Quantizing time: Interacting clocks and systems

TL;DR

This paper extends the conditional probability interpretation of time in quantum gravity by analyzing how interactions between a clock and system, including gravitational effects, lead to a time-nonlocal Schrödinger equation.

Contribution

It introduces a formalism for interacting clocks and systems within the Wheeler-DeWitt framework, deriving a perturbative solution for the resulting time-nonlocal Schrödinger equation.

Findings

Interaction terms modify the system's Hamiltonian with gravitational corrections.

The formalism applies to Newtonian gravity interactions between clock and system.

Perturbative solutions reveal nonlocal effects in the system's evolution.

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 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 Schr\"{o}dinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schr\"{o}dinger 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.