Fermion Systems in Discrete Space-Time - Outer Symmetries and Spontaneous Symmetry Breaking

TL;DR

This paper develops a method to construct fermion systems in discrete space-time with specific symmetries and demonstrates that permutation symmetry is spontaneously broken in systems with many particles and space-time points.

Contribution

A systematic procedure for constructing fermion systems with prescribed outer symmetries and analysis of symmetry breaking in complex systems.

Findings

Permutation symmetry is always spontaneously broken in systems with many particles and space-time points.

Constraints on particle number are derived for systems with symmetric group symmetry.

Illustrative examples demonstrate the construction method.

Abstract

A systematic procedure is developed for constructing fermion systems in discrete space-time which have a given outer symmetry. The construction is illustrated by simple examples. For the symmetric group, we derive constraints for the number of particles. In the physically interesting case of many particles and even more space-time points, this result shows that the permutation symmetry of discrete space-time is always spontaneously broken by the fermionic projector.