Chasing Individuation: Mathematical Description of Physical Systems

TL;DR

This paper explores the mathematical foundations of Classical and Quantum Mechanics, emphasizing the role of algebraic and geometric structures in describing properties as quantities and transformations, and investigates techniques for system individuation.

Contribution

It introduces a unified algebraic and geometric framework for classical and quantum systems and analyzes methods, including group theory, to address the individuation of physical systems.

Findings

Quantum mechanics differs from classical by a compatibility condition.

A tension exists between abstract mathematical structures and state specification.

Group theory offers partial solutions for system individuation.

Abstract

This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the set of observables of a physical system, be it classical or quantum, is described by a \emph{Jordan-Lie algebra}. From the geometric point of view, the space of states of any system is described by a \emph{uniform Poisson space with transition probability}. Both these structures are here perceived as formal translations of \emph{the fundamental twofold role of properties in Mechanics}: they are at the same time quantities and transformations. The question becomes then to understand the precise articulation between these two roles. The analysis will show that Quantum Mechanics can be thought as distinguishing itself from Classical Mechanics by a compatibility condition between properties-as-quantities and properties-as-transformations. Moreover, this dissertation shows the existence of a tension between a certain "abstract way" of conceiving mathematical structures, used in the practice of mathematical physics, and the necessary capacity to specify particular states or observables. It then becomes important to understand how, within the formalism, one can construct a \emph{labelling scheme}. The "Chase for Individuation" is the analysis of different mathematical techniques which attempt to overcome this tension. In particular, we discuss how group theory furnishes a partial solution.