Are minimizers of causal variational principles discrete?

We begin by introducing an action principle defined on a finite set of points. This action principle is causal in the sense that it generates a relation on pairs or points which distinguishes between spacelike and timelike separation. In this way, minimizing the action gives rise to a "discrete causal structure". We generalize our action principle to include continuum space-times and review existence results. We outline how the same action principle can be formulated in Minkowski space to obtain a formulation of quantum field theory.

In the second part of the talk, we consider as a special case a variational principle for Borel measures on the two-sphere. We prove that the support of every minimizing measure has no interior. This can be understood that when minimizing the action, a spontaneous symmetry breaking effect leads to the formation of a discrete structure.