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Towards a Group-Theoretical Interpretation of Mechanics

Catren, Gabriel (2013) Towards a Group-Theoretical Interpretation of Mechanics. [Preprint]

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Abstract

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 (notably, Souriau's moment map and the Mardsen-Weinstein symplectic reduction formalism) and group representation theory (notably Kirillov's orbit method). The proposed argumentative line strongly relies on the conjecture proposed by Guillemin and Sternberg according to which ``quantization commutes with (symplectic) reduction''. By using the generalization of this conjecture to non-zero coadjoint orbits, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This fact points towards a gauge-theoretic interpretation of Heisenberg indeterminacy principle.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Catren, Gabrielgabrielcatren@gmail.com
Keywords: quantum mechanics; gauge theories; moment map; symplectic reduction
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics > Quantum Mechanics
Specific Sciences > Physics > Symmetries/Invariances
Depositing User: Dr. Gabriel Catren
Date Deposited: 03 Dec 2013 16:59
Last Modified: 03 Dec 2013 16:59
Item ID: 10116
Subjects: Specific Sciences > Physics > Classical Physics
Specific Sciences > Physics > Quantum Mechanics
Specific Sciences > Physics > Symmetries/Invariances
Date: 30 November 2013
URI: http://philsci-archive.pitt.edu/id/eprint/10116

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