Communications in Mathematical Physics

, Volume 28, Issue 3, pp 251-257

First online:

The finite group velocity of quantum spin systems

  • Elliott H. LiebAffiliated withDept. of Mathematics, Massachusetts Institute of Technology
  • , Derek W. RobinsonAffiliated withDept. of Physics, Univ. Aix-Marseille IICentre de Physique Théorique C.N.R.S.

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