  To admit a canonically conjugate time operator, the Hamiltonian has to be a
generator of translations (like the momentum operator generates translations in
space), so its spectrum must be unbounded. But the Hamiltonian governing our
world is thought to be bounded from below. Also, judging by the number of
fields and parameters of the Standard Model, the Hamiltonian seems much more
complicated.
  In this article I give examples of worlds governed by Hamiltonians generating
translations. They can be expressed as a partial derivative operator just like
the momentum operator, but when expressed in function of other observables they
can exhibit any level of complexity. The examples include any quantum world
realizing a standard ideal measurement, any quantum world containing a clock or
a free massless fermion, the quantum representation of any deterministic
time-reversible dynamical system without time loops, and any quantum world that
cannot return to a past state.
  Such worlds are as sophisticated as our world, but they admit a time
operator. I show that, despite having unbounded Hamiltonian, they do not decay
to infinite negative energy any more than any quantum or classical world. Since
two such quantum systems of the same Hilbert space dimension are unitarily
equivalent even if the physical content of their observables is very different,
they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking
the observables into account removes the ambiguity of HSF and the clock
ambiguity problem attributed to the Page-Wootters formalism, also caused by
assuming HSF. These results provide additional motivations to restore the
spacetime symmetry in the formulation of Quantum Mechanics and for the
Page-Wootters formalism.
