The finite group velocity of quantum spin systems

Abstract

It is shown that if Φ is a finite range interaction of a quantum spin system,τ _t ^Φ the associated group of time translations, τ _x the group of space translations, andA, B local observables, then

$$\mathop {\lim }\limits_{\begin{array}{*{20}c} {|t| \to \infty } \\ {|x| > \upsilon |t|} \\ \end{array} } ||[\tau _t^\Phi \tau _x (A),B]||e^{\mu (\upsilon )t} = 0$$

wheneverv is sufficiently large (v>V_Φ) where μ(v)>0. The physical content of the statement is that information can propagate in the system only with a finite group velocity.