A notion of local algebras is introduced in the theory of causal fermion systems. Their properties are studied in the example of the regularized Dirac sea vacuum in Minkowski space. The commutation relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators associated to a Cauchy surface satisfy a time slice axiom. It is proven that the algebra generated by operators in an open set is irreducible as a consequence of Hegerfeldt's theorem. The light cone structure is recovered by analyzing expectation values of the operators in the algebra in the limit when the regularization is removed. It is shown that every spacetime point operator commutes with the algebras localized away from its null cone, up to small corrections involving the regularization length.

中文翻译：

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Minkowski 空间中因果费米子系统的局部代数

在因果费米子系统的理论中引入了局部代数的概念。在闵可夫斯基空间中正则化狄拉克海真空的例子中研究了它们的性质。求解了对易关系，讨论了与正则对易关系的区别。结果表明，与柯西表面相关联的时空点算子满足时间片公理。已经证明，作为 Hegerfeldt 定理的结果，由开集算子生成的代数是不可约的。通过分析去除正则化时极限中代数中算子的期望值来恢复光锥结构。结果表明，每个时空点算子都与远离其零锥的局部代数交换，直到涉及正则化长度的小修正。