Quantum Clocks: Gravitation and Relativity*

Time enters quantum theory through its appearance as an external classical parameter in the Schrödinger equation. In general relativity, time is defined operationally as what is indicated by a clock and the spacetime metric encodes the relationship between different clocks. Reconciling these two different notions of time in a quantum theory of gravity leads to the problem of time, one aspect of which is the disappearance of time in the Wheeler-DeWitt equation.

Motivated by this problem, the conditional probability interpretation (CPI) posits that time evolution emerges from entanglement shared between a clock and system of interest, the joint state of which does not evolve with respect to a background time and satisfies a Wheeler-DeWitt equation. After reviewing the CPI, I will present a generalization in which the clock and system interact — we should expect such a coupling when the gravitational interaction between the clock and system is taken into account. I will demonstrate how such clock-system interactions result in a time-nonlocal modification to the Schrödinger equation. Furthermore, I will demonstrate how time dilation becomes probabilistic within the CPI framework and recover on average the special relativistic result.