Quantum clocks, conditional probabilities, and probabilistic time dilation

What allowed Einstein to transcend Newton's conception of an absolute time was his insistence on an operational definition of time as that which is measured by a clock. Quantum theory has yet to be liberated from this absolute notion of time as evidenced by the Schrödinger equation in which time appears as an external classical parameter.

In this talk I will introduce an operational formulation of quantum theory known as the conditional probability interpretation of time (CPI) in which time is defined in terms of an observable on a quantum system functioning as a clock; in some contexts, the CPI is known as the Page and Wootters mechanism. This clock and a system of interest do not evolve with respect to an external time, but instead, they are entangled and as a consequence a relational dynamics between the system and clock emerges. I will present a generalization of the CPI relevant when the clock and system interact, which should be expected when the gravitational interaction between them is taken into account. I will demonstrate how such clock-system interactions result in a time-nonlocal modification to the Schrödinger equation. I will then examine relativistic particles with internal degrees of freedom which constitute a clock that tracks their proper time. By examining the conditional probability associated with two such clocks reading different proper times, I will show that these clocks exhibit both classical and quantum time dilation effects. Moreover, I will show that the Helstrom-Holevo lower bound requires that these clocks satisfy a time-energy uncertainty relation between the proper time they estimate and their rest mass.