Numerical Analysis of the Causal Action Principle in Low Dimensions

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.