The finite group velocity of quantum spin systems

Abstract

It is shown that if Ф is a finite range interaction of a quantum spin system, τ t Ф the associated group of time translations, τ x the group of space translations, and A, B local observables, then $$ \mathop {{\text{lim}}}\limits_{\mathop {\left| t \right| \to \infty }\limits_{\left| x \right| > v\left| t \right|} } \left\| {\left[ {\tau _t^\varphi \tau \left( A \right),B} \right]} \right\|e^{\mu \left( v \right)t} = 0$$ (1) whenever v is sufficiently large (ν > V Ф ,) where μ(ν) > 0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.