Numerical Analysis of the Causal Action Principle in Low Dimensions

TL;DR

This paper employs differentiable programming to numerically analyze the causal action principle in low-dimensional causal fermion systems, introducing discrete Dirac spheres as minimizers and exploring their properties.

Contribution

It introduces a numerical framework for analyzing the causal action principle in low dimensions and identifies discrete Dirac spheres as potential minimizers for large systems.

Findings

Minimizers approximate discrete Dirac spheres for large system sizes.

Numerical methods visualize minimizers via projected spacetime plots.

The approach extends to settings lacking known analytic minimizers.

Abstract

The numerical analysis of causal fermion systems is advanced by employing differentiable programming methods. The causal action principle for weighted counting measures is introduced for general values of the integer parameters $f$ (the particle number), $n$ (the spin dimension) and $m$ (the number of spacetime points). In the case $n=1$, the causal relations are clarified geometrically in terms of causal cones. Discrete Dirac spheres are introduced as candidates for minimizers for large $m$ in the cases $n=1, f=2$ and $n=2, f=4$. We provide a thorough numerical analysis of the causal action principle for weighted counting measures for large $m$ in the cases $n=1,2$ and $f=2,3,4$. Our numerical findings corroborate that all minimizers for large $m$ are good approximations of the discrete Dirac spheres. In the example $n=1, f=3$ it is explained how numerical minimizers can be visualized by projected spacetime plots. Methods and prospects are discussed to numerically investigate settings in which hitherto no analytic candidates for minimizers are known.