Study of Quantum Set Theory Aiming at an Interpretation of Elements of Reality in Quantum Theory

如何对量子可观测值给出一个现实的解释，是物理学和哲学领域长期存在的开放性问题之一。在这项研究中，我们提出了一种基于量子集合理论的新方法来解决这个问题。研究了量子集合论与量子理论的模态解释之间的关系，证明了von Neumann代数在给定状态下是可成立代数的条件与量子集合论的子域在该状态下满足ZFC集合论所有定理的条件之间的等价性。利用量子集合理论中两个观测值的量子完全相关和两个实数的量子集合理论等价，将传统的射影测量的无坍缩解释推广到任意广义测量的无坍缩解释。

It is one of long-standing open problems in the field across physics and philosophy to give a realistic interpretation for values of quantum observables. In this research we proposed a new approach to this problem based on quantum set theory. We studied the relationship between quantum set theory and modal interpretation of quantum theory, and we showed the equivalence between the condition for a von Neumann algebra to be a beable algebra for a given state and the condition for a subuniverse of quantum set theory to satisfy all the theorems of ZFC set theory in that state. We also extended the conventional no collapse interpretation for projective measurements to a no collapse interpretation for arbitrary generalized measurements by using the equivalence between quantum perfect correlation for two observables and the quantum-set-theoretical equality for two real numbers in quantum set theory.