This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time $t$. Doing so assigns a conditional state to the rest of the Universe $|\psi_S(t)\rangle$, referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state $|\psi_S(t)\rangle$ satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schrödinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of $G/c^4$ and inversely proportional to the distance between the clock and system.
Such a reformulation of quantum theory was put forward by Don Page and William Wootters in 1982, which begins by partitioning the Universe into two parts: a clock and a system comprised of everything else. Then any statement we would ordinarily make regarding the time dependence of the system takes the form, “The quantum state of the system is $\psi(t)$, if a measurement of the clock reads the time $t$”. The crucial difference in this formulation of quantum theory is that it makes no reference to an external background time; instead, time appears as an operational concept defined in terms of the measurement of a physical clock. An entangled state is assigned to the clock and system and the usual dynamics of quantum theory, as described by the Schrödinger equation, emerge from quantum correlations between the clock and system.
This article explores the effects of when the Universe cannot be partitioned cleanly into two parts, that is, when the clock and system interact. In some situations we should expect such an interaction given that gravity couples to everything! It is shown that such clock-system interactions lead to a more general quantum dynamics described by a time-nonlocal modification to the Schrödinger equation. This temporal nonlocality implies that the way in which the system changes in time depends not just on its current state, but also on its state in the future and the past. Three different clock-system interactions are examined to explore the effects of this temporal nonlocality. The first interaction considered leads to the system’s Hamiltonian developing a particular time dependence. The second example supposes that the clock and system interact via Newtonian gravity, which leads to corrections to the system’s Hamiltonian at order $G / c^4$ and is inversely proportional to the distance separating the clock and system. The third interaction gives an example of a qubit clock interacting with another qubit.
 K. V. Kuchař, Proceedings of the 4th Canadian Conference on General Relativity and Relativistic Astrophysics, edited by G. Kunstatter, D. Vincent and J. Williams (World Scientific, Singapore, 1992).
 D. N. Page, in Physical Origins of Time Asymmetry, edited by J. J. Halliwell, J. Pérez-Mercader, and W. H. Zurek, 287 (Cambridge University Press, Cambridge, 1994).
 R. Gambini, R. A. Porto, and J. Pullin,.
 C. Rovelli, Quantum Gravity, (Cambridge University Press, Cambridge, 2004).
 C. Kiefer, Quantum Gravity, (Oxford University Press, Oxford, 2012).
 P. A. M. Dirac, Lectures on Quantum Mechanics, (Dover Publications, New York, 1964).
 P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics, (Springer-Verlag, Berlin, Heidelberg, 1995).
 B. S. DeWitt, On Recent Developments in Theoretical and Experimental General Relativity, Gravitation, and Relativistic Field Theories, edited by T. Piran, 6, (World Scientific, Singapore, 1999).
 D. Šafránek, M. Ahmadi, and I. Fuentes, New J. Phys. 17, 033012 (2015).
 S. Massar and P. Spindel, Phys. Rev. Lett. 100, 190401 (2008).
 Maximilian P E Lock and Ivette Fuentes, "Quantum and classical effects in a light-clock falling in Schwarzschild geometry", Classical and Quantum Gravity 36 17, 175007 (2019).
 Ismael L. Paiva, Amit Te’eni, Bar Y. Peled, Eliahu Cohen, and Yakir Aharonov, "Non-inertial quantum clock frames lead to non-Hermitian dynamics", Communications Physics 5 1, 298 (2022).
 David Edward Bruschi and Andreas Wolfgang Schell, "Gravitational Redshift Induces Quantum Interference", Annalen der Physik 535 1, 2200468 (2023).
 Christophe Goeller, Philipp A. Hoehn, and Josh Kirklin, "Diffeomorphism-invariant observables and dynamical frames in gravity: reconciling bulk locality with general covariance", arXiv:2206.01193, (2022).
 Marcello Rotondo and Yasusada Nambu, "Clock Time in Quantum Cosmology", Universe 5 2, 66 (2019).
 Lorenzo Maccone, "A Fundamental Problem in Quantizing General Relativity", Foundations of Physics 49 12, 1394 (2019).
 Jürg Fröhlich and Alessandro Pizzo, "The Time-Evolution of States in Quantum Mechanics according to the ETH-Approach", Communications in Mathematical Physics 389 3, 1673 (2022).
 K. L. H. Bryan and A. J. M. Medved, "No time for isolated clocks", Journal of Physics Conference Series 1275 1, 012037 (2019).
The above citations are from Crossref's cited-by service (last updated successfully 2023-05-31 15:02:59) and SAO/NASA ADS (last updated successfully 2023-11-29 01:32:15). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch Crossref cited-by data during last attempt 2023-11-29 01:32:13: Encountered the unhandled forward link type postedcontent_cite while looking for citations to DOI 10.22331/q-2019-07-08-160.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.