18 found
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  1. Geometric Foundations of Classical Yang–Mills Theory.Gabriel Catren - 2008 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 39 (3):511-531.
    We analyze the geometric foundations of classical Yang-Mills theory by studying the relationships between internal relativity, locality, global/local invariance, and background independence. We argue that internal relativity and background independence are the two independent defining principles of Yang-Mills theory. We show that local gauge invariance -heuristically implemented by means of the gauge argument- is a direct consequence of internal relativity. Finally, we analyze the conceptual meaning of BRST symmetry in terms of the invariance of the gauge fixed theory under general (...)
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  2. On Classical and Quantum Objectivity.Gabriel Catren - 2008 - Foundations of Physics 38 (5):470-487.
    We propose a conceptual framework for understanding the relationship between observables and operators in mechanics. To do so, we introduce a postulate that establishes a correspondence between the objective properties permitting to identify physical states and the symmetry transformations that modify their gauge dependant properties. We show that the uncertainty principle results from a faithful—or equivariant—realization of this correspondence. It is a consequence of the proposed postulate that the quantum notion of objective physical states is not incomplete, but rather that (...)
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  3.  82
    On the Relation Between Gauge and Phase Symmetries.Gabriel Catren - 2014 - Foundations of Physics 44 (12):1317-1335.
    We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state and \ to \ or \ in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s moment (...)
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  4.  12
    Grothendieck’s theory of schemes and the algebra–geometry duality.Gabriel Catren & Fernando Cukierman - 2022 - Synthese 200 (3):1-41.
    We shall address from a conceptual perspective the duality between algebra and geometry in the framework of the refoundation of algebraic geometry associated to Grothendieck’s theory of schemes. To do so, we shall revisit scheme theory from the standpoint provided by the problem of recovering a mathematical structure A from its representations \ into other similar structures B. This vantage point will allow us to analyze the relationship between the algebra-geometry duality and the structure-semiotics duality. Whereas in classical algebraic geometry (...)
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  5.  50
    Towards a Galoisian Lnterpretation of Heisenberg Lndeterminacy Principle.Julien Page & Gabriel Catren - 2014 - Foundations of Physics 44 (12):1289-1301.
    We revisit Heisenberg indeterminacy principle in the light of the Galois–Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois–Grothendieck duality between finite K-algebras split by a Galois extension \ and finite \\) -sets can be reformulated as a Pontryagin duality between two abelian groups. We define a Galoisian quantum model in which the Heisenberg indeterminacy principle can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \ of (...)
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  6.  42
    On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois–Grothendieck Theory.Gabriel Catren & Julien Page - 2014 - Synthese 191 (18):4377-4408.
    We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities between the roots of a (...)
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  7.  19
    Klein-Weyl's Program and the Ontology of Gauge and Quantum Systems.Gabriel Catren - 2018 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 61:25-40.
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  8. A Throw of the Quantum Dice Will Never Abolish the Copernican Revolution.Gabriel Catren - 2009 - Collapse: Philosophical Research and Development 5.
  9.  15
    On the Galoisian Structure of Heisenberg Indeterminacy Principle.Julien Page & Gabriel Catren - unknown
    We revisit Heisenberg indeterminacy principle in the light of the Galois-Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois-Grothendieck duality between finite K-algebras split by a Galois extension L and finite Gal-sets can be reformulated as a Pontryagin-like duality between two abelian groups. We then define a Galoisian quantum theory in which the Heisenberg indeterminacy principle between conjugate canonical variables can be understood as a form of Galoisian duality: the larger the group of (...)
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  10.  49
    Outland Empire: Prolegomena to Speculative Absolutism.Gabriel Catren - 2011 - In Levi R. Bryant, Nick Srnicek & Graham Harman (eds.), The Speculative Turn: Continental Materialism and Realism. re.press.
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  11.  24
    Quantum Ontology in the Light of Gauge Theories.Gabriel Catren - unknown
    We propose the conjecture according to which the fact that quantum mechanics does not admit sharp value attributions to both members of a complementary pair of observables can be understood in the light of the symplectic reduction of phase space in constrained Hamiltonian systems. In order to unpack this claim, we propose a quantum ontology based on two independent postulates, namely the phase postulate and the quantum postulate. The phase postulate generalizes the gauge correspondence between first-class constraints and gauge transformations (...)
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  12.  76
    Quantic Fibers for Classical Systems: An Introduction to Geometric Quantization.Gabriel Catren - 2013 - Scientiae Studia 11 (1):35-74.
    En este artículo, se introducirá el formalismo de cuantificación canónica denominado "cuantificación geométrica". Dado que dicho formalismo permite entender la mecánica cuántica como una extensión geométrica de la mecánica clásica, se identificarán las insuficiencias de esta última resueltas por dicha extensión. Se mostrará luego como la cuantificación geométrica permite explicar algunos de los rasgos distintivos de la mecánica cuántica, como, por ejemplo, la noconmutatividad de los operadores cuánticos y el carácter discreto de los espectros de ciertos operadores. In this article, (...)
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  13. Correction: Grothendieck’s theory of schemes and the algebra–geometry duality.Gabriel Catren & Fernando Cukierman - 2022 - Synthese 200 (4):1-1.
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  14.  31
    On the Notion of Indiscernibility in the Light of Galois-Grothendieck Theory.Gabriel Catren & Julien Page - unknown
    We analyze the notion of indiscernibility in the light of the Galois theory of field extensions and the generalization to K-algebras proposed by Grothendieck. Grothendieck's reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a duality between G-spaces and the minimal observable algebras that separate theirs points. In order to address the Galoisian notion of indiscernibility, we propose what we call an epistemic reading of the Galois-Grothendieck theory. According to this viewpoint, the Galoisian (...)
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  15.  57
    Can Classical Description of Physical Reality Be Considered Complete?Gabriel Catren - unknown
    We propose a definition of physical objects that aims to clarify some interpretational issues in quantum mechanics. We claim that the transformations generated by the objective properties of a physical system must be strictly interpreted as gauge transformations. We will argue that the uncertainty principle is a consequence of the mutual intertwining between objective properties and gauge-dependant properties. The proposed definition implies that in classical mechanics gauge-dependant properties are wrongly considered objective. We will conclude that, unlike classical mechanics, quantum mechanics (...)
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  16.  27
    Towards a Group-Theoretical Interpretation of Mechanics.Gabriel Catren - unknown
    We argue that the classical description of a symplectic manifold endowed with a Hamiltonian action of an abelian Lie group G and the corresponding quantum theory can be understood as different aspects of the unitary representation theory of G. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory. The proposed argumentative line strongly relies on the conjecture proposed by Guillemin and Sternberg according to which ``quantization commutes with reduction''. By using (...)
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  17.  16
    Introduction to the Special Issue Hermann Weyl and the Philosophy of the ‘New Physics’.Silvia De Bianchi & Gabriel Catren - 2018 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 61:1-5.
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  18.  1
    On Gauge Symmetries, Indiscernibilities, and Groupoid-Theoretical Equalities.Gabriel Catren - 2022 - Studies in History and Philosophy of Science Part A 91:244-261.
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