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What happens when we quantize time?

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The lesson of general relativity is background independence: a physical theory should not be formulated in terms of external structures. This motivates a relational approach to quantum dynamics, which is necessary for a quantum theory of gravity. Using a covariant POVM to define a time observable, I will introduce the so-called trinity of relational quantum dynamics comprised of three distinct formulations of the same relational quantum theory: evolving constants of motion, the Page-Wootters formalism, and a symmetry reduction procedure. The equivalence between these formulations yields a temporal frame change map that transforms between the dynamics seen by different clocks. This map will be used to illustrate a temporal nonlocality effect that results in a superposition of time evolutions from the perspective of a clock indicating a superposition of different times. Then, a time-nonlocal modification to the Schrödinger equation will be shown to manifest when a system is coupled to the clock that is tracking its evolution. Such clock-system interactions should be expected at some scale when the gravitational interaction between them is taken into account. Finally, I will examine relativistic particles with internal degrees of freedom that constitute a clock that tracks their proper time. By evaluating the conditional probability associated with two such clocks reading different proper times, I will show that these clocks exhibit a novel quantum time dilation effect when moving in a superposition of different momenta.