  Time in quantum mechanics is peculiar: it is an observable that cannot be
associated to an Hermitian operator. As a consequence it is impossible to
explain dynamics in an isolated system without invoking an external classical
clock, a fact that becomes particularly problematic in the context of quantum
gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in
1983. PaW showed that dynamics can be an emergent property of the entanglement
between two subsystems of a static Universe. In this work we first investigate
the possibility to introduce in this framework a Hermitian time operator
complement of a clock Hamiltonian having an equally-spaced energy spectrum. An
Hermitian operator complement of such Hamiltonian was introduced by Pegg in
1998, who named it "Age". We show here that Age, when introduced in the PaW
context, can be interpreted as a proper Hermitian time operator conjugate to a
"good" clock Hamiltonian. We therefore show that, still following Pegg's
formalism, it is possible to introduce in the PaW framework bounded clock
Hamiltonians with an unequally-spaced energy spectrum with rational energy
ratios. In this case time is described by a POVM and we demonstrate that Pegg's
POVM states provide a consistent dynamical evolution of the system even if they
are not orthogonal, and therefore partially un-distinguishables.

Quantum Physics