ar X iv : h ep - t h / 06 01 14 0 v 3 4 J un 2 00 7 Fermion Systems in Discrete Space - Time

Fermion systems in discrete space-time are introduced as a model for physics on the Planck scale. We set up a variational principle which describes a non-local interaction of all fermions. This variational principle is symmetric under permutations of the discrete space-time points. We explain how for minimizers of the variational principle, the fermions spontaneously break this permutation symmetry and induce on space-time a discrete causal structure. It is generally believed that the concept of a space-time continuum (like Minkowski space or a Lorentzian manifold) should be modified for distances as small as the Planck length. We here propose a concise model where we assume that space-time is discrete on the Planck scale. Our notion of “discrete spacetime” differs from other discrete approaches (like for example lattice gauge theories or spin foam models) in that we do not assume any structures or relations between the space-time points (like the nearest-neighbor relation on a lattice or a causal network). Instead, we set up a variational principle for an ensemble of quantum mechanical wave functions. The idea is that for mimimizers of our variational principle, these wave functions should induce relations between the discrete space-time points, which, in a suitable limit, should go over to the topological and causal structure of a Lorentzian manifold. The concepts outlined here are worked out in detail in a recent book [1]. Furthermore, in this book the connection to the continuum theory is made precise by introducing the notion of the continuum limit, and mathematical methods are developed for analyzing our variational principle in this limit. More specifically, in the continuum limit the fermionic wave functions group to a configuration of Dirac seas; for details see [5]. Analyzing our variational principle in the continuum limit gives concrete results for the effective continuum theory; see [1] and the review article [4]. In this short article we cannot enter the constructions leading to the continuum limit. Instead, we introduce the mathematical framework in the discrete setting (Sections 1 and 2) and discuss it afterwards, working out the underlying physical principles (Section 3). We finally describe the spontaneous symmetry breaking and the appearance of a “discrete causal structure” (Section 4).