Study of the Probabilistic Interpretation of Quantum Set Theory

2016 Fiscal Year Final Research Report

• Project/Area Number: 15K13456
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Nagoya University
• Principal Investigator: OZAWA Masanao 名古屋大学, 情報科学研究科, 特任教授 (40126313)
• Project Period (FY): 2015-04-01 – 2017-03-31
• Keywords: 量子集合論 / 公理的集合論 / ブール代数値モデル / 量子論理 / 含意結合子 / 移行原理 / オーソモデュラー束 / フォン・ノイマン代数

Outline of Final Research Achievements

The study of quantum set theory, set theory based on quantum logic, aims to reconstruct quantum theory and to extend its probabilistic interpretation based on logical methods. In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional. To reconcile arbitrariness, in this research, we studied the difference of quantum set theories based on different conditionals. We proved that there are exactly 6 polynomially definable binary operations, for which the transfer principle of quantum set theory holds. For three of them called material conditional we clarified the difference of the probabilistic interpretations of the order relation between quantum observables defined by quantum set theories based on those conditionals, and gave their experimentally testable characterizations. By this research we can expect the emergence of a new research field between foundations of mathematics and physics and its applications to quantum information technology.

Free Research Field

数物系科学