Categorical Foundation of Quantum Mechanics and String Theory

TL;DR

This paper explores how category theory and Peircean relational logic can serve as foundational frameworks for quantum mechanics and string theory, unifying these theories through algebraic structures.

Contribution

It introduces a categorical algebra based on Peircean logic that derives quantum laws and connects to string theory via group structures like W_infinity.

Findings

Categorical algebra reproduces fundamental quantum laws.

Relational logic generates group structures linked to string theory.

Category theory may serve as a new paradigm for physical theories.

Abstract

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.