  Schroedinger's equation says that the Hamiltonian is the generator of time
translations. This seems to imply that any reasonable definition of time
operator must be conjugate to the Hamiltonian. Then both time and energy must
have the same spectrum since conjugate operators are unitarily equivalent.
Clearly this is not always true: normal Hamiltonians have lower bounded
spectrum and often only have discrete eigenvalues, whereas we typically desire
that time can take any real value. Pauli concluded that constructing a general
a time operator is impossible (although clearly it can be done in specific
cases). Here we show how the Pauli argument fails when one uses an external
system (a "clock") to track time, so that time arises as correlations between
the system and the clock (conditional probability amplitudes framework). In
this case, the time operator is not conjugate to the system Hamiltonian, but
its eigenvalues still satisfy the Schroedinger equation for arbitrary
Hamiltonians.
