Categorical Foundation of Quantum Mechanics and String Theory

Abstract

The unification of QuantumMechanics 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 Rij among two individual terms Si and Sj , 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∞. The group W∞ 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. The raison d’ être of physics is to understand the wonderful variety of nature in a unified way. A glance at the history of physics is revealing: the unification of terrestrial and celestial Mechanics by Newton in the 17th century; of optics with the theories of electricity and magnetism by Maxwell in the 19th century; of space-time geometry and the theory of gravitation by Einstein in the years 1905 to 1916; and of thermodynamics and atomic physics through the advent of Quantum Mechanics in the 1920s [1]. The next leap in this on-going process is the unification of the two pillars of modern physics, quantum mechanics and general relativity. String theory, in this respect, appears as the most promising example of a candidate unified theory [2]. Strings emerged in the study of strong interactions, modelling the flux tubes between quark-antiquark pairs in hadronic collisions, in the Regge limit, nicely described by the Veneziano amplitude [3], which can be reproduced from a relativistic string theory [4].

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Cite this paper

@inproceedings{Nicolaidis2008CategoricalFO, title={Categorical Foundation of Quantum Mechanics and String Theory}, author={A. Nicolaidis}, year={2008} }