From Discrete Space-Time to Minkowski Space: Basic Mechanisms, Methods and Perspectives

TL;DR

This survey explores how discrete space-time models relate to Minkowski space, focusing on mechanisms like spontaneous symmetry breaking and methods such as lattice models and continuum limits to understand the transition between these frameworks.

Contribution

It provides a comprehensive overview of recent approaches and methods for connecting discrete space-time systems with Minkowski space, including new models and analytical techniques.

Findings

Spontaneous symmetry breaking leads to emergent causal structures.

Lattice models can simulate static, isotropic space-times.

Continuum limit methods analyze interactions in discrete systems.

Abstract

This survey article reviews recent results on fermion system in discrete space-time and corresponding systems in Minkowski space. After a basic introduction to the discrete setting, we explain a mechanism of spontaneous symmetry breaking which leads to the emergence of a discrete causal structure. As methods to study the transition between discrete space-time and Minkowski space, we describe a lattice model for a static and isotropic space-time, outline the analysis of regularization tails of vacuum Dirac sea configurations, and introduce a Lorentz invariant action for the masses of the Dirac seas. We mention the method of the continuum limit, which allows to analyze interacting systems. Open problems are discussed.