More really is different (Q2389362)

This paper tries to support the claim that \textit{More is different} [\textit{P. W. Anderson}, ``More is different'', Science 393 (1972)] in the sense that the global behaviour and macroscopic properties (magnetisation, spin-spin correlation, pressure, entropy etc.) of complex physical systems cannot be fully understood only in terms of the fundamental rules its microscopic particles are subject to. To back up the above mentioned paradigm, the authors suggest an elementary 2-dimensional Ising model for which they claim so called \textit{emergent} behaviour. The model resembles the action of an arbitrary Turing machine simulating a universal cellular automaton (CA) by coding the standard operations from logic (NAND, XOR as well as SWAP and FANOUT) in the ground state of a finite, specially designed block of the Ising lattice. Following from the classical halting problem, undecidability of the computational CA-model implies unpredictability of (macroscopic) observables, which can not be deduced from knowing the (microscopic) lattice Hamiltonian. This is related to undecidability results in the theory of multidimensional symbolic dynamics and tilings [R.\ Berger, R.\ Robinson], thus does not come as a surprise for mathematicians working is this field. Nevertheless from a physical point of view the argument is not as convincing as the paper claims, especially as the authors have to use the mathematical idealization of an infinite system instead of a physically existing, large but finite one for which the undecidable questions become decidable. Furthermore its specific realization -- choosing a particular, more than artificial Hamiltonian to code the universal CA-rule which need not be realized in nature -- probably distinguishes this theoretical limit from any real world system. Moreover, from the viewpoint of mathematical logic it is a hard, most often undecidable problem to know whether a system we see in nature resembles exactly the particular model one constructed. Hence the paper by no means contradicts the existence of a \textit{theory of everything}, but rather demonstrates the expected difficulty to calculate quantities and understand the interplay of particles in a system with a large or even infinite number of constituents.