Alexander R. H. Smith

Time observables, relational dynamics, and quantum time dilation

General relativity demands that spacetime not be treated as a fixed background structure but as a dynamical entity. In the canonical formulation, this manifests as a Hamiltonian constraint, which appears to “freeze” physical states and gives rise to the notorious problem of time in quantum gravity: if the total Hamiltonian annihilates all states of matter and geometry, how does our familiar notion of time evolution emerge?

In this talk, I will review a class of time observables described by covariant positive‐operator‐valued measures (POVMs) [1]. These POVMs evade Pauli’s objection to the existence of a time operator, saturate the time-energy uncertainty relation, and serve as the keystone for two equivalent formulations of relational quantum dynamics [2-5]:

The Page-Wootters formalism, in which evolution is encoded in entanglement between a clock and the rest of the system;
The evolving constants of motion formalism, in which a family of gauge-invariant Dirac observables is constructed that evolve relationally with respect to a chosen clock variable.

Using these formalisms, I will show how dynamics emerges from conditional probabilities and the kinematical structure of quantum theory alone [3]. I will also outline extensions to interacting clock systems [2] and quantum field theory [6].

Finally, I will apply this machinery to relativistic particles carrying internal degrees of freedom that function as clocks measuring their proper time [7]. Remarkably, a novel quantum time-dilation effect arises between two clocks when one is placed in a superposition of different momenta or a superposition of locations in a gravitational field. Using the lifetime of a hydrogen‐like atom as a concrete clock, I will argue that this effect is within reach of current high-precision spectroscopic experiments, thus offering a new test of relativistic quantum mechanics [8,9]. Moreover, by invoking the Helstrom-Holevo bound, I will derive a fundamental time-energy uncertainty relation linking the precision of proper‐time measurements to the clock’s rest mass [7].

References:

Smith, A. R. H. Time in Quantum Physics. in Encyclopedia of Mathematical Physics 254–275 (Elsevier, 2025).
Smith, A. R. H. & Ahmadi, M. Quantizing time: Interacting clocks and systems. Quantum 3, 160 (2019).
Höhn, P. A., Smith, A. R. H. & Lock, M. P. E. Trinity of relational quantum dynamics. Phys. Rev. D 104, 066001 (2021).
Höhn, P. A., Smith, A. R. H. & Lock, M. P. E. Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings. Frontiers in Physics 9, 181 (2021).
Ali Ahmad, S., Galley, T. D., Höhn, P. A., Lock, M. P. E. & Smith, A. R. H. Quantum Relativity of Subsystems. Phys. Rev. Lett. 128, 170401 (2022).
Höhn, P. A., Russo, A. & Smith, A. R. H. Matter relative to quantum hypersurfaces. Phys. Rev. D 109, 105011 (2024).
Smith, A. R. H. & Ahmadi, M. Quantum clocks observe classical and quantum time dilation. Nature Communications 11, 5360 (2020).
Grochowski, P. T., Smith, A. R. H., Dragan, A. & Dębski, K. Quantum time dilation in atomic spectra. Phys. Rev. Research 3, 023053 (2021).
Paczos, J., Dębski, K., Grochowski, P. T., Smith, A. R. H. & Dragan, A. Quantum time dilation in a gravitational field. Quantum 8, 1338 (2024).