  We investigate the continuous extension of discrete shift translations on one
dimensional quantum lattice systems. We focus on a specific construction
provided by a quasi-free C*-flow on the one-dimensional fermion lattice system.
This quasi-free C*-flow is heuristically generated by a long-range Hamiltonian
consisting of two-body translation invariant interactions with
$\frac{1}{r}$-decay. We explore statistical-mechanical interpretation of such a
long-range Hamiltonian, which may be more naturally associated with a momentum
operator rather than the Hamiltonian. Through its explicit dynamical formulas,
wherein notably the sinc function appears, the continuous shift translations
reveal violations of causality and locality. Furthermore, we demonstrate that
this quasi-free C*-flow, implementing the shift translations, cannot be
extended to the one-dimensional quantum spin system via the Jordan-Wigner
transformation.
