Chasing Individuation: Mathematical Description of Physical Systems

📝 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.

💡 Analysis

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.

📄 Content

Université Sorbonne Paris Cité Université Paris.Diderot (Paris 7) ÉCOLE DOCTORALE : Savoirs scientifiques (ED 400) Laboratoire SPHERE, UMR 7219 Basic Research Community for Physics Projet ERC – Philosophie de la Gravitation Quantique Canonique Chasing Individuation: Mathematical Description of Physical Systems par Federico ZALAMEA Thèse de doctorat en Histoire et Philosophie des Sciences Dirigée par Gabriel CATREN Présentée et soutenue publiquement à l’Université Paris Diderot le 23 Novembre 2016 Président du jury: M. Olivier DARRIGOL, Directeur de recherche au CNRS Rapporteurs: M. James LADYMAN, Professeur à l’Université de Bristol M. Nicolaas P. LANDSMAN, Professeur à l’Université Radboud de Nijmegen Examinateurs: M. Marc LACHIÈZE-REY, Directeur de recherche au CNRS M. Thomas RYCKMAN, Professeur de l’Université de Stanford Directeur de thèse: M. Gabriel CATREN, Chargé de recherche au CNRS Université Sorbonne Paris Cité Université Paris.Diderot (Paris 7) ÉCOLE DOCTORALE : Savoirs scientifiques (ED 400) Laboratoire SPHERE, UMR 7219 Basic Research Community for Physics ERC Project – Philosophy of Canonical Quantum Gravity Chasing Individuation: Mathematical Description of Physical Systems by Federico ZALAMEA Ph.D. in Philosophy of Physics Under the supervision of Gabriel CATREN Presented and defended publicly at Paris Diderot University on November 23rd 2016 President of the jury: M. Olivier DARRIGOL, Directeur de recherche at CNRS Rapporteurs: M. James LADYMAN, Professor at the University of Bristol M. Nicolaas P. LANDSMAN, Professor at Radboud University Nijmegen Examinators: M. Marc LACHIÈZE-REY, Directeur de recherche at CNRS M. Thomas RYCKMAN, Professor at the University of Stanford Doctoral supervisor: M. Gabriel CATREN, Chargé de recherche at CNRS “All great insights and discoveries are not only usually thought by several people at the same time, they must also be re-thought in that unique effort to truly say the same thing about the same thing.” Martin Heidegger Acknowledgments The four years of research culminating with this dissertation were almost entirely funded by the European Research Council, under the European Community’s Seventh Framework Programme (Project Philosophy of Canonical Quantum Gravity, FP7/2007- 2013 Grant Agreement n° 263523). First, and foremost, I must thank my supervisor, Gabriel Catren, for providing me with the opportunity to be part of his project. During these past years, I have had the chance of evolving in the blurry borders between Physics, Mathematics and Philosophy. To be sure, the demand of navigating at ease within this triangle has oftentimes been a source of difficult challenges, and even of anguishing questions of identity. But it has also significantly enlarged my curiosity and consolidated my thinking. I particularly appreciate the freedom I have enjoyed to explore many different fields and develop my own questions. I know this freedom to be the sign of a rare confidence in my work, for which I am the more grateful. I also would like to thank warmly James Ladyman and Klaas Landsman for ac- cepting to be the rapporteurs of my dissertation, for their careful reading and for the detailed comments which will greatly help me in improving my work; Olivier Darrigol, Marc Lachièze-Rey and Thomas Ryckman for accepting to be part of my jury, for their comments, suggestions and encouragements in the various occasions I have had the luck of meeting them. It is for me an honor to have my work evaluated by such a high-level group of thinkers whose several works I admire. Because my research would have been exceedingly more difficult without a vi- brant institution favoring exchanges and discussions, I am indebted to the laboratory ii Acknowledgments SPHERE that hosted me during my graduate studies. In particular, I thank the former and current directors, David Rabouin and Pascal Crozet, as well as Roberto Angeloni for our discussions on the history of Quantum Mechanics, Karine Chemla for her inspir- ing and indefatigable enthusiasm, Nadine de Courtenay, Michel Paty and Jean-Jacques Szczeciniarz for their numerous advises and their generosity in sharing their valuable knowledge. The assistance of all the administrative staff has also been crucial all along the way. I thank Sandrine Pellé and Patricia Philippe for their help, Nad Fachard for her joyful presence and for her interest in all aspects of life, and Virginie Maouchi for her kindness and her unshakeable efficiency which rendered simple the otherwise labyrinthine administrative procedures. My colleagues of the ERC Project Philosophy of Canonical Quantum Gravity de- serve a special mention. In these four years, we spent countless hours together in informal discussions, meetings, seminars and workshops. My work would be signifi- cantly poorer had it not been confronted with their critical eye. I am extremely grateful to Alexandre Afgoustidis and Mathieu Anel for their patience and their generosity in guiding me through

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