# The Page-Wootters mechanism 36 years on: a consistent formulation which accounts for interacting systems

This is a Perspective on "Quantizing time: Interacting clocks and systems" by Alexander R. H. Smith and Mehdi Ahmadi, published in Quantum 3, 160 (2019).

By Mischa Woods (Institute for Theoretical Physics, ETH Zürich, Switzerland).

Do we need dynamics to explain quantum systems which change over time? In closed systems, this is indeed the case — quantum systems evolve according to the Schrödinger equation in non-relativistic quantum mechanics, and in relativistic systems; time dependent quantum fields are predicted. However, what if we are allowed an ancillary system in addition to the system $S$ of interest? Could we then have a setup in which the global system is time independent, yet system $S$ is time dependent and obeys the Schrödinger equation? This is the question which Page and Wooters answered in 1983 and is now known as the Page-Wooters mechanism [1]. They were inspired by a previous paper [2], in which an equation — now known as the Wheeler-DeWitt equation — was introduced with the aim of studying canonical quantum theories of gravity.

The mechanism works by using a quantum clock as the ancillary system. In this context, a quantum clock is a quantum system which evolves in time according to the Schrödinger equation for which there is an appropriately chosen time observable which can be measured to reveal what the time is, i.e. its measurement outcome is approximately $t$ — the time parameter in the Schrödinger equation. When the clock is used as the ancillary system, its initial state can be correlated with the initial state of $S$ so as to “internalize” the time parameter $t$ coming from the Schrödinger equation. In other words, one can find a Hamiltonian and initial state over $S$ and the clock which does not evolve in time, yet has the property that when one measures the clock and conditions on obtaining outcome time $t$, the system $S$ collapses into the state one would have found, had they evolved the initial system $S$ state for a time $t$ according to the Schrödinger equation.

However, their mechanism employs a Hamiltonian over the clock and system $S$ which is local, i.e. does not have any interaction terms coupling the clock to $S$. Moreover, the time independency of the global system induced by the mechanism, is solely a consequence of the initial correlations of the state on $S$ and the clock. However, physical systems usually interact with each other, but allowing system $S$ and the clock to interact while maintaining a global time independent state is challenging.

In [3] Alexander Smith and Mehdi Ahmadi have come up with a solution to this problem. Consider an interaction term in the Hamiltonian which couples the dynamics of the clock and system. The resulting dynamics of system $S$ after conditioning on the clock is governed by a time-nonlocal modification to the Schrödinger equation. It is widely believed that gravity couples to everything. If this is the case, in any Page Wooters style model the clock must interact with the system. The work [3] has opened up the possibility to study the implications of such models.
Finally, another interesting aspect of the study of [3] is that they do not assume the clock to constitute a perfect reference for time. Such clocks — known as “idealized clocks” — are often considered to be unphysical, since they necessitate infinite energy either in the form of a Hamiltonian which is unbounded from below, or requiring infinitely strong potential functions. Despite this fact, they are often used in studies of the Page Wooters mechanism due to their mathematical simplicity.

### ► References

[1] Don N. Page and William K. Wootters, Phys. Rev. D 27, 2885 (1983), 10.1103/​PhysRevD.27.2885.
https:/​/​doi.org/​10.1103/​PhysRevD.27.2885

[2] Bryce S. DeWitt, Phys. Rev. 160, 1113 (1967), 10.1103/​PhysRev.160.1113.
https:/​/​doi.org/​10.1103/​PhysRev.160.1113

[3] Alexander R. H. Smith and Mehdi Ahmadi, Quantum 3, 160 (2019), 10.22331/​q-2019-07-08-160.
https:/​/​doi.org/​10.22331/​q-2019-07-08-160

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