios involving more complex interactions and synchronization involving relativistic effects in spacetime.

Implications and Future Research

This research paves the way for further investigations into the integration of spacetime dynamics within the PW framework. Given that Kuypers and Rijavec's model does not inherently describe space, an extension incorporating spatial dimensions could reveal how relative movement impacts clock synchronization and ultimately the perception of time within quantum mechanics.

The authors adeptly situate the PW construction within rigged Hilbert space formalism to handle technical challenges associated with clocks having continuous spectra. Such formalism is crucial for rigorous handling of the eigenstates involved in the model, thereby ensuring the mathematical and physical consistency of their propositions.

Conclusion

In sum, "Measuring time in a timeless universe" enriches the ongoing discourse on the nature of time in quantum mechanics. By demonstrating that clocks can be measured and synchronized without violating the timeless premise, the paper provides valuable insights into emergent time within quantum systems. Future research endeavors may well build on these findings to explore the intricate interplay between quantum mechanics and spacetime, a domain replete with both theoretical and practical implications for the physics community.