量子力学和弦论的范畴基础
Categorical Foundation of Quantum Mechanics and String Theory
摘要：
统一量子力学和广义相对论仍然是理论物理学的主要目标，而弦论则是唯一可能的统一方案。在目前的工作中，在弦论的概念基础的搜索，我们分析的关系逻辑开发的C。S.皮尔士在19世纪后期。皮尔士逻辑具有范畴的数学结构，其中两个独立项$S_i$和$S_j$之间的关系$R_{ij}$充当箭头（或态射）。我们介绍了一个实现相应的范畴代数的组成，这自然会产生的基本量子定律，从而表明范畴理论的基础量子力学。同一个关系代数生成了许多组结构，其中包括$W_{\infty}$。$W_{\infty}$群由矩阵模型具体化和实现，矩阵模型本身与弦论紧密相连。本文认为，关系逻辑和广义范畴论可能为发展现代物理学理论提供一种新的范式。
英文摘要：
The unification of Quantum Mechanics and General Relativity remains the primary goal of Theoretical Physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations of string theory, we analyze the relational logic developed by C. S. Peirce in the late nineteenth century. The Peircean logic has the mathematical structure of a category with the relation $R_{ij}$ among two individual terms $S_i$ and $S_j$, serving as an arrow (or morphism). We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of Quantum Mechanics. The same relational algebra generates a number of group structures, among them $W_{\infty}$. The group $W_{\infty}$ is embodied and realized by the matrix models, themselves closely linked with string theory. It is suggested that relational logic and in general category theory may provide a new paradigm, within which to develop modern physical theories.
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