Title: Scale invariant transfer matrices and Hamiltionians,
Abstract: Given a direct system of Hilbert spaces $s\mapsto \mathcal H_s$ (with
isometric inclusion maps $\iota_s^t:\mathcal H_s\rightarrow \mathcal H_t$ for
$s\leq t$) corresponding to quantum systems on scales $s$, we define notions of
scale invariant and weakly scale invariant operators. Is some cases of quantum
spin chains we find conditions for transfer matrices and nearest neighbour
Hamiltonians to be scale invariant or weakly so. Scale invariance forces
spatial inhomogeneity of the spectral parameter. But weakly scale invariant
transfer matrices may be spatially homogeneous in which case the change of
spectral parameter from one scale to another is governed by a classical
dynamical system exhibiting fractal behaviour.