More really is different

In 1972, P.W. Anderson suggested that ‘More is Different’, meaning that complex physical systems may exhibit behavior that cannot be understood only in terms of the laws governing their microscopic constituents. We strengthen this claim by proving that many macroscopic observable properties of a simple class of physical systems (the infinite periodic Ising lattice) cannot in general be derived from a microscopic description. This provides evidence that emergent behavior occurs in such systems, and indicates that even if a ‘theory of everything’ governing all microscopic interactions were discovered, the understanding of macroscopic order is likely to require additional insights.