  Entertaining the possibility of time travel will invariably challenge dearly
held concepts of fundamental physics. It becomes relatively easy to construct
multiple logical contradictions using differing starting points from various
well-established fields of physics. Sometimes, the interpretation is that only
a full theory of quantum gravity will be able to settle these logical
contradictions. Even then, it remains unclear if the multitude of problems
could be overcome. Yet as definitive as this seems to the notion of time travel
in physics, such a recourse to quantum gravity comes with its own,
long-standing challenge to most of these counter-arguments to time travel:
These arguments rely on time, while quantum gravity is (in)famously stuck with
and dealing with the problem of time. One attempt to answer this problem within
the canonical framework resulted in the Page-Wootters formalism, and its recent
gauge-theoretic re-interpretation - as an emergent notion of time. Herein, we
will begin a programme to study toy models implementing the Hamiltonian
constraint in quantum theory, with an aim towards understanding what an
emergent notion of time can tell us about the (im)possibility of time travel.
