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.