BOUNDARY-CONDITIONS FOR QUANTUM-LATTICE SYSTEMS

For classical lattice systems, the Dobrushin-Lanford-Ruelle theory of boundary conditions states that the restriction of a global equilibriu m state to a subsystem can be obtained as an integral over equilibrium states of the subsystem alone. The Hamiltonians for the subsystem are obtained by fixing a configuration for the variables in the complemen t of the subsystem, or more generally, by evaluating the full interact ion Hamiltonian with respect to a state for the complement. We provide examples showing that the quantum mechanical version of this statemen t is false. It fails even if the subsystem is classical, but embedded into a quantum environment. We suggest an alternative characterization of the local restrictions of global equilibrium states by inequalitie s involving only local data.