On the representation of differentiations by Hamiltonians, operating from an algebra of local observable spin systems

Abstract

Let$\mathop \mathfrak{A}\limits^ -$ and$\mathfrak{A}$ be algebras of local and quasilocal observable spin systems corresponding to the group Zr,$D:\mathfrak{A} \to \mathop \mathfrak{A}\limits^ -$ be a differentiation invariant with respect to displacements. The question of representation of D in the form of formal Hamiltonian$H = \sum _{k \in Z^r } T_k X$ formed by the displacements of an elementx ε$\mathop \mathfrak{A}\limits^ -$ is considered. It is shown that such a representation exists if the condition$\mathop \mathfrak{A}\limits^ -$ holds, where$a \in \mathfrak{A}$ means an element obtained from the elements [T_kX,a] by some r-multiple process of summation.