Communications in Mathematical Physics

, Volume 28, Issue 3, pp 251–257 | Cite as

The finite group velocity of quantum spin systems

  • Elliott H. Lieb
  • Derek W. Robinson


It is shown that if Φ is a finite range interaction of a quantum spin system,τtΦ 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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Group Velocity 
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  1. 1.
    Robinson, D. W.: Commun. math. Phys.6, 151 (1967).Google Scholar
  2. 2.
    Robinson, D. W.: Commun. math. Phys.7, 337 (1968). -- See also; Streater, R. F.: Commun. math. Phys.7, 93 (1968). -- Ruskai, M. B.: Commun. math. Phys.20, 193 (1971).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  • Derek W. Robinson
    • 2
    • 3
  1. 1.Dept. of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Dept. of PhysicsUniv. Aix-Marseille IIMarseille-LuminyFrance
  3. 3.Centre de Physique Théorique C.N.R.S.Marseille 9°France

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