论文标题
在低维度中的因果行动原理的数值分析
Numerical Analysis of the Causal Action Principle in Low Dimensions
论文作者
论文摘要
通过采用可区分的编程方法来提出因果费系统的数值分析。为整数参数的一般值$ f $（粒子号），$ n $（旋转尺寸）和$ m $（时空点的数量）引入了加权计数措施的因果行动原理。在$ n = 1 $的情况下，因果关系在几何上阐明了因果关系。在$ n = 1，f = 2 $和$ n = 2，f = 4 $的情况下，引入了离散的狄拉克球作为最小值的候选者。我们对因果行动原理提供了彻底的数值分析，用于在$ n = 1,2 $和$ f = 2,3,4 $的情况下进行加权计数措施。我们的数值发现证实了所有大型$ m $的最小化器都是离散狄拉克球体的良好近似值。在示例$ n = 1，f = 3 $中解释了如何通过投影的时空图可视化数值最小化。讨论了方法和前景来研究迄今已知最小化的分析候选者的数值设置。
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.
