  In quantum theory, the concept of time rests on shaky ground. One way to
address this problem is to remove the usual background time parameter as a
primitive entity and explain its emergence via correlations between physical
systems. This approach was adopted by Page and Wootters (1983), who showed how
time can emerge in a stationary quantum universe from the correlations between
two of its subsystems, one of them acting as a clock for the other. In this
work, I study the robustness of the Page-Wootters construction across different
pictures, states of the universe and clock interactions, clarifying the role
and the nature of the correlations between the subsystems of the universe. I
start by showing how to formulate the Page-Wootters construction in the
Heisenberg picture via a unitary change of basis. I consider both pure and
mixed states of the universe and extend the analysis to include interactions
between the clock and the other subsystem of the universe. The study reveals
what kind of correlations are necessary for the construction to work.
Interestingly, entanglement is not required as long as there are no
interactions with the clock. The study also shows that these interactions can
lead to a non-unitary evolution for some mixed states of the universe. In a
simple two-level system, this aspect becomes relevant at scales where one would
expect strong relativistic effects. At these scales, I also observe an
inversion in the system's direction of time.
