Continuous extension of the discrete shift translations on one-dimensional quantum lattice systems

We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free C ∗ {C} -flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free C ∗ {C} -flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation.