Paul Baird

Our objective is to construct an elementary universe populated entirely by graphs from which geometry and dynamics emerge. The universe is based on a binary relation between objects: are they connected by an edge or not? It is only this relation that matters; the nature of the objects being irrelevant. Our perspective is that, out of the graphs so formed, further implicit structure is present, defined by a geometric spectrum and corresponding fields. This implicit structure comes into play when graphs correlate. Thus we propose ways in which graphs can interact and so dynamics, that is change, appears. With a suitable definition of time, this change can be ordered to give a universe endowed with local geometry and time.

Quantum mechanics is a linear theory. This is a strange fact. Why should nature be so convenient? Perhaps linearity is simply a consequence of small perturbations against a relatively uniform background. This can be formalized within the mathematical notions of weak solution and linearization. The true picture is unlikely to be so and some more appropriate framework may be required to deal with theories where this uniformity is lost, such as for example quantum gravity. A geometric model of quantum interaction provides a useful illustration. In this case the equations are quadratic, but both their weak form and linearization leads to a striking analogue of the Schrödinger equation.

In a relational universe, the only properties of a system that have meaning are those that arise from correlations with other systems. No state exists in an absolute sense independently of other states. Our basic premise is that the universe is built from elementary combinatorial (or relational) structures. These structures change and if we are not to suppose the existence of laws over and above the universe, then they must encode the rules for their own change. Geometry, time and consciousness, it is argued, emerge from such a universe as relational Kantian phenomena.