Communications in Mathematical Physics

, Volume 28, Issue 3, pp 251–257

The finite group velocity of quantum spin systems


  • Elliott H. Lieb
    • Dept. of MathematicsMassachusetts Institute of Technology
  • Derek W. Robinson
    • Dept. of PhysicsUniv. Aix-Marseille II
    • Centre de Physique Théorique C.N.R.S.

DOI: 10.1007/BF01645779

Cite this article as:
Lieb, E.H. & Robinson, D.W. Commun.Math. Phys. (1972) 28: 251. doi:10.1007/BF01645779


It is shown that if Φ is a finite range interaction of a quantum spin system,τtΦ the associated group of time translations, τx 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.

Copyright information

© Springer-Verlag 1972