从因果作用原理对爱因斯坦方程的几何推导
A Geometric Derivation of the Einstein Equations from the Causal Action Principle
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中文总结 AI 辅助
研究从因果作用原理推导爱因斯坦方程，通过对因果费米子系统的特定分析，引入密切真空概念，利用拉格朗日量诱导洛伦兹度量，得出里奇张量满足爱因斯坦方程及引力耦合常数，还提供了推导方程修正的系统程序。
中文摘要 AI 辅助
对因果费米子系统的因果作用原理进行分析，针对一种最小化测度，其支撑集假定具有光滑流形$\tilde{M}$的结构。引入了密切真空的概念。结果表明拉格朗日量在$\tilde{M}$上诱导出一个洛伦兹度量。此外，因果作用的欧拉 - 拉格朗日方程意味着对于以正则化长度的幂次展开给出的能量动量张量，里奇张量必须满足广义相对论的爱因斯坦方程。引力耦合常数被发现是正则化长度的平方。我们的方法为推导爱因斯坦方程的修正提供了一个系统程序。本文还对因果变分原理和因果作用原理进行了自包含的介绍。大多数几何结构（联络、黎曼度量和曲率）在$\tilde{M}$任意维度的因果变分原理的一般设定下被引入和分析。洛伦兹设定仅适用于因果费米子系统，且仅在四个时空维度中进行了研究。
英文摘要
The causal action principle for causal fermion systems is analyzed for a minimizing measure whose support is assumed to have the structure of a smooth manifold $\tilde{M}$. The concept of osculating vacua is introduced. It is shown that the Lagrangian induces on $\tilde{M}$ a Lorentzian metric. Moreover, the Euler-Lagrange equations of the causal action imply that the Ricci tensor must satisfy the Einstein equations of general relativity for an energy momentum tensor given in terms of a power expansion in the regularization length. The gravitational coupling constant is found to be the square of the regularization length. Our methods provide a systematic procedure for deriving corrections to the Einstein equations. The paper includes a self-contained introduction to causal variational principles and the causal action principle. Most geometric structures (connection, Riemannian metric and curvature) are introduced and analyzed in the general setting of causal variational principles for an arbitrary dimension of $\tilde{M}$. The Lorentzian setting works only for causal fermion systems and is worked out only in four spacetime dimensions.
