Extremal variety as the foundation of a cosmological quantum theory
  Dynamical systems of a new kind are described, which are motivated by the problem of constructing diffeomorphism invariant quantum theories. These are based on the extremization of a non-local and non-additive quantity that we call the variety of a system. In these systems all dynaqmical variables refer to relative coordinates or, more generally, describe relations between particles, so that they are invariant under discrete analogues of diffeomorphisms in which the labels of all particles are permutted arbitrarily. The variety is a measures of how uniquely each of the elements of the system can be distinguished from the others in terms of the values of these relative coordinates. Thus a system with extremal variety is one in which the parts are related to the whole in as distinct a way as possible.   We study numerically several dynamical systems which are defined by setting the action of the system equal to its variety. We find evidence that suggests that such systems may serve as the basis for a new kind of pregeometry theories in which the geometry of low dimensional space emerges in the thermodynamic limit from a system which is defined without the use of any background space.   The mathematical definition of variety may also provide a quantitative tool to study self-organizing systems, because it distinguishes highly structured, but asymmetric, configurations such as one finds in biological systems from both random configurations and highly ordered configurations. 
Julian Barbour and Lee Smolin