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 | ||||||
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| 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 |
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| 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 |
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| Date: | 30 November 2013 | ||||||
| URI: | https://philsci-archive.pitt.edu/id/eprint/10116 |
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