The Finite Group Velocity of Quantum Spin Systems
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
- PDF / 745,124 Bytes
- 7 Pages / 439.37 x 666.142 pts Page_size
- 106 Downloads / 182 Views
© by Springer-Verlag 1972
The Finite Group Velocity of Quantum Spin Systems Elliott H. Lieb* Dept. of Mathematics, Massachusetts Institute of Technology Cambridge, Massachusetts, USA
Derek W. Robinson** Dept. of Physics, Univ. Aix-MarseiIle II, MarseiIle-Luminy, France Received May 15, 1972 Abstract. It is shown that if ~ is a finite range interaction of a quantum spin system, t~ the associated group of time translations, t x the group of space translations, and A, B local observables, then lim II[t~ t x (A). 8] II
e"(u), =
0
I'I~"
Ixl>ul'l
whenever v is sufficiently large (v > V",) where II(V) > O. The physical content of the statement is that information can propagate in the system only with a finite group velocity.
1. Introduction
In [2] it was demonstrated that for a large class of translationally invariant interactions, time translations of quantum spin systems can be defined as automorphisms of a C*-algebra, .91, of quasi-local observables,
i.e. the abstract algebra generated by the spin operators. This should allow one to discuss features of the dynamical propagation of physical effects in an algebraic manner independent of the state of the system, i.e. independent of the kinematical data. It is expected that this propagation has many features in common with the propagation of waves in continuous matter and the point of this paper is to demonstrate such a feature, namely a finite bound for the group velocity of a system with finite range interaction. This result is obtained by a simple estimation derived from the equations of motion and it is possible that more detailed estimations would give more precise information of the form of spin-wave propagation. We briefly discuss this possibility at the end of Section 3. * Work supported by National Science Foundation Grant N°: GP-31674 X. ** Work supported by National Science Foundation Grants N°: GP-31239 X and GP-30819 X.
B. Nachtergaele et al. (eds.), Statistical Mechanics © Springer-Verlag Berlin Heidelberg 2004
425
With D.W. Robinson in Commun. Math. Phys. 28, 251-257 (1972) E. H. Lieb and D. W. Robinson:
252
2. Basic Notation
We use the formalism introduced in [1] and [2]. For completeness we recall the basic definitions which will be used in the sequel. The kinematics of a quantum spin system constrained to a v-dimensional cubic lattice, ZV, are introduced by associating with each point x E 71': an N-dimensional vector space Jfx and with each finite set A c 71: the direct product space xeA
The algebra of strictly local observables, .9IA, of the subsystem A, is defined to be the algebra of all matrices acting on ~. If Al C A 2 , the algebra .9IA, acting on~, can be identified with the algebra .9IA, ® lIA2\A' acting on ~2 (llA2\A' is the identity operator on ~2\A,) and with this identification .9IA, C .9IA2 . Due to this isotony relation the set theoretic union of all .9IA, with A C ZV finite is a normed *-algebra and we define the completion of this algebra to be the C*-algebra .91 of (quasi-)local kinematical observables of the spin system
Data Loading...
