Quantizing time: Interacting clocks and systems
Abstract
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 WheelerDeWitt 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 WheelerDeWitt equation contains an interaction term coupling the clock and system, the conditional state $\psi_S(t)\rangle$ satisfies a timenonlocal Schrödinger equation in which the system Hamiltonian is replaced with a selfadjoint integral operator. This timenonlocal Schrödinger equation is solved perturbatively and three examples of clocksystem 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 DOI:
 10.48550/arXiv.1712.00081
 arXiv:
 arXiv:1712.00081
 Bibcode:
 2017arXiv171200081S
 Keywords:

 Quantum Physics;
 General Relativity and Quantum Cosmology
 EPrint:
 Two new examples of clocksystem interactions have been added and a few points clarified. Comments welcome