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Numerical Analysis of the Causal Action Principle in Low Dimensions

Felix Finster, Robert H. Jonsson, Niki Kilbertus

TL;DR

The paper investigates numerical minimization of the causal action principle for weighted counting measures in low-dimensional causal fermion systems, focusing on $n=1,f=2$ and $n=2,f=4$ to understand asymptotics as the number of spacetime points $m$ grows. It develops differentiable-programming-based optimization to handle nonconvex, high-dimensional problems, and demonstrates that discrete Dirac sphere configurations emerge as absolute minimizers for large $m$ in the minimal Hilbert-space cases, with precise asymptotics for the action and boundedness functionals. The analysis also reveals causal cone structures for spin dimension one and introduces projected spacetime plots to visualize minimizers, aiding intuition and guiding further analytic work. Beyond the minimal cases, the authors begin exploring $n=1,f=3$ and other settings, outlining prospects for convex-relaxation approaches and higher $f$ to broaden understanding of the minimizers' geometry and scaling, with significant implications for the structure of spacetime in causal fermion systems.

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.

Numerical Analysis of the Causal Action Principle in Low Dimensions

TL;DR

The paper investigates numerical minimization of the causal action principle for weighted counting measures in low-dimensional causal fermion systems, focusing on and to understand asymptotics as the number of spacetime points grows. It develops differentiable-programming-based optimization to handle nonconvex, high-dimensional problems, and demonstrates that discrete Dirac sphere configurations emerge as absolute minimizers for large in the minimal Hilbert-space cases, with precise asymptotics for the action and boundedness functionals. The analysis also reveals causal cone structures for spin dimension one and introduces projected spacetime plots to visualize minimizers, aiding intuition and guiding further analytic work. Beyond the minimal cases, the authors begin exploring and other settings, outlining prospects for convex-relaxation approaches and higher to broaden understanding of the minimizers' geometry and scaling, with significant implications for the structure of spacetime in causal fermion systems.

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 (the particle number), (the spin dimension) and (the number of spacetime points). In the case , the causal relations are clarified geometrically in terms of causal cones. Discrete Dirac spheres are introduced as candidates for minimizers for large in the cases and . We provide a thorough numerical analysis of the causal action principle for weighted counting measures for large in the cases and . Our numerical findings corroborate that all minimizers for large are good approximations of the discrete Dirac spheres. In the example 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.
Paper Structure (22 sections, 11 theorems, 146 equations, 6 figures)

This paper contains 22 sections, 11 theorems, 146 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The Causal Action Principle for Weighted Counting Measures
  3. The Causal Action Principle on a Two-Dimensional Hilbert Space
  4. The Lagrangian and the Causal Structure
  5. Qualitative Description of Underlying Mechanisms
  6. Minimizing in the Class of Isotropic Measures
  7. The Two-Dimensional Discrete Dirac Sphere
  8. Causally Trivial Measures as Local Minimizers
  9. The Four-Dimensional Discrete Dirac Sphere
  10. Causal Cones in Spin Dimension One
  11. Numerical Analysis
  12. Overview and Optimization Choices
  13. Parametrization of the Optimization Problem
  14. Initialization
  15. The Asymptotics for Fixed m and Large f
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $\tau,\tau'\in[1,\infty)$ and $\vec{x},\vec{x}'\in S^2$. We denote the angle between $\vec{x}$ and $\vec{x}'$ by $\vartheta \in [0, \pi]$ (i.e. $\cos \vartheta = \vec{x}\cdot \vec{y}$). Then the operator product $F_\tau(\vec{x})F_{\tau'}(\vec{x}')$ has the eigenvalues Moreover, where $\chi_{[\vartheta_-,\vartheta_+]}$ is the characteristic function of the interval defined by

Figures (6)

  • Figure 1: Causal structure on ${\mathscr{F}}$ in the case $f=2$.
  • Figure 2: Causal cones in spin dimension one. On the right, conditions \ref{['eq:sideconditions']} on the coordinates $(y_0,y_1,y_2,y_3)$ representing $\pi_xy\pi_x$ are visualized. On the left, for fixed value of $y_0$ the causal cone in the $(y_1,y_2,y_3)$-hyperplane is shown, using rotational symmetry around the $y_3$-axis.
  • Figure 3: In the left column we show the numerically found minimal causal action for increasing values of $f$ and $m$ at $n=1$ (top) and $n=2$ (bottom). The right column then displays the scaling behavior of the minimal causal action at the largest value of $f$ for each $m$. A simple linear least squares fit (in log-log space) confirms the analytic asymptotic behavior of ${\mathcal{S}}$ as $f$ becomes large for both $n=1$ and $n=2$.
  • Figure 4: The minimal causal action (left) and value of the boundedness functional (right) found numerically for $n=1, f=2$ (top row) and $n=2, f=4$ (bottom row) as $m$ increases.
  • Figure 5: Clustering in the numerical configuration for $n=1,f=2,m=768$. Red lines indicate expectations from discrete Dirac sphere.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Proposition 3.6
  • ...and 13 more