量子逻辑中一个形式化的状态-性质对偶关系（英文）

A Formal State-Property Duality in Quantum Logic

本文展示了量子物理中一个状态一性质对偶关系的形式化。在性质方面,Piron证明了Piron格(最初被称为不可分解的命题系统)刻画了量子系统的可测试性质所组成的结构。在状态方面,我们定义量子Kripke框架来刻画量子系统的状态在非正交关系之下所组成的结构。而且,我们定义了Piron格之间的线性态射,并把Piron格所组成的类组织成一个范畴。我们也定义了量子Kripke框架之间的连续同态,并把量子Kripke框架所组成的类组织成一个范畴。最后,我们证明了在范畴论的意义上Piron格所组成的范畴和量子Kripke框架所组成的范畴是对偶的,这样我们就用数学的语言描述了量子物理里面一个直观上的状态—性质对偶关系。这个形式化的对偶关系在代数结构和关系结构之间建立了联系,这将会有助于研究关于量子物理的逻辑。

This paper presents a formalization of the state-property duality in quantum physics.On the side of properties,Piron shows that Piron lattices,originally called irreducible propositional systems,capture the essential structure formed by the testable properties of quantum systems.On the side of states,we define quantum Kripke frames to capture the essential structure formed by the states of quantum systems under the non-orthogonality relation.Moreover,we define linear morphisms between Piron lattices,and then organize the class of Piron lattices into a category.We also define continuous homomorphisms between quantum Kripke frames,and then organize the class of quantum Kripke frames into a category.Finally,we will show a duality,in the sense of category theory,between the category of Piron lattices and the category of quantum Kripke frames,and thus capture the conceptual state-property duality in quantum physics in a mathematical language.This formal duality,connecting algebraic structures with relational structures,will be helpful in the study of logics of quantum physics.