"Candidate theories of quantum gravity must answer the questions: how can the dynamics of quantum states of matter and geometry be defined in a diffeomorphism-invariant way? How are the quantum states related to probabilities in the absence of a preferred time? To address these issues, we discuss the construction and interpretation of relational observables in quantum theories with worldline diffeomorphism invariance, which serve as toy models of quantum gravity. In this context, we present a method of construction of quantum relational observables which is analogous to the construction of gauge-invariant extensions of noninvariant quantities in usual gauge (Yang-Mills) theories. Furthermore, we discuss how the notion of a physical propagator may be used to define a unitary evolution in the quantum theory, which is to be understood in terms of a generalized clock, as is the classical theory. We also discuss under which circumstances this formalism can be related to the use of conditional probabilities in a generalization of the Page-Wootters approach. Finally, we also examine how our formalism can be adapted to calculations of quantum-gravitational effects in the early Universe.
Refs.: L. Chataignier, Phys. Rev. D 101, 086001 (2020); 103, 026013 (2021); 103, 066005 (2021)"