Zalamea, Federico (2016) Chasing Individuation: Mathematical Description of Physical Systems. [Preprint]

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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 Jordan-Lie algebra. From the geometric point of view, the space of states of any system is described by a uniform Poisson space with transition probability. Both these structures are here perceived as formal translations of 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 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.

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Item Type:	Preprint
Creators:	Creators Email ORCID Zalamea, Federico federicozalamea@gmail.com
Keywords:	Quantum Mechanics; Classical Mechanics; Symplectic Geometry; Abstraction; Mathematical Structuralism;
Subjects:	Specific Sciences > Physics > Classical Physics Specific Sciences > Mathematics Specific Sciences > Physics > Quantum Mechanics
Depositing User:	Mr Federico Zalamea
Date Deposited:	08 Dec 2016 15:45
Last Modified:	08 Dec 2016 15:45
Item ID:	12681
Journal or Publication Title:	Ph.D. Thesis- University Paris Diderot
Subjects:	Specific Sciences > Physics > Classical Physics Specific Sciences > Mathematics Specific Sciences > Physics > Quantum Mechanics
Date:	December 2016
URI:	https://philsci-archive.pitt.edu/id/eprint/12681

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